h
*Ia,[eSimp
SWfr lFormgfrysics
CapeUrtit 2
PrestonKissoon
DiplomaEducation,
BSc.(Physics),
Engineering
DiplomaEngineering(UK
Council)
A n y u n a u t h o r i z ecdo p y i n go f t h e
c o n t e n t so f t h i sb o o kw i l l b e i n
v i o l a t i o no f t h e I n t e l l e c t u aPl r o p e r t y
R i g h t sa c t o f T r i n i d a da n dT o b a g o
T o o r d e rc o p i e so f t h i s b o o k ,p l e a s e
c a l l 7 8 7 - 9 1 5 0o r e m a i l:
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CONTENTS
TOPIC
PAGE
E l e c t r o m o t i vfeo r c ea n d P o t e n t i adl i f f e r e n c e
Resistivity
ElectriC
c ircuita
s n dC h a r a c t e r i s t i c s
12
Kirchoff'sLaws
18
Resistance
28
Electrostatics
44
Capacitors
61
M a g n e t i cF i e l d s
78
M a g n e t i cF o r c e s
86
Electromagnetism
93
HallEffect
97.
E l e c t r o m a g n e tIincd u c t i o n
99
Alternating
current
113
Semiconductors
121
O p e r a t i o n aAl m p l i f i e r s
I29
D i g i t aE
l lectronics
159
Photoelectricity
185
X-Rays
197
L i n eS p e c t r a
204
AtomicStructure
209
Radioactivity
222
%of,ufe1
tr fectricityanf,%-agnetism
I M P O R T A NFTO R M U L A E
(1-)Quanti.tyof Electri"c charge Q : It
W
=, *
( 2 ) P o t e n t i a l D t f f e r e n c eV Q
^v2
(3) Electri.calPower P : IV : I'R :
T
RA
(4)Resi.stivi.ty
p :
I
I
((s)Dri.ft uelocity v 71EA
QtQ'
4rctsrz
(6)Electrostattc Force on each of 2 potnt charge F Q
(7)Etectrtcf teld due to apoint charge
' Av E
Lr :- ,
ttesQ
( B ) E L e c t r i .Pco t e n t t a ld u e t o a p o i n t c h a r g e , #
( 9 ) C a p a c i t a n c eC - ?
V
( 1 0 ) F o r a p a r a l l e l p l a t e c a p a c t t o r , C-
( 1 1 )E n e r gsyt o r e d
t na c a p a c i . tW
o r- r y
eA
z
-y
222C
-y
- : +: +:
\.r.
cI
c2 c3
(1,3)Threeapaci.torsi"nParallel C7 Cr * Cz* Cz
(Iz)TlveeCapaci,tors
tn rrrrc, !
( 1 4 ) W h e no c a p o c i t o r d i s c h a r g e s Q
: -
Q o e - ' l c n; V :
Voe-'lcn;I :
I o e - t l c n'
ELECTRICITY
A b a s i ce l e c t r i c asly s t e mh a sf o u r ( 4 )m a i np a r t s :
( 1 ) T h es o u r c ee ' g ' b a t t e r y w
- h i c hi s u s e dt o p r o v i d ee l e c t r i c aeln e r g y ;
t h i si s c a l l e da n
ACTIVE
D E V I C,Ea si t s u p p l i e es n e r g y .
( 2 ) T h el o a d -e ' g .a r e s i s t o rl-t a b s o r b o
s r converte
s l e c t r i c aeln e r g ya n d i s c a l l e da PASSIVE
DE V I C E .
( 3 ) T h et r a n s m i s s i osny s t e mw
- h i c hi s a s e to f c o n n e c t i nign s u l a t e w
d i r e st h a t c o n n e c t tsh e
s o u r c e . t toh e l o a d .
(a) The controlsystem-eg.
A switchto controlthe periodsfor whichthe active
device
s u p p l i etsh e p a s s i v d
ee v i c e .
The e.m.f.in a circuitis associated
with the totaramount
of energy that is availablefrom the
activedevicein the circuit'.The e.m.f.of a sourceis
definedas the TOTALenergyper coulomb
it deliversround a circuitjoinedto it.
Theterminalpotentialdifferencein a circuitis associated
with the amountof electrical
energyavailableto the passivedevicesin the circuit.
The terminatpotentialdifferenceof a- .
sourceis'definedas the amountof energyper coulomb
it deliversto the EXTERNAL
circuitto
which it is connected.
I l n t h e c a s eo f a n e l e c t r i cf i e l d ,t h e p . d .b e t w e e nt w o p o i n t si n t h e f i e l d
i s t h e w o r k d o n eo n u n i t
c h a r g ea s i t i s m o v e db y t h e f i e l df r o mo n e p o i n tt o t h e
other)
A n ya c t i v ed e v i c es u c ha sa b a t t e r y h, a si n t e r n arl e s i s t a n c e
t o t h e i n t e r n am
l o v e m o not f c h a r g e .
w h e n t h e d e v i c ei s n o t c o n n e c t e tdo a c i r c u i t , , a ntdh u s
d o e sn o t h a v ea c u r r e n t t h r o u g iht , t h e
p o t e n t i adl i f f e r e n c b
e e t w e e ni t st e r m i n a l si s e q u a lt o i t s e m f .
H o w e v e rw, h e nt h a t d e v i c eh a sa
c u r r e n t h r o u g hi t , t h e p o t e n t i adl i f f e r e n c b
e e t w e e ni t st e r m i n a l si s l e s st h a n i t s e m f .
T h i si s
becausp
e a r to f t h e e m f i s d r o p p e da c r o s st h e i n t e r n a l
resistanca
en
, dt h e r e s ti s d r o p p e d
a c r o s st h e l o a do r e x t e r n arl e s i s t a n c e .
L e t t h ei n t e r n arl e s i s t a n cbee r a n dt h e e x t e r n arl e s i s t a n coer l o a db e R .l f t h e e m f o f t h e s o u r c e
i s E a n dt h e c u r r e n it n t h e c i r c u i its l , t h e n :
E=lr+lR
l o a da n di s t h e t e r m i n apl . d . a n d i s u s u a l l y
H e r e t, h e q u a n t i t yl R r e p r e s e n t s t hpe. d .a c r o s s t h e
of the
the voltageacrossthe internalresistance
by V. The quantity(lr), represents
represented
battery.
H e n c ew e c a nw r i t e E = V + l r .
lf however,a currentis beingsuppliedto the battery, then V = E + lr. Thiscan occurwhen the
battery in questionis connectedto anotherone of a highere.m.f.
Electriccharge:
a
l f a c o n s t a nct u r r e n tI ( A )f l o w si n a c i r c u i ft o r a g i v e nt i m e t ( s )i n a c i r c u i tt,h e n t h e q u a n t i t yo f
p a s ta n yp o i n ti n t h e c i r c u i ti,s g i v e nb y Q = I t .
c h a r g eQ t h a t w o u l dh a v e f l o w e d
T h eu n i t o f e l e c t r i c h a r g ei s t h e C o u l o m b( C ).
The Coulombis definedas the total chargethat would haveflowed pasta point in a circuit
after one secondif a constantcurrentof 1 Ampereflows throughthat point.
T h eC o u l o m bi s t h e r e f o r ed e f i n e da st h e A m p e r es e c o n d .
i.e.1C= l-As.
or loadbe R. lf the emf of the source
be r and the externalresistance
Letthe internalresistance
i s E a n dt h e c u r r e n ti n t h e c i r c u i ti s l , t h e n :
E=lr+lR
l o a da n di s t h e t e r m i n apl . d . a n d i s u s u a l l y
H e r e t, h e q u a n t i t yl R r e p r e s e n t s t hpe. d .a c r o s s t h e
of the
the voltageacrossthe internalresistance
by V. Thequantity(lr), represents
represented
battery.
Hencewe canwrite E = V + lr .
tf however,a currentis beingsuppliedto the battery, then V = E + lr. Thiscan occurwhen the
battery in questionis connectedto anotherone of a highere.m.f.
Electriccharge:
a
l f a c o n s t a nct u r r e n tI ( A )f l o w si n a c i r c u i ft o r a g i v e nt i m e t ( s )i n a c i r c u i tt,h e nt h e q u a n t i t yo f
c h a r g eQ t h a t w o u l dh a v ef l o w e dp a s ta n yp o i n ti n t h e c i r c u i ti,s g i v e nb y Q = I t .
T h eu n i t o f e l e c t r i c h a r g ei s t h e C o u l o m b( C ).
The Coulombis definedas the total chargethat would haveflowed pasta point in a circuit
after one secondif a constantcurrentof 1 Ampereflows throughthat point.
T h eC o u l o m bi s t h e r e f o r ed e f i n e da st h e A m p e r es e c o n d
i . e .1 C= 1 A s .
PotentialDifference:
The unit of potentialdjfferenceis the Volt (V).
1 Volt = l- Jouleper coulomb,or LV=l-JCThe potentialdifferencebetweentwo pointsin a circuitis definedas the amount of electrical
energyconvertedto other forms of energyper coulombof chargepassingfrom one point to
the other.
T h ev o l t i s d e f i n e da sf o l l o w s :
The potentialdifferencebetweentwo pointsin a circuitis 1 volt if l Jouleof electricalenergy
is convertedto other forms of etectricalenergywhen 1 Coulombof chargepassesfrom one
point to the other.
P otenttal di"ff erence =
energy converted
W
char g e
a
R e c a lal n d u s et h e f o l l o w i n ge q u a t i o n s :
V : I R , P _ I V , P_ I Z RP
, -+,
E _I(r+R)
RESISTIVITY:
F o ra u n i f o r ml e n g t ho f c o n d u c t ohr a v i n ga l e n g t hl , c r o s ss e c t i o n aal r e aA a n d r e s i s t a n cRe,t h e
resistivity
of the materialfrom whichthe conductoris made,is givenby the formula:
0-
RA
T
The unitsof resistivity
are Om. Notethat the resistivity
is a propertyof the materialand is
t h e r e f o r ei n d e p e n d e notf t h e d i m e n s i o n sh,e n c ea 2 c m 3s a m p l eo f c o p p e rh a st h e s a m e
r e s i s t i v i toyf a 5 m 3 s a m p l e .
F r o mt h e e q u a t i o na b o v e t, h e R e s i s t i v i t y w o ubl de e q u a lt o t h e r e s i s t a n ci e
f the length'l'was
L m a n dt h e c r o s ss e c t i o n aal r e aw a s 1 m 2 '
Henceresistivity
can be definedasfoilows:
I The Resistivityof a specirnenof materialis the productof its crosssectionalarea
anUits etectricalresistanceper unit length.
i
DRIFTVELOCITY
s c o n d u c t oAr B i n t h e c i r c u ist h o w n :
C o n s i d ear p . d .b e i n ga p p l i e da c r o s a
A
E+
A n e l e c t r i cf i e l di s E i s s e t u p a c r o s sA B i n t h e d i r e c t i o nf r o m A t o B .
E l e c t r ifci e l d sa r e a l w a y sd i r e c t e df r o m p o s i t i v et o n e g a t i v eT. h ee l e c t r i cf i e l di s a v e c t o r
quantity.
n h y ), a n di n t h e i r
n t h e d i r e c t i o n- E , ( e x p l a i w
E l e c t r o nisn t h e c o n d u c t oar r et h e n a c c e l e r a t ei d
) u t t h e y c o n t i n u et o
m o t i o n ,t h e y c o n s t a n t lcyo l l i d ew i t h a t o m s( c a u s i nhge a td i s s i p a t i o n, b
VELOCITY.
movetowardsA with an avbrasevelocitv calledthe DRIFT
Q u e s t i o nW
: h y a r et h e y c o n s i d e r etdo h a v ea n a v e r a g ev e l o c i t y ?
A n s w e r: T h e ya r e a c c e l e r a t ebdy t h e e l e c t r i cf i e l d, a n dt h e y c o l l i d ew i t h c o n d u c t oar t o m s ,
H.e n c et h e v e l o c i t yi s n o t c o n s t a nbt u t i n s t e a dc o n t i n u o u s l y
w h i c hc a u s e s o m ed e c e l e r a t i o n
i n c r e a s easn d d e c r e a s e s .
Derivati"on of the equation I - nAve
C o n s i d ear s e c t i o no f a u n i f o r ml e n g t ho f c o p p e rw i r e o f c r o s ss e c t i o n aal r e aA , t h r o u g hw h i c ha
c u r r e n tI i s f l o w i n g .
<--
Electronflow
C u r r e n It
Assumethat there are 'n' electronsper unit volume,with eachelectroncarrying
a c h a r g e' e ' a n d e d c hh a v i n ga n a v e r a g e
drift velocity'v'.
T h ev o l u m ep e r s e c o n do f e l e c t r o n p
s a s s i ntgh r o u g ht h e p l a n eA i s e q u a lt o A v .
Vol
_-
Av
.s
T h e n u m b e ro f e l e c t r o n p
t f t h e n u m b e ro f
s e r s e c o n dw i l l b e t h e p r o d u c o
e l e c t r o np
s e r u n i tv o l u m ea n dt h e v o l u m ep e r s e c o n d .
H e n c et h e n u m b e ro f e l e c t r o n p
s e r s e c o n dw i l l b e g i v e nb y :
A/
t
T h ec u r r e n tI i s d e f i n e da st h e t o t a lc h a r g ef l o w i n gp e r s e c o n d .
_
t-
T otal cltar g e
a
time
t
Ne
f
L
- nAve
sheet
TUTORIAL
( 1 ) A c o i lo f w i r e i s c o n n e c t e d
a c r o s sa 1 . 2 Vs u p p l yw h i c hc a nd e l i v e r1 5 0 0 0 Ci n a o n e h o u r
period.
C a l c lua t e
b y t h e p o w e rs u p p l y .
(i)
c u r r e n td e l i v e r e d
T h ea v e r a g e
(ii)
T h et o t a l c h a r g et h a t w o u l dh a v ef l o w e dt h r o u g ht h e w i r e d u r i n gt h e f i r s t8
(iii)
m i n u t e sa f t e rt h e s w i t c hi s c l o s e d
T h et i m e f o r w h i c ht h e s t e a d yc u r r e n tm u s tf l o w i n o r d e rf o r t h e t o t a l e l e c t r i c
c h a r g et o d r a w nf r o m t h e s u p p l yt o b e 2 2 5 C .
;5 4 s e c o n d s )
@ . I 7A ; 2 0 0 0 C
l . d .d r o p st o z e r ow h e n a c u r r e n to f 2 5 A
( 2 ) A d r y c e l lh a sa n e m f o f 1 , . 5 2 Vl t st e r m i n a p
p a s s etsh r o u g hi t . W h a t i s i t i n t e r n arl e s i s t a n c e( ?r = 0 . 0 6 1 O )
( 3 ) A d i r e c tc u r r e n tg e n e r a t o rh a sa n e m f o f 1 2 0 V t; h a t i s ,i t st e r m i n avl o l t a g ei s 1 2 0 V
l . d .i s 1 1 5 V .( a )
w h e n n o c u r r e n ti s f l o w i n gf r o m i t . A t a n o u t p u to f 2 0 At h e t e r m i n a p
W h a t i s t h e i n t e r n arl e s i s t a n creo f t h e g e n e r a t o r(?b )W h a tw i l l b e t h e t e r m i n a l
voltageat an outputof 40A?
(r=0.250V=110V)
. o wm a n ym e t r e so f
( 4 ) N u m b e r1 0 A l u m i n i u m
w i r e h a sa d i a m e t eor f 2 . 5 8 m mH
n u m b e r l 0a l u m i n i u mw i r e a r e n e e dt o g i v ea r e s i s t a n coef L O ,i f t h e r e s i s t i v i toyf
i s 2 . 8x 1 0 - B O m(. A N S t = ( 1 8 6 m )
aluminium
(b)
t h r o u g ha n e x t r u d e sr o a st o m a k ei t
A w i r e h a sa r e s i s t a n coef o f 5 0 a n d i s p a s s e d
. h a ti st h e n e w r e s i s t a n c e ?
i n t o a n e w w i r e t h r e et i m e sa s l o n ga st h e o r i g i n a lW
ANS(r =45O)
( H i n t :T h ev o l u m eo f t h e w i r e i s u n c h a n g e ds ,o t h a t t h e a r e aw i l l c h a n g ef;i n dt h e n e w
a r e ai n t e r m so f t h e o r i g i n aol n e )
(5) Two resistors
of 40 and 12Oare connectedin parallelacrossa 22Vbattery,havingan
i n t e r n arl e s i s t a n coef l O . C a l c u l a t(ea ) T h eb a t t e r yc u r r e n t (, b )t h e c u r r e n ti n t h e 4 0
the terminalvoltageof the battery,(d)the currentin the 12Oresistor.
resistor,(c)
(Ans 5.5A;/+:.A;t7V; 1.4A)
10
(6) Threeresistors
of 40O,60Q,and1200,areconnected
in parallel
andthisparallel
groupisconnected
in series
witha 15Oin series
witha 25O.Thewholesystemisthen
(a)Thecurrentin the 25O,(b)the p.d.across
connected
to.a120Vsource.
Determine
group,(c)the p,d.across
the parallel
the 25O,(d)thecurrentin the 600, (e)the
currentin the40O.
(2A;4Ov;50V;0.67A;1A.)
(7) Two coilsof wire are madefrom l.0m of wire eachhavinga diameterof 0.05mm.
Oneof the coilsis madeusingcopperwire, while the other is madefrom
aluminiumwire.
Ifthe resistivityofcopperis 1.59x10-8Om
and the resistivityof aluminiumis 2.7x
10-80rn,
find the resistanceof eachcoil.(Ans:81O;138O)
( 8 ) A l e n g t ho f w i r e h a sa d i a m e t e o
r f 0 . 4 5 m ma n da r e s i s t a n coef 4 0 0 0 . F i n dt h e
r e s i s t a n coef t h e s a m el e n g t ho f w i r e i f t h e d i a m e t e irs :
(i)
Doubled
'
(ii)
T r ip l e d.
(Ans1000; 44.40)
( 9 ) H i g ht e n s i o nc a b l e sh a v ea d i a m e t e o
r f 2 c ma n d a 3 0 k ml e n g t hh a sa r e s i s t a n coef - 4 . O .
'
Determine:
(i)
T h er e s i s t i v i toyf t h e m a t e r i aul s e di n m a k i n gt h e c a b l e s .
(ii)
T h er e s i s t a n coef a n o t h e rc a b l eo f t h e s a m em a t e r i ahl a v i n ga d i a m e t e o
r f 1.5cm
a n d a l e n g t ho f 1 5 0 m .
( A n s4 . I x 1 0 - 8 O m0;. 0 3 6 0 )
(10)
O n ep i e c eo f w i r e P h a sa l e n g t hI a n dd i a m e t e d
r a n d r e s i s t a n cRe .W h a t i s t h e
r e s i s t a n ci en t e r m so f R f o r a n o t h e rp i e c eo f w i r e Q o f t h e s a m em a t e r i ahl a v i n ga
l e n g t h4 . 5 1a n dd i a m e t e2r . 4 d ( A n s0 . 7 8 R )
TT
ELECTRICAL
CIRCUITS
A g r a p ho f c u r r e n ta g a i n svt o l t a g ef o r a n yp a r t i c u l acro m p o n e n ti ,s n o r m a l l yc a l l e dt h e I - V
c h a r a c t e r i s tfiocr t h a t c o mp o n e n t .
T h e r ea r ea f e w g r o u p so f c o m p o n e n ttsh a t h a v et h e i rd i s t i n cIt- V c h a r a c t e r i s t i casn,dt h e s ea r e
( a ) M e t a l l i c o n d u c t o r s I-V characteristic
for a METALLIC
CONDUCTOR
at constant
( b ) S e m i c o n d u c tD
o ri o d e temperature
( c ) F i l a m e nJta m p s .
p.d./v
T h eg r a p hi n d i c a t etsh a t t h e c u r r e n ti s a l w a y sd i r e c t l yp r o p o r t i o n at ol t h e a p p l i e dv o l t a g e
a n dt h a t w h e nt h e v o l t a g ei s r e v e r s e dt h
, e r ei s n o c h a n g ei n t h e b e h a v i oor f t h e m e t a l l i c
c o n d u c t oirn t e r m so f i t s c o n d u c t i o n
T h i st y p e o f c o n d u c t oirs a l s or e f e r r e dt o a sa n O H M I CC O N D U C T OsRi n, c ei t s g r a p h
i n d i c a t etsh a t i t o b e y sO h m ' sL a wa t a l lt i m e s ,P R O V I D ETD
H A TT H ET E M P E R A T Ul R
SE
C O N S T A N Tt .hI ef t e m p e r a t u r e
i n c r e a s ehso w e v e rt,h e r e s i s t a n coef t h e c o n d u c t ow
r ill
a l s oi n c r e a s ea,n dt h e g r a d i e not f t h e g r a p hw i l l d e c r e a s eT.h i sd e c r e a s ien g r a d i e n its
d u e t o t h e f a c tt h a t t h eg r a d i e n t
-
1.2
:\ / R- :
I - V characteristic for a FITAMENT LAMP.
p.d./v
a
T h i sg r a p hs h o w st h a t t h e c u r r e n ti s p r o p o r t i o n a l t toh e a p p l i e dv o l t a g eo n l yw h e nt h e r ei s a
r e l a t i v e lsym a l a
l p p l i e dv o l t a g eF. r o mt h e p o i n t ' a ' t ot h e p o i n t ' b ' ,t h e g r a p hi s l i n e a a
r n dt h e
'
a
'
f i l a m e n tl a m po b e y sO h m ' sl a w o n l yi n t h e r e g i o nf r o m t o ' b ' .
B e y o p d ' a ' a n d ' b ' t hg
e r a d i e not f t h e g r a p hD E C R E A SaEnSdt h i si n d i c a t etsh a t t h e R E S I S T A N C E
I SI N C R E A S I N G .
T h i si n c r e a s ien r e s i s t a n caet t h e h i g h e rc u r r e n t si s e x p l a i n e a
d sf o l l o w sA
: h i g h e rc u r r e n t
m e a n st h a t t h e r ea r e m o r ee l e c t r o nfsl o w i n gp e rs e c o n dw
, h i c hm e a n sm o r ee l e c t r o n cs o l l i d e
w i t h t h e l a t t i c es t r u c t u r eo f t h e t u n g s t e n .
T h en u m b e ro f c o l l i s i o npse r s e c o n di s a l s oa m e a s u r eo f t h e r e s i s t a n coef t h e m a t e r i a l .
A t e a c hc o l l i s i o nt h
, e r ei s a c o n v e r s i oonf s o m eo f t h e k i n e t i ce n e r g yo f t h e e l e c t r o ni n t o h e a t
e n e r g yh, e n c et h e t e m p e r a t u r e
rises.
13
DIODE
for a SEMICONDUCTOR
I-V characteristic
I/mA
I n t h i sc a s et h e c o n d u c t i ocnu r r e n its i n t h e o r d e ro f m i l l i a m pws h i l et h e a p p l i e d
p o t e n t i adl i f f e r e n c ies i n t h e o r d e ro f V o l t s .
T h ec o n d u c t i o cnu r r e n ti n t h e r e v e r s ed i r e c t i o ni s z e r oo r i t m a yb e s o s m a l tl h a t i t i s
c o n s i d e r etdo b e n e g l i g i b l e .
l n t h e f o r w a r dd i r e c t i o nt ,h e c o n d u c t i o cnu r r e n tr e m a i n sa t z e r ou n t i lt h e a p p l i e d
whichis
VOLTAGE
voltagereachesa particularvaluecalledthe SATURATION
a p p r o x i m a t e 0l y. 6 V .
t h e nt h e a p p l i e dv o l t a g er e a c h e as n d e x c e e dtsh e
T h ed i o d eo n l yb e g i n st o c o n d u c w
s a t u r a t i o vn o l t a g e .
e f t h e i n f i n i t e l yh i g h
T h ed i o d ed o e sn o t c o n d u c itn t h e r e v e r s ed i r e c t i o nb e c a u s o
r e s i s t a n ci en t h e r e v e r s ed i r e c t i o n .
I n t h e f o r w a r dd i r e c t i o nt h e r e s i s t a n ci es i n i t i a l l yi n f i n i t eb u t b r e a k sd o w nt o a v e r yl o w
v a l u ew h e nt h e a p p l i e dp o t e n t i adl i f f e r e n c ies i n c r e a s eudn t i lt h e s a t u r a t i o vno l t a g ei s
a t t ai n e d .
t4
THERMISTORS
T h e r m i s t o rasr es p e c i asl o l i dt e m p e r a t u r e
s e n s o rw
s h o s er e s i s t a n cceh a n g e rsq p i d l ya n d
p r e d i c t a b lwy i t h t e m p e r a t u r eT. h e i rn a m ei s a c o n t r a c t i oonf " t h e r m a l "a n d " r e s i s t o r , , .
There
are basically
two broadtypes,NTC-Negative
Temperature
Coefficient,
usedmostlyin
temperaturesensingand PTC-Positive
Temperature
Coefficient,
usedmosttyin electriccurrent
control.
T h ec o n v e n t i o n as ly m b o fl o r a t h e r m i s t o irs :
/
+
J
a
A g r a p ho f R e s i s t a n cvee r s u sT e m p e r a t u ries c a l l e da n R - Tc h a r a c t e r i s t i c
R-Tcharacteristic
for a
p . t . c .t h e r m i s t o r
R-Tcharacteristic
for a
n . t . c t. h e r m i s t o r
15
THERMISTORS
T h e r m i s t o rasr es p e c i asl o l i dt e m p e r a t u r se e n s o r w
s h o s er e s i s t a n cceh a n g e rsq p i d l ya n d
p r e d i c t a b lwy i t h t e m p e r a t u r eT. h e i rn a m ei s a c o n t r a c t i oonf " t h e r m a l "a n d " r e s i s t o r , , .
There
are basically
two broadtypes,NTC-Negative
Temperature
Coefficient,
usedmostlyin
temperaturesensingand PTC-Positive
Temperature
Coefficient,
usedmostlyin electriccurrent
control
T h ec o n v e n t i o n as ly m b o fl o r a t h e r m i s t o irs :
/
+
-J
a
A g r a p ho f R e s i s t a n cvee r s u sT e m p e r a t u ries c a l l e da n R - Tc h a r a c t e r i s t i c
R-Tcharacteristic
for a
p.t.c.thermistor
R-Tcharacteristic
for a
n . t . c t. h e r m i s t o r
15
Typesof NTCThermistors:
(1) Beadin glass
T h e m o s tc o m m o nf o r m o f t h e t h e r m i s t oirs a b e a dw i t h t w o w i r e sa t t a c h e dT. h eb e a d
d i a m e t ecr a nr a n g ef r o m a b o u t0 . 5 m m( 0 . 0 2 "t)o 5 m m ( 0 . 2 " ) .
Lii::;*t lfifii
!"h
Ti-r*l-f-rii::;[t:if ,r'y'i
t-ll;3.1"i,:":l-t
E il,:-:;::.f:.,:j
M e c h a n i c a ltl hy e t h e r m i s t o irs s i m p l ea n d s t r o n g p, r o v i d i n tgh e b a s i sf o r a h i g hr e l i a b i l i t y
s e n s o rT. h e m o s tl i k e l yf a i l u y em o d ei s f o r t h e l e a dt o s e p a r a t fer o m t h e b o d yo f t h e t h e r m i s t o r
- a n u n l i k e l ye v e n ti f t h e s e n s o ri s m o u n t e ds e c u r e l yT. h es i n t e r e dm e t a lo x i d em a t e r i ails
i n g l a s so r e p o x ye n c a p s u l a t i o n .
p r o n et o d a m a g eb y m o i s t u r es, o i t i s e n c a s e d
, e r m i s t o ras r eo f t e nm o u n t e di n s t a i n l e ssst e e lt u b e s ,t o
L i k eo t h e r t e m p e r a t u rsee n s o r st h
. r e a s ei s t y p i c a l l yu s e dt o
p r o t e c t h e m f r o m t h e e n v i r o n m e ni tn w h i c ht h e ya r et o o p e r a t eG
r n dt h e t u b e .
t e t w e e nt h e s e n s o a
i m p r o v et h e t h e r m a lc o n t a c b
T h e R - Tc h a r a c t e r i s toi cf a t h e r m i s t o cr a nb e m a d em o r el i n e a rb y c o n n e c t i nag r e s i s t oirn
p a r a l l ewl i t h i t .
T l-r+r nt i::lttr Fl+:iti r:rns+
rl
t-
-:r
'',tiiIrErlrri.llEl
5 :l thernri,:tut
!l[tt_lrt
re:ri*tr]r
._._,
thermistor
rl'
'=
qr
+]
ii
=\=\_
1
iL'
|I
l'-
I tlel tfrliitLrf ul-llU
10
T h ep l o t i n t h e d i a g r a ms h o w st h e
impactof a 2200ohm resistorin
parallelwith a 2252ohm (at 25'C)
ill
fiJ
TernPer+tr.tre
1.6
(2) DiscThermistors:
D i s ct h e r m i s t o ras r e m a d eb y p r e p a r i n tgh e v a r i o u sm e t a lo x i d ep o w d e r sb, l e n d i n g
them
w i t ha s u i t a b lb
e i n d e ra, n dt h e nc o m p r e s s i ns gm a l a
l m o u n t so f t h e m i x t u r ei n a d i e u n d e r
s e v e r a l t o nos f p r e s s u r eT.h ed i s c sa r et h e n f i r e da t h i g ht e m p e r a t u r etso f o r m s o l i dc e r a m i c
bodies.
A t h i c kf i l m e l e c t r o d em a t e r i a lt,y p i c a l l ys i l v e ri,s a p p l i e dt o t h e o p p o s i t es i d e so f t h e d i s ct o
p r o v i d et h e c o n t a c t fso r t h e a t t a c h m e not f l e a dw i r e s .A c o a t i n go f e p o x y p
, h e n o l i co, r g l a s s
i s a p p l i e dt o e a c hd e v i c et o p r o v i d ep r o t e c t i o n
f r o m m e c h a n i c a ln de n v i r o n m e n t a l
s t r e s s eT
s .y p i c aul n c o a t e d i s cs i z e sr a n g ef r o m 0 . 0 5i n .t o 0 . 1 0i n .( 1 . 3m m t o 2 . 5m m ) i n
y e a s u r 0e . 1 0i n .t o 0 . 1 5i n . ( 2 . 5m m t o 3 . 8m m ) i n
d i a . ;c o a t e dd i s c t h e r m i s t ogr e
s n e r a l lm
di a .
€
-\'
H-
h=
F
;*ri
.LJ*
(3) Rod-shapedthermistors:
a viscouo
s x i d e - b i n d emr i x t u r et h r o u g ha d i e ,
R o d - s h a p et h
d e r m i s t o ras r e m a d eb y e x t r u d i n g
e l e c t r o d e sa,n d a t t a c h i n g
l e a d sR
. od
h e a t - t r e a t i ni gt t o f o r m a c e r a m i cm a t e r i a la, p p l y i n g
t h e r m i s t o rasr e u s e dp r i m a r i l yf o r a p p l i c a t i o nr se q u i r i n vg e r yh i g hr e s i s t a n caen d / o rh i g hp o w e r
di s s i p a t i o n .
17
LAWS
KIRCHOFF'S
Kirchoff's FirstLaw :
T h i si s n o r m a l l yc a l l e dt h e j u n c t i o nr u l e ,a n d i t s t a t e st h a t t h e a l g e b r a iscu mo f t h e c u r r e n t sa t a
j u n c t i o ni s z e r o .T h i sm e a n st h a t t h e s u mo f t h e c u r r e n t se n t e r i n ga j u n c t i o n ,i s e q u a lt o t h e
thejunction.
s u mo f t h e c u r r e n t sl e a v i n g
Thisrule impliesthat there is neithera build-up of chargenor a depletionof chargeat a
junction.The rule is thereforean applicationof the conservationof charge.
or negative)
of
Sincethe law refersto the algebraic
sum of the currents,then the signs(positive
t h e c u r r e n t sm u s tb e t a k e ni n t o c o n s i d e r a t i oC
n .u r r e n t es n t e r i n gt h e j u n c t i o na r eg i v e na
p o s i t i v es i g n w
t h e j u n c t i o na r eg i v e na n e g a t i v e
sign.
, h i l ec u r r e n t sl e a v i n g
H e n c ei n t h e a b o v ej u n c t i o n, t h e f o l l o w i n ge q u a t i o nc a nb e a p p l i e d :
1 ,+ I s -I +- I s - l r = 0 o r 1 , + I s = I ++ I : + l r
18
Kirchoff'sSecondlaw
Kirchoff's
2nd Lawstates:
In any closedloop of a circuit,the algebraicsum of the emf's of the sourcesis equalto the
algebraicsum of the potentialdifferencesof the loads;ORthe algebraicsum of all the
voltagesin the loop is zero.
ThissecondLawis an applicationof the conservation
of energy.In its simplestform the
f o l l o w i n gc i r c u i tc a nb e u s e dt o d e m o n s t r a t iet s m e a n i n g .
T h ea l g e b r a iscu mo f t h e e m f ' s i s ( E 1- E z ) .T h em i n u ss i g ni s u s e da s t h ec e l l so p p o s ee a c h
other.
T h eA l g e b r a iscu mo f t h e p o t e n t i adl i f f e r e n c eiss V r + V , + V r .
Henceby Kirchoff's
secondlaw, (Et - Er) = Vr + V2 + V3
o r ( E t - E r ) = I R r + 1 P 2+ I R a
T h es e c o n dl a w i n d i c a t etsh a t t h e t o t a le n e r g ys u p p l i e db y t h e s o u r c e si s e q u a lt o t h e t o t a l
energyconvertedby the loads.
I n m o s tq u e s t i o n sb,o t h l a w sm u s tb e a p p l i e ds i m u l t a n e o u st loy s o l v eu n k n o w nc u r r e n t s .
19
W o r k e de x a m p l e s
U n k n o w nc u r r e n t sa n d p o t e n t i adl i f f e r e n c ei sn a c i r c u i ct o n t a i n i n g t w o r m o r el o o p sc a nb e
c a l l e dM E S HA N A L Y S I S .
s o l v e db y K i r c h o f f 'lsa w su s i n ga t e c h n i q u e
( 1 ) F i n dt h e c u r r e n t h r o u g he a c hr e s i s t oirn t h e f o l l o w i n gc i r c u i t :
a
6V
s r E P( 1 ):
AssigncurrentsIr and Iz to the individualloops.The directionof the currentsis not important here.
S T E P( 2 ) ,
Apply Kirchoff s secondlaw to eachloop separatelyso that two sets of
simultaneousequationsinvolving Ir and lz canbe obtained.
S T E P[ 3 ) '
Solvefor Ir andIz.
20
OV
For loop 1, imagineyourselfwalking aroundthe loop in the direction
of the current
Ir'Remeberthat current flows from a high potential to a low potential and
a negative
sign is usedto indicatea drop in potentialand a positivesign to indicate
a rise in
potential.
.
starting from a ,movingto b then to c and back to a, we get:
- 7 l t - 4 - 6= 0 > - 7 l r : 1 0 ; t ' : # . 4
H e n c et , :
T A (anti.clockwi.se)
For loop 2, startingfrom b to d to c and backto b and movingin the direction
of lr*"
get :
-5lz+B+4=0 p 5Iz- 12; Iz = 2.4A
[clockwise)
The current through the 4v battery is equalto lz-[- Ir)=lz+lr=3.83A
The minus sign for Ir indicatesthat the wrong directionwas assignedto it,
and hence
its correctdirectionshouldhavebeen anticlockwisein the loop.
2I
what happensif we reversethe assumedcurrentsIr and Iz:
Investigate
F o rl o o p( 1 ), i n g o i n g f r o ma t o b t o c a n db a c k t oa :
- 7 h + 6 + 4 = 0 ; H e n c e- 7 l r - - 1 0 D 1 r - | A
Here we get a positivevalue for Ir,which indicatesour assumeddirection was correct.
Considerloop (2) ,in going from d to a to c and back to d:
-5lz -4-B'=0 ; Hence -5lz='J.2> Iz :
+
- -2.4A
The minus sign indicatesthat the assumedanticlockwisedirection of Iz was incorrect,and
that it should have been clockwise.
This mesh analysistells us that Ir= +A [anticlockwise)and 12=2.4 A [clockwise)which is
identicalto the result obtainedwhen we initially assumedthat both currentswere
clockwise.
22
Considernow the casewherewe assumeIr to be clockwise,and lztobe anticlockwise
initiallv
F o r l o o p[ 1 J , w eg e t - 7 h - 4 - 6 - 0 ; H e n c e T l l _ - 1 0 , s o t h a t I t :
]e.
Rememberthat the minr, ,ign is used as we drop from a high potential to a low a potential.
Thismeansthat Ir :
I e(anti"clockwi.se)
F o rl o o p( 2 ) , w eg e t- 5 I z- 4 - B= 0 ; H e n c e- 5 / , - t 2 , s o t h a t I z :
-1")
;A
Thismeansthat Iz =?,4 [clockwise)
Theseresultsare identicalto those obtainedwhen we assumeddifferent directionsfor the
currents.
It is thereforesafeto assumethat regardlessof which initial directionswe assumefor the
currents,the results are the same,and that the negativesigns in the answerstell us that we
must reverseour initial directionsof the currents.
23
SHEET
TUTORIAL
q u e s t i o ntso d e t e r m i n teh e c u r r e n t tsh r o u g he a c ho f t h e b a t t e r i e s :
N o wt r y t h e f o l l o w i n g
(1)
I r =0 . 7 5 A ;
Iz =
-T-
1,1,4
rS
BV
||l
tl
(2)
I r =0 . 7 5 4; b :
*A
9
lz:lz-2sA
12V
( 3 ) D e t e r m i n teh e c u r r e n ti n e a c hr e s i s t o r :
S o l u t i o no n p g 2 6
12V
24
(4)
_
,I E
O-g
/rso :
1so:
2
A(anttclockwise)
1,1,
4SA
(clockwi"se)
7
(upwards)
1,SA
(s)
I:cr:
24
clockwtse
BA
lrso : 1rca0.29Aclockwise
1so: 0'27A
(6)
'10Q
Izsa
: 0.37 7A(anticlo ckwise)
lrso : lrocr
- 0.1 37Aanttclockwi.se
lscr: O.24Aupwards
25
f o r q u e s t i o n( 3 )
Solution
12V
F o r l o o p [ 1 J ' - 2 0 1 r- 1 0 l 1+ 1 0 1 2- 1 2 = 0 ; - 3 0 1 r+ 1 0 l z= 1 ' 2- t ( 1 )
F o r l o o p [ 2 ) - 2 5 1 2 ' + 6 - 1 , 0+1120 l r - 0 ; - 3 5 1 2+ 1 0 l r = - 6 - s ( 2 )
Solvingsimultaneouslygives:
Multiplyingeq'n (2)by 3 ; 30h -10512= -LB --> (3)
R e w r i t i n g[ 1 )
: - 3 0 h + 1 0 1 2= L 2
t1)
Adding [1J and [3J gives -9512--6; henceIz = 0.063A
Ir= -0.379.The minus sign here indicatesthat the clockwise
direction that we assignedfor Ir was actuallyin the reverse
directionof what it was supposedto be. HenceIrwas 0.379Ain
an anticlockwisedirection.
26
lroo : Izoa: 0.21A anttclockwi"se
l r s o : / s o : 0 . 0 1 7 Ac l o c k w i s e
I z s a: 0 ' 2 2 7A
(8)
I +e : I oa : 0'77A anttclockwise
1go: lrso : 0'0764 clockwtse
/so : 0.8464 upwards
1 r s 0: 1 s o: 0 . 2 1 Ac l o c k w i s e
It aa : It+a : 0.0066Aclockwise
lrso : 0'2034 downwards
27
RESISTANCE
S E R I ECSO M B I N A T I O N :
R1
A current Ir leavesthe battery and enters into resistor R1.The current that leavesRL i s I r
and this is equal to Ir by Kirchoff s first law. [The current enteringa componentmust b e
equalto the current leavingit by the principle of conservationof charge)
When Ir enters R2,it leavesasIz,but Iz = It by Kirchoffs First Law.
Iz entersR: and the emergingcurrent is I3,but I:= Izby Kirchoffs first Law. Ir = Ir Iz = l :
Kirchoff'sF i r s t
Hencein the seriescircuit,all of the resistorshavethe SAMECURRENT.[By
Law)
By Kirchoff s secondLaw, the e.m.f.of the battery is equalto the algebraicsum of the
voltagesacrosseachof the resistors.
H e n c eE = V t + V z + V : a n d V r = I t R r = l r R r ; V z= l z R z= l r R z ; V : = I : R : = I r R :
By Ohm'slaw, E = IrRr where Ir and Rr representthe total current and total resistance
respectively.
IrRr = IrRr+lrRz+lrR:
If this equationis divided by Ir, w€ get Rr = Rr + Rz + Rs (You must learn this
derivation).
28
PARALTELCOMBINATION:
T h ed i f f e r e n lto o p sa r e i d e n t i f i e d
w i t ht h e n u m b e r s
h a v i n ga n o v a ls h a p ea r o u n dt h e m .H e n c et h e r ea r e 3
loops.
C o n s i d et h
r e p o i n ta .
T h ec u r r e n te n t e r i n ga i s e q u a lt o I r a n d t h e t o t a l
currentleavinga is equalto
h + Iz+ I: .
By Kirchoffs first law, the total current enteringa
point, must be equalto the total current leavingthe
p o i n t .H e n c e
Ir=lr+lz+l:
l f K i r c h o f f 'sse c o n dL a wi s n o w a p p l i e dt o L o o p1 , t h e nw e g e t E = I r R i
A p p l y i n gK i r c h o f f ss e c o n dl a w t o l o o p z , w e g e t 0 = I r R r - I z R z ;I r R r = I z R z= E
If the secondlaw is now appliedto loop 3 we get,0= IzRz-IsR: I:R: = IzRz=E
;
This applicationof the secondlaw tells us that in a parallelcombination
of resistorsor
other components,the potentialdifferenceacrosseachbranch is the same.
Henceif Vr ;Vz i and V: are the potentialdifferencesacrossRr Rz; and R:
;
respectively,then Vr = Vz=Vs = E
By Ohm'sLaw,
-V:2_- E
,VtE
It:
&:
^'t'-
R2
,
-J
R2'
Sincelr=lt+lz+
E
Rr
-E+E E- + - ' R 3
R1
R2
And hence
1,1,1,I
Rr
Rl
29
R2
'R3
:
_V :E_
R3
E
R3
and T, ,R:T
E
DLI V I D E R
T H EP O T E N T I A
connectedin seriesand
of threeterminals,two or more resistors
The potentialdividerconsists
a v o l t a g ea p p l i e da c r o s tsh e e n t i r ec o m b i n a t i oonf r e s i s t o r s .
'tap off' fractionsof the appliedvoltage,and in so
Thefunctionof the potentialdivideris to
d o i n g t, o p r o d u c ea v a r i a b l es u p p l yf r o m a s i n g l ed . c .s o u r c e '
a ss h o w n :
c o n s i d etrh e f o l l o w i n gp o t e n t i adl i v i d e rc i r c u i t sc o n t a i n i ntgh r e er e s i s t o r s
dividedby the total
I n c i r c u i t( 1 ) ,t h e c u r r e n t h r o u g he a c hresistoris givenby the total voltage
r e s i s t a n ci e. e .
' I J
tr
Ir : - :
Rr
E
Rr * Rz+ R3
A n dt h e v o l t a g eV r t h a t i s t a p p e do f f i s e q u a lt o :
Vt : IrRt:
ERt
Rr*R2+R3
s i m i l a r l yi ,n c i r c u i t( 2 )t h e v o l t a g eV zt h a t i s t a p p e do f f i s e q u a lt o :
Vz:
.
It(Rt * Rr) :
E [ R 1+ R 2 ]
R;-
&
s u c ha sV r a n d
T i r . .s S o v ee x a m p l es h o w st h a t w h i l et h e s o u r c eh a sa v o l t a g eE ,s m a l l evr o l t a g e
V 2c d nb e o b t a i n e df r o m i t .
30
THEPOTENTIOMETER
l f i n s t e a do f u s i n gt h r e er e s i s t o risn seriesa one metrelengthof resistance
wire is usedand
a
s l i d i n gc o n t a c it s u s e d t, h e n s e v e r atla p p e do f f v o l t a g e cs a nb e o b t a i n e da, n d a v a r i a b l ed . c .
s u p p l yc a nt h u s b e p r o d u c e d .
A st h e s l i d i n gc o n t a c it s m o v e d a l o n gA B ,
t h e r e s i s t a n ci en c r e a s ef sr o m A t o B a n d
hencethe tappedoff voltageincreases
s i n c eV = l R ,w i t h t h e c u r r e n tI b e i n g
c o n s t a ntth r o u g h o utth e e n t i r el e n g t ho f
resistor.
D r i v e rc e l l
\.
T h ew h o l er e s i s t ocr a nb e c o n s i d e r eadsa n
i n f i n i t en u m b e ro f t i n y r e s i s t o rasl t
connectedin series,so that the current
t h r o u g ht h e m i s c o n s t a n t .
W h e na s l i d ew i r e i s u s e da s i n t h e a b o v ea p p l i c a t i o nt h, e p o t e n t i adl i v i d e ri s r e f e r r e dt o a sa
POTENTIOMETER.
T h e b a t t e r yo r c e l lt h a t i s a p p l i e da c r o s st h e e n t i r el e n g t ho f t h e p o t e n t i o m e t ewr i r e , i s c a l l e d
the
CELL
.D R I V E R
D r i v e rc e l l
S t a n d a r cde l l
31
ONFT H EP O T E N T I O M E T E R
APPLICATIO
( 1 ) M e a s u r e m e not f e . m . f .
i f a n o t h e rc e l l
O n eo f t h e p r i m a r yu s e so f t h e p o t e n t i o m e t eirs i n t h e m e a s u r e m e notf t h e e . m . f .o f a c e l l
c e l l )i s a v a i l a b l e
o f k n o w ne . m . f .( c a l l e da s t a n d a r d
c e l l . S a fyo r e x a m p l et h e s t a n d a r cd e l lh a sa n e . m ' f
l e n g t hi s f o u n df o r t h e s t a n d a r d
F i r s t l yt ,h e b a l a n c e
o f 1 . 1 5 Va, n dt h e b a l a n c ep o i n tw a sf o u n dt o b e 3 6 c m .
the
T h e s t a n d a r dc e l li s t h e n t a k e no u t o f t h e c i r c u i ta n d r e p l a c e db y t h e c e l lo f u n k n o w ne . m . f .a n d
b a l a n c ep o i n ti s a g a i nf o u n d .S a yf o r e x a m p l ea g a i n t, h e b a l a n c ep o i n tw a sf o u n dt o b e 5 4 c m .
D r i v e rc e l l
D r i v e rc e l l
4
C e l lo f u n k n o w n
S t a n d a r cd e l l
e.m.f.
T h ec a l c u l a t i ofno r t h e u n k n o w ne . m . f .i s a sf o l l o w s :
l e n g t ho f 3 6 c m .
s a balance
1 . 1 5 Vc o r r e s p o n dt o
1.15
H e n c e1 c m w i l l h a v ea n e q u i v a l e nvt o l t a g eo f E ;
1.15
x 54 =1,.725V
5 4 c m w i l l h a v ea n e q u i v a l e nvt o l t a g eo f E
H e n c et h e e . m .f . o f t h e u n k n o w nc e l li s 1 ' . 7 2 5 V
32
( 2 1 M e a s u r e m e not f t h e i n t e r n a lr e s i s t a n c oe f a c e l l :
T h eb a l a n c el e n g t ho f t h e e . m . f .o f t h e c e l li s f i r s tf o u n dw h e nt h e c e l li s o n o p e nc i r c u i ti,. e .n o
l e n g t hi s 6 8 c m
s c o n n e c t etdo i t . S a yf o r e x a m p l et ,h i sb a l a n c e
e x t e r n arl e s i s t a n ci e
D r i v e rc e l l
E
T h e n e x ts t e pi s t o i n s e r ta K n o w nr e s i s t a n cien s e r i e sw i t h t h e c e l lo f e . m . f .E .U s ef o r
e x a m p l ea 2 0 Or e s i s t o rT.h en e w b a l a n c el e n g t hw i l l d e c r e a steo s a y4 6 c m ,f o r t h e
n e w v o l t a g eV
D r i v e rc e l l
T h e e . m . f .E o f t h e c e l ln e e d
not be known in this case
E
R
R e c a ltlh e f o r mu l aV : 1 R - R*r
VR46
Hence-:
E
R*r
:68
i
68R=46R+46r
22R -- 46r, but R =20C), hence 46r = 440Q ;
r = 9.560
33
THEWHEATSTONE
BRIDGE
T h i sb r i d g ec i r c u i w
t a sd e s i g n e tdo p r o d u c ea v e r ya c c u r a t em e a n so f m e a s u r i nrge s i s t a n c e .
I n n o r m a l a b o r a t o r py r a c t i c e
t h e r e s i s t a n coef a c o m p o n e nits d e t e r m i n e d
byapplying
a
p o t e n t i adl i f f e r e n c a
e c r o s si t , m e a s u r i n g t hceu r r e n t t h r o u g iht , a n dt h e n a p p l y i n gR - y .
H o w e v e rt,h e a b o v em e t h o di n t r o d u c eus n c e r t a i n t i ei ns t h e m e a s u r e m e ndtu et o U o t ht f ' r .
i n s t r u m e n tasn dt h e o b s e r v e r .
T h eW h e a t s t o n e
B r i d g eu s e sa n u l lm e t h o d w
, h i c hm e a n st h a t t w o o p p o s i n g
c u r r e n t sa r e
p r o d u c e dw h o s em a g n i t u d easr e a d j u s t e du n t i lt h e y b a l a n c e a c ho t h e r ,s o t h a t t h e r e s u l t a n t
c u r r e n ti s Z E R Oa, n d a t t h i s p o i n t ,a n u n k n o w nr e s i s t a n ci en t h e c i r c u i ct a nb e d e t e r m i n e d .
A current I leavesthe battery and entersthe junction
of A and B, where it splits into Ir and Iz.
The resistorsA and C are in seriesand thereforethe
samecurrent Ir flows through both of them. Similarly,
the resistorsB and D are in series,and the same
current Iz flows through both of them.
If the resistancesare adjustedso that no current
flows from X to Y then the potential at X is equalto
the potential at Y and a BALANCEPOINTis reached.
This would also mean that the potential difference
acrossA is equalto the potential differenceacrossB.
The p.d.acrossA is equalto IrA and the p.d.acrossB
is equalto IzB.
Henceat the balancepoint IrA = IzB
t1)
A l s oa t t h e b a l a n c ep o i n t ,t h e p . d .a c r o s sC i s e q u a lt o t h e p . d .a c r o s sD , s o t h a t : I r C = l z D ( 2 )
If equation [1) is divided by equation[2) we get:
AB
CD
I
I
I If the rraluesof three of the resistancesare known, then the fourth one can be calculated.
I
I
34
BRIDGE
T H EM E T R E
e r i d g ea, n d i t i s u s e dt o m e a s u r e
T h em e t r eb r i d g ei s a m o d i f i e dv e r s i o no f t h e W h e a t s t o n B
u n k n o w nr e s i s t a n c ewsi t h m u c hm o r ee a s et h a nt h e W h e a t s t o n vee r s i o n .
DLI V I D E R .
EO T E N T I A
I t o p e r a t e sa s a D O U B L P
e r i d g et,w o r e s i s t o rasn da
I n s t e a do f u s i n g f o u r e s i s t o rass i n t h e c a s eo f t h e W h e a t s t o n b
wire are used.
metreof resistance
and N is
M is a knownresistance,
M
lz
t h e u n k n o w no n e .
T h es l i d ew i r ef r o m t h e
g a l v a n o m e t ei sr m o v e da l o n gt h e
r e s i s t a n cwei r e A B u n t i la b a l a n c e
p o i n ti s o b t a i n e d .
At the balancepoint,the lengthof
'c'can
be consideredas one
wire
'd'
resistorand the length as
another resistor.
A currentI leavesthe battery,enterspoint A and splits into Ir and Iz.Letthe resistanceof
'c'
be kc and the resistanceof d' be kd, where k is the resistanceper unit length of the
wire.
At the balancepoint Ir kc= IzM t1)
; and Irkd = IzN (2)
When equation (1) is divided by equation(2J we get :
CM
dAl
'd'
are measuredlengthson the resistance wire while M is a known resistance,
and
henceN can be calculated.
'c'
35
TUTORIAL
on CurrentElectricity
( 1 ) A b a t t e r yo f e m f 6 V a n d i n t e r n arl e s i s t a n c1e. 8 Oi s c o n n e c t e a
d c r o s sa 1 2 Or e s i s t o r .
C a l c u l a t eh e t e r m i n apl . d .o f t h e b a t t e r y . ( H i nTth: et e r m i n apl . d .i s t h e v o l t a g ea c r o s s
t h e e x t e r n arl e s i s t a n cie. e .a c r o s st h e 1 2 Or e s i s t o r )
(s.22V)
( 2 ) W h e na b a t t e r yo f e m f E a n d i n t e r n arl e s i s t a n cre i s c o n n e c t e a
d c r o s sa 6 O r e s i s t o rt ,h e
t e r m i n ap
l . d .i s 8 . 4 4 V . l ft h e 6 Q r e s i s t oirs n o w r e p l a c e d
b y a 1 0 Or e s i s t o rt ,h e t e r m i n a l
p . d .i s n o w 8 . 6 5V . D e t e r m i n teh e e m f a n dt h e i n t e r n arl e s i s t a n coef t h e b a t t e r y .
( H i n t :D r a w2 s e p a r a t cei r c u i t as n da p p l yt h e e q u a t i o nE =V + l r f o r b o t ho f t h e m ;t h e n
u s et h e s i m u l t a n e o uesq u a t i o ntso f i n dt h e r e q u i r e d
quantities)
( r =0 . 3 9 Q E
, =9V)
( 3 ) A b a t t e r yo f e . m . f .1 - 8 V
a n d i n t e r n arl e s i s t a n crei s c o n n e c t e tdo a n e t w o r ko f r e s i s t o r s
h a v i n gt w o r e s i s t o rish p a r a l l e(l8 Oa n d 1 2 O) i n s e r i e sw i t h a 6 0 r e s i s t oar ss h o w n :
l f t h e c u r r e n it n t h e 8 O r e s i s t oirs 0 . 4 A c, a l c u l a t e :
(i)
T h ec u r r e n ti n t h e 1 2 Or e s i s t o r ( Y of iur s tn e e dt o f i n d t h e p . d .a c r o s st h e
(ii)
l 2 Q r e s i s t owr h i c hi s t h e s a m ep . d .a c r o s st h e 8 O r e s i s t o r )
T h ec u r r e n ti n t h e 6 e r e s i s t o r
T h ei n t e r n arl e s i s t a n coef t h e b a t t e r y
(iii)
A n s .0 . 2 6 7
A , 0 . 6 6 7 A1, , 6 . 2 Q ., Z
7V
36
(41A b a t t e r yP o f e . m . f . 1 , 2 Va n di n t e r n arl e s i s t a n c4eO i s c o n n e c t e idn s e r i e sw i t h a b a t t e r y
e with a 20Oresistorconnectedacrossthe
Q o f e . m . f .8 V a n di n t e r n arl e s i s t a n c90,
c o m b i n a t i oans ' s h o w n :
L 2 V , 4n
x
|l l r-l
V, 9Q
|
I
200
C a l c u l a t eh e t e r m i n apl . d .o f e a c hb a t t e r y .
A n s :9 . 5 8 V1
, 3.45V)
( Ad e t a i l e d
s o l u t i o ni s g i v e no n p a g e3 5 )
( 5 ) D e t e r m i nteh e
below:
24V
( T h ep . d .a c r o s X
s Yi st h e p . d .a c r o s tsh e 8 O m i n u st h e p . d .a c r o s tsh e r z r ) )
( A n s 0: V )( A d e t a i l e d
s o l u t i o ni s g i v e no n p a g e3 5 )
(Ans.3.2V)
Determine
thevoltmeter
r e a d i n gi n t h e g i v e n
rzv
20v
c i r c ui t .
A ns :1 5. 2 V
( S e ep a g e 3 6 )
37
(7) In the followingcircuit,a voltmeterof resistance
8000 is firstlyconnectedacrossthe
6000 resistorand then acrossthe 2200O resistorseparately.
F i n dt h e r e a d i n go n t h e v o l t m e t e ri n e a c hc a s e .
12V
A n s1. 6 2 V ; 5 . 9 3 V
I
I
I
L__O_-,
(8) ln the followingcircuit,a voltmeterof resistance
400Ois firstlyconnectedacrossthe
16000 resistorand then acrossthe 200 O resistorseparately.
F i n dt h e r e a d i n go n t h e v o l t m e t e ri n e a c hc a s e .
18V
A n s1 1 . 0 8 V 1
; .38V
r__o-,
38
FORQUESTION
4:
SOLUTION
1,2V,
4A
8V,
9C)
200
S i n c et h e b a t t e r i e sh a v et h e i r e m f ' si n t h e s a m ed i r e c t i o nt ,h e n e t e m f i n t h e c i r c u i ti s 1 2 V+ 8 V =
20v
T h e 1 2 V b a t t e r yh a sa g r e a t e re m f t h a n t h e 8 V b a t t e r ys o t h a t a c u r r e n tl e a v e st h e 1 2 V a n d
entersthe 8V battery.
20
The total resi-stancels 330, and the current i"s
TA
T h e t e r m t n a l p . d . o f t h e I z V b a t t e r y t s o b t a t n e df r o m t h e e q n V t : E - I r
v:rz-2fn>:e.sBV
55
T h ec u r r e net n t e r st h e 8 Vb a t t e r ys ot h a ti t st e r m i n apl . d .i sg i v e nb yV 2 =E+ l r
Vz:B+*(9):1,3.45V
55
FORQUESTTON
5:
soLUTroN
T h e r ea r e2 p a r a l l ebl r a n c h e s o t h a t t h e p . d .a c r o s se - a c h
b r a n c hi s 1 2 V .
T h e c u r r e n ti n t h e t o p b r a n c hi s g i v e nb y :
t' R: 2Y4 - 1 2 : 0 . 5 ^ 4
H e n c et h e v o l t a g ea c r o s st h e 8 Q r e s i s t o irs g i v e nb y
V : I R : 0 . 5 x B - 4 V ; H e n c eV e v: l V
N o w t h e n o t a t i o nV a ym e a n st h e p o t e n t i adl i f f e r e n c e
a c r o s sA Y ,
:
that is, VA Vy 4
T h ec u r r e n ti n t h e l o w e rb r a n c hi s g i v e nb y :
, _v _
- 3r 2
6 _1 n
a c r o s st h e 1 2 O
T h e p o t e n t i adl i f f e r e n c e
r e s i s t o irs V s xa n d i s :
T h e p o t e n t i aal t A i s e q u a lt o t h e p o t e n t i aal t B s i n c et h e y a r e
connectedto eachother by a conductoronly;Va- Vn
H e n c eV A - V x : 4 a n d V A - V v : 4
F r o mt h e s et w o e q u a t i o n sw, e g e t :
1
Vsx:lR:;X12:4V
Vx-Vy:0,soVyy:0
J
VB-v,r::4
39
S O L U T I OF
NO RQ U E S T I O5N:
T h ev o l t m e t e r e a d i n gi s
t h e p o t e n t i adl i f f e r e n c e
acrose
s i t h e rt h e b r a n c h
1,2V
c o n t a i n i ntgh e 1 2 Vb a t t e r y
t o g e t h e rw i t h t h e 4 0
r e s i s t oor r t h e b r a n c h
c o n t a i n i ntgh e 2 0 Vb a t t e r y
withthe 60 resistor.
C o n s i d ef ri r s tt h e b r a n c hw i t h t h e j _ 2 Vb a t t e r y .
T h en e t e m f i s 8 V a n dt h e c u r r e n tw o u l db e
VB
I-R:10:o'BA
V : E t I r : 1 , 2 +( 0 . 8x 4 ) : I 5 . 2 V
I f w e u s e dt h e b r a n c h
w i t ht h e2 0 Vb a t t e r yw, e w o u l dg e t :
V:E-lr
V -20 -(0.8x6):I5.2V
Remember
that whencurrententersthe battery,we use
V=E+lr, andwhencurrentleaves
the batterywe use
V =E - l r
40
WorkedExamples
on Potentiometerand Metre Bridge
( 1 ) T h ee m f o f a b a t t e r yX i s b a l a n c e d
b y a l e n g t ho f 5 8 c mo n a p o t e n t i o m e t ewr i r e w h i l s ta
s t a n d a r dc e l lo f e m f 1 . 0 4 Vi s b a l a n c e b
d y a l e n g t ho f 4 6 c m .
(i)
D e t e r m i nteh e e m f o f X
(ii)
T h ec e l lX h a sa n i n t e r n arl e s i s t a n coef 4 C ) , a n ids c o n n e c t e d
i n s e r i e s t oa l - 2 O
r e s i s t o rC. a l c u l a tteh e n e w b a l a n c el e n g t h .
( A n s .1 , . 5 4 V ; 5 0 . 9 c m
Solution:
(i)1.04h
Va sa b a l a n c lee n g t ho f 4 6 c m .
Lcm correspondsto a p. d. of
6 8 c mh a sa n e q u i v a l e npt. d .o f
1,.04
46
( i i )T h ec i r c u i th a sa c e l lo f e m f 1 . 5 4 Vi n s e r i e s
w i t h a 4 Q r e s i s t oar n da 1 2 Or e s i s t o r .
T h ev o l t a g ea c r o s st h e 1 2 Or e s i s t oirs t h e
t e r m i n apl . d .o f t h e b a t t e r ya n d i s e q u a tl o :
1,2
1,.54,
1,.1,5V
tO-
68xry-Ls4v
+6
T h eb a l a n cpeo i n tc o r r e s p o n d itnogl - . 1 , 5i V
s
g i v e nb y :
( 2 )A s t a n d a r dc e l lo f e m f 1 . 3 4 Vi s b a l a n c e b
d y t h e p . d .a c r o s sa r e s i s t a n coef 2 8 0 0 0i n s e r i e sw i t h a
1 - ml e n g t ho f p o t e n t i o m e t ewr i r e P Qh a v i n ga r e s i s t a n coef 6 0 . A t h e r m o c o u p l e
p r o d u c i n ga n e m f o f
1 . 2 m Vi s n o w c o n n e c t e idn s e r i e st o t h e s t a n d a r d
c e l la s s h o w n C
. a l c u l a tteh e l e n g t hP Ro n t h e
p o t e n t i o m e t ewr i r et h a t p r o d u c etsh e n e w b a l a n c e
point.
Thetotal resistance
i s e q u a lt o 2 8 0 6 0 .
T h ep . d .a c r o s tsh e
1 m o f w i r ei s
a6
x 1.34
2806
- 1,.6mV
1 . 6 m Vh a sa l e n g t h
of 100cm
1..34V
l - . 2 m Vw i l l h a v ea
lengthof
loox
1.2:75cm
1..6
41,
on the Potentiometerand the metre bridge
TUTORIAL
( 1 ) T h e e m f o f a b a t t e r y X i s b a l a n c e db y a l e n g t h o f 4 4 c m o n a p o t e n t i o m e t e r w i r e w h i l s t a
s t a n d a r dc e l l o f e m f 1 . 2 0 Vi s b a l a n c e db y a l e n g t ho f 2 8 c m .
(iii)
D e t e r m i n et h e e m f o f X
(iv)
T h e c e l l X h a s a n i n t e r n a lr e s i s t a n c eo f 2 C ) , a n di s c o n n e c t e di n s e r i e s t o a l - 8 O
: h e c kw o r k e d e x a m p l e )
resistorC
. a l c u l a t et h e n e w b a l a n c el e n g t h . ( H i n tC
( A n s1 . 8 9 V ; 3 9 . 6 c m )
( 2 ) T h e e m f o f a b a t t e r yZ i s b a l a n c e db y a l e n g t ho f 7 4 c m o n a p o t e n t i o m e t e rw i r e w h i l s t a
standard.ceo
l l f e m f 1 . t z V i s b a l a n c e db y a l e n g t ho f 3 8 c m .
(v)
D e t e r m i n et h e e m f o f Z
(vi)
T h e c e l l Z h a s a n i n t e r n a lr e s i s t a n c eo f 6 0 , a n d i s c o n n e c t e di n s e r i e st o a 4 4 O
resistorC
. a l c u l a t et h e n e w b a l a n c el e n g t h .
(Ans2
: .18V;65.2cm)
( 3 ) T h e e m f o f a b a t t e r y . Yi s b a l a n c e db y a l e n g t ho f 8 8 c m o n a p o t e n t i o m e t e rw i r e w h i l s t a
s t a n d a r dc e l l o f e m f 0 . 9 6 Vi s b a l a n c e db y a l e n g t ho f 3 6 c m .
(vii)
D e t e r m i n et h e e m f o f Y
( v i i i ) T h e c e l l Y h a s a n i n t e r n a lr e s i s t a n c eo f I . 2 Q , a n d i s c o n n e c t e di n s e r i e s t o a 3 6 O
resistorC
. a l c u l a t et h e n e w b a l a n c el e n g t h
( A n s2 . 3 5 V 8
; 5cm
( 4 ) T h e d r i v e rc e l lo f a p o t e n t i o m e t ewr i r e h a sa n e . m . f .o f 1 . 8 Va n d n e g l i g i b l ien t e r n a rl e s i s t a n c e .
T h e p o t e n t i o m e t ewr i r e h a sa r e s i s t a n coef 6 0 . C a l c u l a tteh e r e s i s t a n cteh a t i s n e e d e di n s e r i e s
r i r e i f a p . d .o f 2 5 0 m Vi s r e q u i r e da c r o s st h e w h o l e l e n g t ho f t h e
w i t h t h e p o t e n t i o m e t ew
p o t e n t i o m e t ewr i r e .
( H i n t :D r a wa c i r c u i td i a g r a mo f t w o r e s i s t o r isn s e r i e sw i t h t h e d r i v e rc e l l ,t h e n f i n d t h e p . d
s
a c r o s st h e e x t r ar e s i s t a n c et h, e n u s eV = l Rt o d e t e r m i n eR , r e m e m b e r i ntgh a t f o r r e s i s t a n c ei n
: 7.2O)
s e r i e st,h e c u r r e n ti s t h e s a m e . () A n s 3
- ( 5 ) T h ed r i v e rc e l lo f a p o t e n t i o m e t ewr i r e h a sa n e . m . f o
. f 4 . 0 Va n d n e g l i g i b lien t e r n arl e s i s t a n c e .
T h e p o t e n t i o m e t ewr i r e h a sa r e s i s t a n coef 2 0 . C a l c u l a tteh e r e s i s t a n cteh a t i s n e e d e di n s e r i e s
w i t h t h e p o t e n t i o m e t ewr i r e i f a p . d .o f 1 2 m Vi s r e q u i r e da c r o s st h e w h o l el e n g t ho f t h e
p o t e n t i o m e t ewr i r e
(Ans6
: 650)
T.h e
( 6 ) T h e d r i v e rc e l !. r f a p o t e n t i o m e t ewr i r e h a sa n e . m . f .o f 6 V a n d n e g l i g i b l ien t e r n a rl e s i s t a n c e
p o t e n t i o m e t ewr i r e h a sa r e s i s t a n coef 3 . 5 O .C a l c u l a tteh e r e s i s t a n cteh a t i s n e e d e di n s e r i e s
w i t h t h e p o t e n t i o m e t ewr i r e i f a p . d .o f 8 m V i s r e q u i r e da c r o s st h e w h o l el e n g t ho f t h e
potentiorneter
wire.(Ans:26220)
42
( 7 ) I nt h e c i r c u i b
t e l o wt h e b a l a n c ep o i n ti s o b t a i n e df o r t h e l e n g t h si n d i c a t e od n t h e d i a g r a mF. i n d
t h e r e s i s t a n coef R .( A n s3 2 . 6 O )
20o.
(8)
I n t h e a b o v ed i a g r a mc, a l c u l a tteh e b a l a n c el e n g t h sc a n d d .
( A n s :c = 2 7. 5 c m ,d = 7 2 . 5 c m \
43
ELECTROSTATICS
E l e c t r o s t a tpi ch e n o m e n a
i n c l u d em a n ye x a m p l eas ss i m p l ea st h e a t t r a c t i o no f t h e p l a s t i cw r a p
t o y o u r h a n da f t e ry o u r e m o v ei t f r o m a p a c k a g et o, t h e a p p a r e n t lsyp o n t a n e o uesx p l o s i o on f
g r a i ns i l o st,o d a m a g eo f e l e c t r o n ic o m p o n e n tdsu r i n gm a n u f a c t u r i ntgo, t h e o p e r a t i o no f
photocopiers.
E l e c t r o s t a t iicnsv o l v etsh e b u i l d u po f c h a r g eo n t h e s u r f a c eo f o b j e c t sd u et o c o n t a c w
t ithother
s u r f a c e sA. l t h o u g hc h a r g ee x c h a n g h
ea p p e nw
s h e n e v ear n y t w o s u r f a c ecso n t a c a
t n ds e p a r a t e ,
t h e e f f e c t so f c h a r g ee x c h a n g ae r e u s u a l l yo n l yn o t i c e dw h e na t l e a s to n e o f t h e s u r f a c ehs a sa
h i g hr e s i s t a n cteo e l e c t r i c af l o w .T h i si s b e c a u s teh e c h a r g e tsh a t t r a n s f e tro o r f r o m t h e h i g h l y
r e s i s t i v seu r f a c ea r e m o r eo r l e s st r a p p e dt h e r ef o r a l o n ge n o u g ht i m e f o r t h e i re f f e c t st o b e
o b s e r v e dT. h e s ec h a r g e tsh e n r e m a i no n t h e o b j e c tu n t i l t h e ye i t h e rb l e e do f f t o g r o u n do r a r e
q u i c k l yn e u t r a l i z ebdy a d i s c h a r g e : e . tgh. e, f a m i l i a pr h e n o m e n oonf a s t a t i c ' s h o c k 'ci sa u s e db y
t h e n e u t r a l i z a t i oonf c h a r g eb u i l tu p i n t h e b o d yf r o m c o n t a c w
t i t h n o n c o n d u c t i vseu r f a c e s .
Coulomb'sLaw:
.
T h ee l e c t r o s t a tfi o
c r c eo f a t t r a c t i o no r r e p u l s i o a
n c t i n go n e a c ho f t w o s t a t i o n a rpy o i n t
c h a r g e si s, d i r e c t l yp r o p o r t i o n at ol t h e p r o d u c to f t h e c h a r g e sa,n d i n v e r s e lpy r o p o r t i o n at ol t h e
s q u a r eo f t h e i rd i s t a n c oe f s e p a r a t i o n .
i . e .l f t w o p o i n tc h a r g eQ
s l a n de z a r ep l a c e da t a d i s t a n c rea p a r t t, h e nt h e f o r c ea c t i n go n e a c h '
o f t h e c h a r g e si s g i v e nb y ;
l:x
Thiscanbe re-written
as
QtQz
-
F : !3+
Byexperiment,
the constant
of proportionality
wasfoundto U"
g i v e nb y :
F-
fr, sothatthefinalformulais
QrQz
4nesrz
w h e r ee i s c a l l e dt h e p e r m i t t i v i t oy f t h e m e d i u mi n w h i c ht h e c h a r g e e
s x i s tT
. h ep e r m i t t i v i t iys a
p h y s i c apl r o p e r t yo f t h e r n e d i u mM
. o s tq u e s t i o n isn v o l v et h e m e d i u mb e i n ga v a c u u ma n dt h e
permittivityis writtehds rs where ro=8.85x !0-t2c2N-tm-2.
1
T n e. , ' , ' r roef _
i s n o r m a l l tya k e na s9 x 1 0 ' N m ' C -.'
4ne.
44
I
In orderto determinethe resultantforceactingon any chargethat is within
an arrangement
of
charges,
a vEcroR approachmust be used,that is,if a pointchargeis acteduponelectrostatic
forcesfrom two or more charges,then the force DUETo EACHCHARGE
is calculatedfrom the
Coulomb'sLawformuraand then the resultantforcecanbe found by means
of VECToR
ADDITION.
Considera simplecasewherea singrepointchargeB is actedupon by forcesfrom
two other
electricchargesas shown:
Suppose
we want to find the resultantforceactingon B
A
20;rC @+
8cm
7.
30sC
1,2cm
I n t h i se x a m p l ea, l l
c h a r g e sa r e
positive.
va
50uC
T h ef i r s ts t e pi s t o f i n dt h e f o r c ea c t i n go n B D U ET o A o N L y .
T h i si s d o n eb y a p p l y i n tgh e c o u l o m b ' L
s a wF o r m u l a
i.e.
rtrA -
20 x10-6 x 30 x 10-6x 9 x 10e
:843.75N
0.082
T h ef o r c ed u et o C o n l y i s g i v e nb y :
ttrc -
5 0 x 1 0 - 6x 3 0 x 1 0 - 6x 9 x 1 0 e
: 937.5N
0.1,22
45
T h et w o f o r c e sc a nb e d r a w no n a v e c t o rd i a g r a ma sf o l l o w s :
F" - 937.5N
Fn : 843.75N
T h e h o r i z o n t aclo m p o n e not f F 6= 9 3 7' 5 c o s4 5 = 6 6 3 N '
forcesis 663+843.75=1506.75N
Sothe total of the horizontal
T h ev e r t i c acl o m p o n e not f F 6= 9 3 7' 5 s i n 4 5= 6 6 3 N '
l r c eo f 1 5 0 6 . 7 5 N .
g i t h a h o r i z o n t af o
S o n o w w e h a v ea v e r t i c aflt r c e o f 6 6 3 Nc o m b i n i n w
a nb e u s e dt o g i v ea r e s u l t a n t f o r coef 1 6 4 6 N .
P y t h a g o r a s ' t h e o r ecm
H e n c et h e r e s u l t a nfto r c eo n B i s 1 6 4 6 N
46
ELECTRIC
FIELD
STRENGTH
(AlsocalledElectricfield Intensityor simplyElectricField):
T h eE l e c t r i F
c i e l dS t r e n g t ha t a p o i n ti s d e f i n e da s t h ef o r c ea c t i n gp e r u n i t p o s i t i v ec h a r g e
p l a c e da t t h a t p o i n t .
E l e c t r ifci e l ds t r e n g t hi s a v e c t o rq u a n t i t ya n da s s u c h ,i t s d i r e c t i o nm u s tb e c o n s i d e r e d
in any
calculation.
F r o mt h e d e f i n i t i o n ,
Electrtc Field Strength -
Force
Charge
, h e n c et h e u n i t sa r e N C - 1
l f a p o i n tc h a r g eh a sa m a g n i t u d e
o f c h a r g ee , t h e n a t a d i s t a n c er f r o m
e t h e e l e c t r i cf i e l d
strengthis givenby :
E-
4nes rz
A p a r tf r o m t h e e l e c t r i cf i e l dd u e a p o i n tc h a r g ee, l e c t r i cf i e l d sa l s oe x i s ti n t h e s p a c eb e t w e e n
t w o o p p o s i t e lcyh a r g e dp a r a l l epl l a t e s N
. o t et h a t t h e f i e l di s u n i f o r mb e t w e e nt h e p l a t e s
m e a n i n g t h a t t h eF I E L D
S T R E N G Ti sHc o n s t a nitn t h e s p a c eb e t w e e nt h e p l a t e s .A n o t h e r
i m p o r t a nfte a t u r eo f t h e f i e l di s t h a t t h e f i e l dl i n e sa r e c u r v e da t t h e e n d so f t h e p l a t e s .
++++++
+++
C o n s i d et rw o p a r a l l em
l e t a lp l a t e ss e p a r a t e b
dy a d i s t a n c e
d a n d h a v i n ga p o t e n t i adl i f f e r e n c e
V b e t w e e nt h e m .
Ti'i. ,-'iectric
fieldstrengthbetweentheseplatesis givenby
47
V
E- a
V
Thisquantity
i
(Unitsarevm-l)
Gradient.
,talsocalledthe Potential
for the parallel
Hencethe potentialgradientis numericallyequalto the electricfield strength
platearrangement.
Pleasenote that the formula
a
E_
is usedto find the
4nesrz
CHARGEONLY
FIELDSTRENGTHFORA STATIONARYELECTRIC
ELECTRIC
a
strengthin the
FIELD
is usedto findthe Electric
\
a.
regionbetweentwo chargedparallelmetalplates
TheformulaE :
i st h a t :
A n o t h e rv e r yI M P O R T A NFTO R M U L A
on
Whenevera chargee is placedin a field that hasa field strengthE,the force
that chargeis givenbY:
p= EQ
Hencefor a pair of parallelplatesthat are separatedby a distanced, and
havinga potentialdifferenceV acrossthe plates,the force on a chargeQ
that is placedin the spacebetweenthe platesis givenby:
F:
VQ
d
48
ELECTRIC
POTENTIAL
:
T h eE l e c t r i P
c o t e n t i aal t a p o i n ti s d e f i n e da st h e w o r k d o n ei n b r i n g i n ga u n i t p o s i t i v ec h a r g e
f r o m i n f i n i t yt o t h a t p o i n t .
A t i n f i n i t yt ,h e e l e c t r i p
c o t e n t i ai ls z e r os i n c et h e c h a r g ei s n o t u n d e r t h ei n f l u e n coef a n y f i e l d .
C o n s i d e r au n i t p o s i t i v ec h a r g eb e i n gp l a c e di n t h e f i e l do f a n o t h e rp o s i t i v ec h a r g ea ss h o w n :
B
A s t h e p o s i t i v ec h a r g ei s m o v e df r o m p o i n tB w h i c hi s c o n s i d e r etdo b e a t i n f i n i t yt,o a p o i n tA i n
t h e f i e l d ,w o r k h a st o b e d o n eo n t h e p o s i t i v e( + 1 C c) h a r g et o o v e r c o m e
t h e f o r c eo f r e p u l s i o n
b y t h e o t h e rp o s i t i v ec h a r g eQ + .
T h i sw o r k d o n eo n t h e l - Cc h a r g ec a u s e a
s nincrease
i n i t s e l e c t r i cp o t e n t i ael n e r g yi n m o v i n g
from infinityto A.
T h i si n c r e a s e
i n p o t e n t i ael n e r g yi s a s s i g n ead p o s i t i v ev a l u e .
O n t h e o t h e rh a n d ,i f Q w a sa n e g a t i v e
c h a r g et,h e n t h e i n c o m i n g+ 1 Cc h a r g ew o u l db e
attractedto it, and it would loseits own electricpotentialenergyas it fallsfrom infinitytowards
a.
T h i sd e c r e a s e
i n e l e c t r i cp o t e n t i ael n e r g yi s a s s i g n ead n e g a t i v ev a l u e .
49
Hence,at all pointsin the fieldof a positivecharge,the potentialis positive;and at all pointsin
the fieldof a negativecharge,the potentialis negative.
Potenttal:
potenttal _
Work done per untt charge
Electrtc potential Energy _ Work done
chirge
charge
Let Potential= V
V- +
f or an isolated.charge
*TEtsT
When a chargeQ movesthrough a p.d. of V, the work done on the chargeis W = QV
whichis equivalentto the Volt.
The unit is the JC-1
QUANTITIES.
BothPotentialand PotentialEnergyare SCALAR
A v e r yu s e f u fl o r m u l at o r e m e m b e ri s t h a t :
in electricpotentialenergyof a chargeQ when it is movedthrougha
The CHANGE
AV , is givenby A W= qAV.
DIFFERENCE
FOTENTIAL
V is the differencein potentialsVr and Vz.
DIFFERENCE
The POTENTIAL
of
ENERGY
in ElectricPotentialenergy is equalto the changein the KINETIC
The CHANGE
the charge.HenceA W= QAV=lmvz
2
where m is the massof the chargeand v is its
speed.
Note that smallv is speedand big V is potentialdifference.
(Thesize of the future you actually experiencewill largely be determined by one
factor: the people you choose to connect with. When you choosepeople who are truly
committed to growth in every aspect of your life, your own potential for growth
becomestruly unlimited--'---DanSullivan.
50
ELECTROSTATTCS
(1)
TUTORTAL
SHEET
( 1 ) A h e l i u mn u c l e u s . h aasc h a r g eo f + 2 e a n da n e o nn u c l e u sh a sa c h a r g eo f + 1 0 e ,w h e r ee i s
electroniccharge(1.6x 10 1eC)
Findthe repulsive
forceexertedon one by the otherwhen
t h e y a r e3 n m a p a r ti n a v a c u u m (. A n s5 . 1 2 x 1 0 - 1 0 N )
(2) A chargeof 3;rCis 4mm awayfrom a chargeof 1.41tC
in a vacuum.What is the electrostatic
f o r c ea c t i n go n e a c ho f t h e c h a r g e s(?A n s 2 . 3 6x t 0 4 w )
(3) What is the distanceof separation
of two pointcharges20pCand-45prC
if the forceof
a t t r a c t i o nb e t w e e nt h e m i s 1 5 N ?( A n s7 3 c m )
(4) Two identicalpointchargesare separatedby a distanceof 25cmand a forceof repulsionof
0 . 5 6 Na c t so n e a c hc h a r g eF. i n dt h e m a g n i t u d o
e f e a c ho f t h e c h a r g e s( .A n s .L . 9 7 x1 0 6 C )
( 5 ) T h r e ep o i n tc h a r r " , a r e p l a c e do n t h e x a x i sa ss h o w n .F i n dt h e n e t f o r c eo n t h e - 5 U C
c h a r g ed u et o t h e o t h e rt w o . ( A n s . 0 . 6 2 5 N ) . ( HFi ni nt :dt h e f o r c ed u et o e a c hc h a r g e
s e p a r a t e ltyh e n f i n d r e s u l t a n t )
-5uc
3uc
8pc
ooo
20cm
30cm
( 6 ) F o rt h e s i t u a t i o ns h o w ni n t h e b e l o wd i a g r a mf,i n d (a)Theelectricfield at point P,(b)The
forceon a -4 x 10-8C chargeplacedat P .
-5x108C
-20 x 10-"C
( A n s : 1 . 4 x 1 0 sN C - 1 ' t o w a r dtsh e l e f t ; 5 . 6 x t o 3 l v; )
51
(7) A 20prC
chargeas shown.
chargeis placed50cmawayfrom a 60prC
50cm
+€
(3<-
60PtC
20PrC
D e t e r m i n et h e e l e c t r i cf i e l d s t r e n g t ht o g e t h e rw i t h i t s d i r e c t i o na t t h e f o l l o w i n gp o i n t s :
( i ) Exactlymidwaybetweenthe two charges.
( ii ) L0 cm to the left of the 20prC
charge
( i i ) 4 0 c m t o t h e r i g h to f t h e 6 0 p Cc h a r g e
( A n s :5 . 7 6x L 0 6 N Ct-o1w a r d st h e l e f t ;1 . 9 5x 1 0 7 N Ct-ot w a r d st h e l e f t ;
3 . 6x 1 0 6 N C - l t o w a rtdhse r i g h t )
( S ) A - 4 0 n C c h a r g e i s p l a c e d a t a d i s t aonf cl ,e2 c m a w a y f r o m a + 5 0 n C c h a r g e a s s h o w n .
1'2cm
+g
*.-
+50nC
-40nC
D e t e r m i n teh e e l e c t r i cf i e l ds t r e n g t ht o g e t h e rw i t h i t s d i r e c t i o na t t h e f o l l o w i n gp o i n t s :
(i)
(ii)
M i d w a yb e t w e e nt h e t w o c h a r g e s
5 c mt o t h e l e f t o f t h e - 4 0 n Cc h a r g e
(iii)
8 c mt o t h e r i g h to f t h e + 5 0 n Cc h a r g e
(Ans:2.25
-1
the right;6.13x l-04NC.1towards
- 1w a r d st h e l e f t ;1. 2 8 x1 0 s N C towards
x l - O s N Ct o
the right)
( 9 . I) n t h e f o l l o w i n gd i a g r a mt,h e p o i n tA r e p r e s e n t sh e r e g i o nw h e r et h e r e s u l t a nfti e l d
c h a r g e(.5 . 5 c m )
s t r e n g t hi s z e r o .D e t e r m i n teh e p o s i t i o no f p o i n tA r e l a t i v et o t h e - 2 0 p t C
+ 60;rC
-10uC
52
ELECTROSTATTCS
TUTORTAL
(2)
SHEET
( 1 ) T w o i d e n t i c am
l e t a l b a l l sh a v ec h a r g e Q
s l a n dq 2 T h er e p u l s i v e f o r ct h
e a t o n e e x e r t so n
t h e o t h e r w h e nt h e y a r e 2 0c m a p a r ti s 1 . 3 5x L 0 - 4N . l f t h e c h a r g eo n < i n eb a l ti s 3 . 2x
L 0 6 c ,d e t e r m i n et h e c h a r g eo n t h e o t h e ro n e . ( A N 1
S, . g 7 s x1 0 - 1 0 c )
( 2 ) W h a t i st h e m a g n i t u d o
e f t h e c h a r g eo n e a c ho f t w o m e t a ls p h e r e si f t h e y h a v et h e
s a m em a g n i t u d e
o f c h a r g ea n d e x e r ta f o r c eo f a t t r a c t i o no f 1 . 9 8x L 0 - 4 N
o n e a c ho t h e r
w h e n t h e y a r es e p a r a t e b
dy a d i s t a n c e
o f 2 . 5c m . ( A N S
3nc)
( 3 ) T w o c h a r g e dm e t a lp l a t e si n a v a c u u ma r e 1 5 c m a p a r t .T h ep o t e n t i adl i f f e r e n c e
b e t w e e nt h e p l a t e si s 2 0 0 V A
. n e l e c t r o n( m a s s= 9 . i -x 1 0 - 3 1 kagn dc h a r g e1 . 6x L 0 - 1 e)C
is
r e l e a s e fdr o m r e s ta t t h e s u r f a c eo f t h e n e g a t i v e
plate.
F i n d( a )T h e E l e c t r ifci e l dt h e e l e c t r o ni s s u b j e c t e tdo .
( b )T h ea c c e l e r a t i bonf t h e e l e c t r o n
( c )T h et i m e t a k e nt o r e a c ht h e o t h e rp l a t e
( d ) T h es p e e do f t h e e l e c t r o nj u s t b e f o r ei t h i t st h e p o s i t i v ep l a t e .
( A n s ( a1) . 3 3x L 0 3 V mt ( u )2 . 3 4x 1 0 1 a m s('. ) 3 . 5x 1 - 0 - 8(sd;)g . 3 g xl - o 6 m 1s )
( a ) W h a t i s t h e f o r c ee x p e r i e n c ebdy a p r o t o n( c h a r g e 1
= . 6x 1 0 - 1 e C
w1h e n i t i s i n t h e s p a c e
b e t w e e nt w o p a r a l l epl l a t e ss e p a r a t e b
dy a d i s t a n c e
o f 2 5 m ma n d h a v i n ga p . d .o f 5 0 O V
1
s
N
;
b e t w e e nt h e m (. A N S3 . 2x 1 0
( 5 ) T w o i d e n t i c at li n y m e t a lb a l l sc a r r yc h a r g e o
s f + 3 n C a n d- 1 , 2 n CT.h e ya r e 3 c ma p a r t .
C a l c u l a tteh e f o r c eo f a t t r a c t i o n(.A N S3 . 6x L 0 - 4 N )
( b )T h eb a l l sa r e n o w t o u c h e dt o g e t h e ra n dt h e n s e p a r a t etdo 3 c m .D e t e r m i n teh e f o r c e
o n e a c hb a l ln o w . (H i n t ;W h e nt h e b a l l sa r e n o w t o u c h e dt o g e t h e rt,h e y a c q u i r et h e
s a m ec h a r g ei . e . t o t acl h a r g ei s c o n s e r v e d )
( A n s ( a ) 3 . 6 x1 0 o w( U )2 . 0 2 5 x1 0 4 Nr e p u l s i o n ) .
( 6 ) T w o i d e n t i c asl p h e r e sh a v ec h a r g e q
. h e nt h e y a r e2 0 c m a p a r t ,t h e r e p u l s i v e
s 1a n d q z . W
4
N
.
f o r c eb e t w e e nt h e m i s 1 . 3 5 x1 0
A f t e rt h e ya r et o u c h e dt o g e t h e ra, n ds e p a r a t e d
.
o n c ea g a i nt o 2 0 c m ,t h e r e p u l s i vfeo r c eb e t w e e nt h e m i s 1 . 4 0 6x 1 0 - o N F
i n dQ l a n d
q z( A n s 2: 0 n C , 3 0 n CS. )E ES O L U T I OONN p G5 4 .
53
Solutionfor question3:
0 . 15 m ; E -
(t)V:200V;d:L\cm(tf.)Accelerati"on-
Force
MASS
EQ
MASS
(ttt) Use the equatton .s_ ut l|or';1r
V
d
200v
0.1-5m
(L\9: 200 x L . 6 x
\d/ m
10-1e
0.15x9.1 x10-31
: 0 hencs
e-
- 3 . 5x 1 0 - 8 s
(iv)U se ettlter v : 'tt * at or v2 : lr2 : Zas w h e r eu = 0 .
54
-7
: l-.33x 1.03Vm
1."8t :
tot";t
- 2 . 3 4x l g t a ^ r - z
JT
-
x 1014
Solution
for question6:
- aflz l-9 -\ 1'0e- 1.35x 10-4A/
Initialty,g
'' 4ft€sr2
(o.z)z
H e n c e Q t Q z- 6 x
-(1)
1g-togz
W h e n t h e s p h e r e sa r e t o u c h e dt o g e t h e rt h e c h a r g e sa r e r e - d i s t r i b u t e db u t t h e t o t a l c h a r g ei s
c o n s t a n ta n d e q u a lt o q r + Q z
A f t e r t h e s p h e r e sa r e s e p a r a t e d e, a c h o n e w i l l a c q u i r ea c h a r g eo f
Qr*Qz
2'
(Since
t h e s p h e r e sa r e i d e n t i c a tl h
, e r ei s n o t e n d e n c yf o r o n et o g a i na g r e a t e rc h a r g et h a nt h e
other.)
T h ef o r c ea c t i n go n e a c hc h a r g en o w i s g i v e nb y
(ry)'rexloe
(0.2)'
- 1 . 4 0 6x 1 0 - a
( q r + Q ) 2 - 2 . 5 x 1 0 - 1 s ; l t e n c e( q r + e z ) = 5 x 1 0 - B C
-Q)
F r o me q u a t i o n( 2 ) Q r =5 x l - 0 - 8g-z . S u b s t i t u t i n g t hi n
i se q u a t i o n( 1 )g i v e s
(5xL0-8 -a)Qz:6x10-16
Q z z- ( 5 x 1 0 - B )e z * ( 6 x t o - t o ) - 0
cives;
S o l v i n tgh i sq u a d r a t i g
5x1o-B+(
)
Qz:
-3xl-0-BC:30nC;
l f t h i sv a l u eo f q z i s s u b s t i t u t e idn e i t h e re q u a t i o n( 1 )o r ( 2 ) ,q 1w o u l db e f o u n dt o b e 2 0 n C .
55
Tutorialsheet(3)
Electrostatics
( T h ec h a r g eo f a n e l e c t r o ni s - 1 . 6 x1 0 - 1 e C )
o f 5 m m ,t h e p . d .
( 1 ) F o rt w o c h a r g e dm e t a lp l a t e sA a n d B ,s e p a r a t e bd y a d i s t a n c e
p
o
t e n t i aal n d p l a t eB t o a
p
o
s
i
t
i
v
e
b e t w e e nt h e p l a t e si s 4 0 V .P l a t eA i s c o n n e c t e tdo a
. a l c u l a tteh e w o r k d o n ei n m o v i n ga + 3 Cc h a r g ef r o m ( a )B t o A , ( b )
n e g a t i v ep o t e n t i a l C
A to B. | 120J; -120J1
a fl a t 2 V b a t t e r y
( 2 ) H o w m u c hw o r k i s d o n e t o c a r r ya n e l e c t r o nf r o m t h e p o s i t i v e t e r m i n o
1
.
6
x
(
A
n
s 1 . 9 x1 0 - 1 8 J 1
1
0
1
s
C
.
t o t h e n e g a t i v tee r m i n a l .T h ec h a r g eo f t h e e l e c t r o ni s
( 3 ) A n e l e c t r o ns t a r t sf r o m r e s ta n df a l l st h r o u g ha p o t e n t i arl i s eo f 8 0 V .C a l c u l a tiet sf i n a l
1 ) { H i n tt :h e c h a n g ei n
. . 3 x1 0 6 m s
x g - 3 1(kAsn s 5
s p e e dT. h e m a s so f t h e e l e c t r o ni s 9 . 1 - 1
P . E=. c h a n g ei n K . E . )
l e t a lp l a t e sa r es e p a r a t e bdy a d i s t a n c eo f 4 . 5 f f i f f i ,a n d a p . d o f 1 2 0 0 V i s
( 4 ) T w o p a r a l l em
-1.6x 10-1sc)
a p p l i e da c r o s tsh e r i r .( m a s so f e l e c t r o n= 9 . 1 - x1 . 0 - 3 1 kcgh, a r g e=
(i)
(ii)
(iii)
(iv)
(v)
W h a t i s t h e v a l u eo f t h e e l e c t r i cf i e l db e t w e e nt h e p l a t e s ?
l f a n e l e c t r o nt h a t i s h e l da t r e s ta t t h e n e g a t i v ep l a t ei s s u d d e n l rye l e a s e dw, h a t
o f t h e e l e c t r o n 'as c c e l e r a t i o n ?
w o u l db e t h e m a g n i t u d e
W h a tw o u l db e t h e c h a n g ei n t h e e l e c t r o n ' ps o t e n t i ael n e r g y ?
W h a tw i l l b e t h e e l e c t r o n 'ssp e e da s i t h i t st h e p o s i t i v ep l a t e ?
H o w l o n gw o u l dt h e e l e c t r o nh a v et a k e nt o c r o s sf o r m o n e p l a t et o t h e o t h e r ?
1;
; 8 x1 0 - 1 0 s .
( A n s :z . O l x l g s v m 4 . 6 8 x1 0 1 6 m s; I-.29 2 x 1 0 - 1 62J.;O Sx L O T m s -41. 3
a p a r t .W h e nt h e y a r e r e l e a s e d ,
( 5 ) T w o p r o t o n sa r e h e l da t r e s ta t a s e p a r a t i oonf 3 x 1 0 - 1 2 m
t h e y r e p e le a c ho t h e ra n d m o v ea p a r t .H o wf a s tw i l l t h e y b e m o v i n gw h e nt h e y a r e
( i )8 m m a p a r t
( i i )a p m a p a r t ,
(iii)5pa
mp a r t .
27kg'
= 1.6x101eC;
mass= !.67x10
Protoncharge
s e p a r a t i oann dt h e n a t t h e 8 m m
H i n t y: o un e e dt o f i n dt h ep o t e n t i af ilr s t l ya t t h e3 x L 0 - 1 2 m
ee t w e e n
e f t h e s et w o p o t e n t i a liss t h e p o t e n t i adl i f f e r e n c b
s e p a r a t i o nT.h ed i f f e r e n c o
Vn) d e q u a t et h i st o K . E >t o f i n d
t h e s et w o p o i n t .T h e nf i n d t h e c h a n g ei n P . E . ( W = Q a
speed.
r;
( A n s 2 . I 4 x 1 0 s m s 1 ' . 0 x7 l O s m s - 1 ; 1 . 3 5l -60xs m s - 1 )
56
( 6 ) T w o p o i n t sA a n d B a r e a t L 0 0 c ma p a r t .A c h a r g eq r = 2 0 0 p ci s p l a c e da t A w h i l e a
c h a r g eQ z =- 1 0 0 p Ci s p l a c e da t B . P o i n tC i s o n t h e l i n eA B a n d i s 2 0 c m a w a yf r o m B a n d
p o i n t D i s a l s oo. n A B a n d i s 2 0 c m a w a yf r o m A . ( a )F i n dt h e p o t e n t i a l as t C a n d a t D . ( b )
How muchwork must be doneto transfera -5001.rC
chargefrom C to D
( A n s .-:2 . 2 5 V; 7 . 8 7 5 V ;- 5 . 1 xl - 0 - 3 J )
Qr=2ooPC
Q z =- 1 0 O P C
2Ocm
20cm
_____+B
c<#*r
nrr-D
^\J-
100cm
(7) Fourpointchargesare placedat the four cornersof a squarethat is 30 cm on eachside.
Findthe potentialat the centreof the squareif eachchargeis 2pC.
(Ans.:3.4x10sV)
(8) Charges
of +21tC,+3pC and -SpCare placedat the verticesof an equilateraltriangle
of
-8pC
side10cm. Calculate
the magnitudeof the forceactingon the
chargedue to the
othertwo charges.)
Hint Drawa vectordiagramand resolvethe forcesverticallyand
horizontally,
then find the total verticaland the total horizontalforces.
Afterthis,use
Pythagoras'Theorem.
) (Ans31N)
( 9 ) F o u re q u a lp o i n tc h a r g e +
a r e p l a c e da t t h e f o u r c o r n e r so f a s q u a r et h a t i s 4 0 t m o n
s3prC
: .97N
a s i d e .F i n dt h e f o r c eo n a n yo n e o f t h e c h a r g e ds u et o t h e o t h e rt h r e e .( A n s 0
o u t w a r da l o n ga d i a g o n a H
l ) i n tu s et h e s a m et e c h n i q u a
e s i n t h e p r e v i o uqs u e s t i o n
57
l f t h e + 3 Cc h a r g eh a st o b e m o v e df r o m B t o A , i t m u s tb e
p u s h e da g a i n stth e f o r c eo f r e p u l s i o n
f r o m t h e p o s i t i v ec h a r g e
f o r q u e s t i o n1 :
Solution
o n p l a t eA . T h i sm e a n st h e nt h a t w o r k m u s tb e d o n eO Nt h e
+++++++++++++++
c h a r g ea s i t i s m o v e df r o m B t o A .
5mm
R e c a ltlh a t w o r k d o n e= Q V
W = 3 Cx 4 0 V = 1 2 0 J .
l f t h e c h a r g el s m o v e df r o m A t o B ,t h e n n o e n e r g yn e e d st o b e
a d d e dt o t h e c h a r g eI.n t h i ss i t u a t i o nt,h e c h a r g el o s e si t s o w n
E l e c t r i c aplo t e n t i ael n e r g ya s i t i s a t t r a c t e dt o p l a t eB .T h i sl o s s
o f P Ei s g i v e na n e g a t i v es i g ns o t h e w o r k d o n ef r o m A t o B i s t20J.
Solutionfor question2: UseW =QV
Solutionfor question4:
(i)
E - \ :d
(ii)
F o rc eac c el e ra tto n -
(iii)
A P E =Q V =1 . 6x 1 0
(iv)
-!^u';v
T h el o s si n P E =i n c r e a sien K E ;H e n c e1 , . 9 2 x 1 0 - 1 6
(v)
UsetheequationofmotionV=u*dt,t-,+-ffi=4'38x].0-1os
,
#
4 .-5:x 1 0 -
2 . 6 7x 1 0 sV m - 1
Eq -
vq
d xmass
16J
lex
1 2 0 0 =I . 9 2 x L 0
58
L2aoxL.6x1o-1e
4 . Sx 1 0 - 3 x 9 . 1 x 1 0 - 1 e
: 4 . 6 8x ! 0 1 6 m s - 2
: 2 . 0 5 x 1 0 7 m s1
S o l u t i o nf o r q u e s t i o n5 :
a p a r t ,t h e
I n t h i sq u e s t i o nu s et h e f a c tt h a t t h e l o s so f P E= G a i no f K E .W h e nt h e p r o t o n sa r e 3 x 1 0 - 1 2 m
i s g i v e n0 V , f f i . I n t h i s c d s eQ 1= Q 2 =1 . 6x 1 0 - 1 s C .
P o t e n t i ael n e r g yo f t h e S Y S T E M
=
HencePEof system
(t u'11 'llx(r'axro-'n)
-: -i ' 6 8 x 1 ' 0 - 1 7 J
+n'o*z'to-
n f 8 m m ,t h e n e w PEof the systemis givenby
W h e nt h e c h a r g e sm o v eo u t w a r d st o a s e p a r a t i o o
( 1 . 6x 1 0 - 1 e )x ( 1 . 6x 1 0 - 1 e )
= Z . B Bx 1 , 0 - 2 6 J
4nesxBx10-3
PE2 -
N o t et h a t t h i sn e w P Ei s s e v e r am
l i l l i o nt i m e sl e s st h a nt h e o r i g i n aPl Ea n d h e n c et h e c h a n g ei n P Ew h i c hc a n
17J
7.68x LO
be lvrittenas Af E=PE1-PE2=
F r o mt h e c o n s e r v a t i oonf e n e r g yt h e l o s so f P E= G a i no f K E , b u t t h i st o t a l K Ei s s h a r e de q u a l l ya m o n gt h e
l a s sa n d c h a r g e .
t w o p r o t o n ss i n c et h e y h a v ei d e n t i c am
S ow e c a nw r i t e , ) m v ' + ) m u 2 = 7 . 6 8 x t 0 - 1 7J o r m v 2 : 7 . 6 8 x 1 0 - 1 7
7.68 x L0-17
7.68 x 1,0-17
a
,r2 -
L . 6 7 x ! 0 - 2 7k g
: 4 . 5 8x 1 0 1 o
4 . 5 8x 1 0 1 0: 2 . ! 4 x 1 0 s m s - 1
T h es a m em e t h o di s e m p l o y e df o r p a r t s( i i )a n d ( i i i )
S o l u t i o nf o r q u e s t i o n6 :
Q z =- 1 0 0 P C
Qr=200PC 2ocm
nA{#n
ALJj--
20cm
C
n
n
B
100cm
F i n d i n tgh e p o t e n t i aal t D :
'Firstly
f i n d t h e p o t e n t i aal t D u et o t h e c h a r g ea t A O N L Y :
t/-h'
200x1.0-12x9x1-0e
0.2
4tegr1
:9V
The Potentiaa
l t D d u e t o t h e c h a r g ea t B o n l y :
vz:
Qz
-100x1,0-12x9x10e
: -1,.725V
4rcesr2
0.8
T h ep o t e n t i aal t D i s : h e nf o u n db y a d d i n gV 1a n dV 2; V1 *V2= 9 V + ( - 1 . 1 2 5 V )7=. 8 7 5 V
59
T h ep o t e n t i aal t C i s f o u n di n a s i m i l aw
r a y ,i . e .
( P o t e n t i aal t C d u e t o t h e 2 0 0 p Cc h a r g eo n l y) + ( P o t e n t i a l aCt d u et o - 1 0 0 p Cc h a r g eo n l y ) .
2 0 0 x L 0 - 1 2x 9 x 1 0 e
I
-100 x!0-12 x9 x 10e
0.8
02
2.25V
( i i i )l n t r a n s f e r r i nag - 5 0 0; r Cc h a r g ef r o m C t o D , i t i s n e c e s s a rt yo f i r s tf i n dt h e P O T E N T I A L
differencb
e e t w e e nC a n d D ,
V D- V c : 7 . 8 7 5 V- ( - 2 , 2 5 V ) : 1 0 . 1 2 5 V
T h ew o r k d o n e ' w h e na c h a r g eQ m o v e st h r o u g ha p o t e n t i adl i f f e r e n c e
V i s g i v e nb y W = Q V ;
.
H e n c eW - - 5 0 0 x 1 0 b x ( 1 0 . 1 2 5 V ) =5-. 0 6x 1 0 3 JT
h e n e g a t i v es i g ni n d i c a t etsh a t t h e - 5 0 0 p Cc h a r g el o s e s
P Ea s i t g o e sf r o m a n e g a t i v ep o t e n t i atlo a p o s i t i v ep o t e n t i a
Solutionfor Question7:
30cm
A t t h e c e n t r eo f t h e s q u a r ei,t i s n e c e s s a rtyo f i n d t h e d i s t a n c e
b e t w e e na n yo n e o f t h e c h a r g ea n dt h e c e n t r ep o i n t .
1
U s i n gP y t h a g o r a s ' T h e o r etm
h i,s d i s t a n c ei s f r 5 2 + 1 5 2 -
^,l4so- 2r.zcm.
J
T h e p o t e n t i adl u e t o E A C Hc h a r g ei s t h e n f o u n df r o m
- 2x10--6-x9xloe =8.5 xro4v
rQ'
4rc€sr1
0.212
T h e p o t e n t i aol f a l l f o u r c h a r g e sw o u l dt h e n b e 4 x 8 . 5 x 1 0 4 Vv 'j -
30cm
| 3 . 4 vx l o s Z
60
CAPACITORS
A c a p a c i t oirs a d e v i c et h a t i s u s e dt o s t o r ee l e c t r i c h a r g eT. h i ss t o r e dc h a r g e
c a nt h e n b e u s e d
w h e na r e l a t i v e llya r g eb u r s to f c h a r g ei s r e q u i r e de . g . t o s t a r ta m o t o r . t, o a c t i v a t e
a camera
flasho
, r t o s w i t c ho n a f l u o r e s c e nt u
t be.
T h es t o r e dc h a r g ec a na l s ob e a l l o w e dt o b e r e l e a s e sdl o w l y t h r o u g ah r e s i s t o r
in a smoothing
c i r c u i ta, n dt h i s h a sp r a c t i c aal p p l i c a t i o ni n
s c o n v e r t i n ag r e c t i f i e da l t e r n a t i ncgu r r e n ti n t oa n
a l m o s tc o n s t a ndt i r e c tc u r r e n t .
C a p a c i t o rcso n s i sot f t w o m e t a ls h e e t ss e p a r a t e bd y a n i n s u l a t o rT. h ei n s u l a t o r
i s c a l l e dt h e
DIELECTRIC.
T h ew h o l ea r r a n g e m e notf t h e m e t a ls h e e t sw i t h t h e d i e l e c t r iicn b e t w e e n ,i s
t h e n r o l l e di n t o a
s m a l tl i g h tc y l i n d ear n ds e a l e di n a c a s ew i t h t w o t e r m i n a les x t e n d i nogu t .
T h eC A P A C I T A N C
o fEa c a p a c i t oirs g i v e nb y t h e e q u a t i o nc _
3
W h e r eQ i s t h e c h a r g es t o r 6 da n dV i s t h e p o t e n t i adl i f f e r e n c b
e e t w e e nt h e p l a t e s .
Capacitance
is definedas the chargerequiredto causeunit changein potentialdifference
betweenthe plates.
T h eu n i t o f c a p a c i t a n ci set h e F a r a d( F ).
T h e F a r a di s d e f i n e da st h e c a p a c i t a n coef a c a p a c i t oirf a c h a r g eo f 1 Cc a u s e s
a c h a n g eo f 1 V
b e t w e e nt h e p l a t e s .
T h ef a r a di s a n e x t r e m e l yl a r g ec a p a c i t a n caen, dt y p i c a cl a p a c i t o risn h o u s e h o l d
a p p l i a n c easr e
r a t e di n m i l l i - f a r a d(sm F )o r m i c r o - f a r a c i s ( p F ) .
6I
CAPACITANCE:
AFFECTING
FACTORS
d the
F o rt w o p a r a l l epl l a t e sh a v i n ga n a r e ao f o v e r l a pA a n d s e p a r a t e bd y a d i s t a n c e
c a p a c i t a n coef t h e p a r a l l epl l a t ec a p a c i t oirs g i v e nb y : C : 4
d
Wheree is the permitivityof the medium,ORdielectricconstantof the medium.
lf the spacebetweenthe platesis a vacuumthen we write : C : I A
l yh e nt h e s p a c eb e t w e e nt h e p l a t e si s f i l l e dw i t h
T h ec a p a c i t a n ci n
e c r e a s essi g n i f i c a n tw
i n s u l a t i nm
g a t e r i asl u c ha s p o l y s t y r e noer g l a s so r p o l y t h e n e .
is determinedby 3 factorsi.e
Hencethe maghitudeof the Capacitance
( i ) T h ed i e l e c t r icco n s t a not f t h e m a t e r i abl e t w e e nt h e p l a t e s ,
( i i ) T h eA r e ao f o v e r l a po f t h e p l a t e sa n d
(iii)The
o f s e p a r a t i oonf t h e p l a t e s .
distance
62
DIELECTRICS
RelativePermitivity:
T h i si s t h e r a t i oo f t h e c a p a c i t a n cwei t h a n dw i t h o u t t h ed i e l e c t r ibc e t w e e n
the plates.
T h e r e l a t i v ep e r m i t t i v i t iys d e n o t e db y t h e s y m b o le , . a n di s g i v e nb y t h e
t -' :
or
tr:;
equation:
Cap aci"tance w tth die lectr ic
Capaci"tancewith vacuum
ca
D i e l e c t r i ccsa na l s ob e c l a s s i f i eadc c o r d i ntgo t h e i r D I E L E C T Rs ITcR E N G T H S .
The DIELECTRIC
STRENGTH
of any materialis the electricfieldstrengthor potentialgradientat
w h i c hi t s i n s u l a t i o bn r e a k sd o w n a n d i t b e g i n st o c o n d u c b
t y a l l o w i n gs p a r k st o t r a v e lt h r o u g h
it.
T h ef o l l o w i n gt a b l eg i v e s6 o m et y p i c a dl i e l e c t r i ct so g e t h e rw i t h t h e i r
c o r r e s p o n d i nr g
elative
p e r m i t i v i t i easn d d i e l e c t r isct r e n g t h sw
, h i c hy o u M U S TL E A R N .
MATERIAL
R E L A T I VPEE R M I T I V I T DIELECTRIC
Y
STRENGTH
( n ou n i t s )
Air
ParaffinWax
Gl a s s
M ica
( K i l o v o l tpse r m m )
1.005
2
15-50
5-10
30-150
6
80- 200
Water
81
lce
94
63
lmportanceof the Dielectric
T h eD I E L E C T RhIaCst h r e ev e r yi m p o r t a nftu n c t i o n sT. h e ya r e :
(1) lt servesto keepthe platesat a smallconstantfixed distanceapart.
(2) lt allowsthe capacitorto acquirea much highervoltagewhilst reducingthe possibility
o f s p a r k i n gl.f t h e r ew a s n o d i e l e c t r ipc r e s e n tt,h e a i r b e t w e e nt h e p l a t e sw o u l db e c o m e
i o n i z e dw h e n a s u f f i c i e n t lhyi g hv o l t a g ei s a p p l i e da, n d s p a r k sw o u l db e p r o d u c e da s
o m t h e a i r m o l e c u l etso t h e p o s i t i v e lcyh a r g e dp l a t e sa n d p o s i t i v e
e l e c t r o n s . m i g r af rt e
i o n sm i g r a t et o t h e n e g a t i v e lcyh a r g e dp l a t e .A t t h i s p o i n t ,t h e a i r i s s a i dt o b e
a n dt h e v o l t a g ea t w h i c ht h i so c c u r si,s c a l l e dt h e b r e a k d o w n
e x p e r i e n c i nbgr e a k d o w n
voltage.
D i e l e c t r i ccsa nw i t h s t a n dm u c hh i g h e re l e c t r i cf i e l d st h a n a i r b e f o r ea n yc o n d u c t i o cna n
t a k ep l a c eb e t w e e nt h e p l a t e si.. e .t h e d i e l e c t r i chsa v ea m u c hh i g h e rb r e a k d o w n
v o l t a g et h a nt h a t o f a i r .
( 3 ) T h e d i e l e c t r i icn c r e a s etsh e c a p a c i t a n coef t h e c a p a c i t o rA. n y a i r f i l l e d . c a p a c i two irl l
r i m e n s i o nasn d
e a t i s m a n yt i m e sl o w e rt h a na n o t h e ro n e o f s i m i l a d
h a v ea c a p a c i t a n ct h
m a t e r i a ltsh a t h a sa d i e l e c t r ibc e t w e e nt h e p l a t e sT. h em o l e c u l eisn t h e d i e l e c t r i c
m a t e r i aal c q u i r ei n d u c e dc h a r g eds u et o t h e c h a r g eosn t h e p l a t e s .
c o l e c u l e as c q u i r ea p o s i t i v ec h a r g eo n
S i n c et h e d i e l e c t r i m
o n e e n d a n d a n e g a t i v ec h a r g eo n t h e o t h e re n d ,i t i s s a i dt o
bepolarized.
f e d i e l e c t r i ca,d j a c e ntto t h e
T h ec h a r g e so n t h e s u r f a c e . ot h
p l a t e sa r e o f o p p o s i t es i g nt o t h o s eo n t h e p l a t e sa, n d a s a
h e p o t e n t i adl i f f e r e n c eb e t w e e nt h e
r e s u l t h e y R E D U CtE
plates.
F r o mt h e f o r m u l a :
c- ?,"decreaseinV would causean tncreasetn C
( S i n c et h e t o t a l c h a r g eQ i s c o n s e r v e d )
64
Capacitors
in Series:
Vr
V2
-olo
V3
-lo
C2
V1
A c h a r g eo f - Q l e a v e st h e n e g a t i v tee r m i n a ol f t h e b a t t e r ya n d a c h a r g eo f + q l e a v e st h e
p o s i t i v et e r m i n a l H
. e n c em a t h e m a t i c a ltlhye n e t c h a r g el e a v i n g
t h e b a t t e r yi s z e r o ,a n d s i n c e
t h e c h a r g eo
s n t h e p l a t e so f t h e c a p a c i t oar r e+ Q a n d- e , t h e n e t c h a r g eo n e a c hc a p a c i t oirs
zero.
T h i si n d i c a t etsh a t c h a r g e ' icso n s e r v esdi n c eT O T A LC H A R GlE
eaving
t h e b a t t e r y= T O T A L
C H A R GaEc q u i r e b
d yt h ec a p a c i t o r s .
T h ea c t i o no f t h e c a p a c i t oirs t h e n r e a l l yt o s e p a r a t teh e n e g a t i v e
c h a r g e fsr o m t h e p o s i t i v e
ones.
F o r c o n v e n i e n ci ne c a l c u l a t i o nt sh ,e c h a r g eo n o n e p l a t ei s c o n s i d e r e id. .e .l f a c h a r g eo f + e
a c c u m u l a t eosn t h e p o s i t i v ep l a t ea n da c h a r g eo f - Q a c c u m u l a t eos n t h e n e g a t i v ep l a t e t, h e n
i n t h e c a l c u l a t i oanc h a r g eo f Q i s u s e d T
. h es i g no f t h e c h a r g ei s n o t i m p o r t a n t .
H e n c ef o r t h e s e r i e sc i r c u i ta b o v et,h e c h a r g eo n e a c hc a p a c i t oirs Q a n d t h e t o t a lc h a r g e
leaving
t h e b a t t e r yi s a l s oQ .
l f V 1 , V z , d n d V 3 r e p r e s e n t h e v o l t a g e sa c r o s sC t , C z a n dC 3r e s p e c t i v e l yt ,h e n b y t h e
Vr - v1 + Vz + V:
c o N S E R V A T I OO
N F E NE R G y ,
But
Ilence Vr
::
n
Y
L7
-
v-
0
C
a
a,Q,
C7
tl
tl
C2
C.
_ J
i f w e d i v i d et h e i a t : e re q u a t i o nb y q t h e n w e g e t :
65
11,,1_,L
C7
Cr'Cr'C=
Capacitorst n P a r a l l e l :
V
of charge,the total chargeQ 1 leavingthe battery,is equalto the sum of
Bythe conservation
t h e c h a r g ea
s c c u m u l a t i nogn t h e c a p a c i t o r s .
Therefore,Qr = Qr +Q2+Q3l
of energy,the voltagesacross
loop rulewhichis a statementof the conservation
By Kirchoff's
e a c hc a p a c i t oirs t h e s a m ea n d i s e q u a lt o t h e b a t t e r yv o l t a g e .
S i n c eQ = ! V , t h e n Q r = C r V ; Q z= C z V ; a n dQ : = C i V
Q, =CrV=C1V+CzV+CgV
W h e n3 c a p a c i t o rasr ec o n n e c t e idn P a r a l l etlh, e t o t a l
is givenby Cr= Cr+ C,+ C,
capacitance
66
ENERGY
STORED
IN A CAPACITOR
Energycanbe storedin a capacitorvia a batteryin the f o l l o w i ncgi r c u i t :
I n t h e g r a p ho f V v e r s u sQ , t h e
a r e au n d e rt h e g r a p h
represents
the energystored
in the capacitor
a
W h e nt h e c a p a c i t ohr a si n i t i a l l yz e r oc h a r g eo n i t , t h e p o t e n t i adl i f t e r e n c b
e e t w e e np o i n t s A
a n d B i s V . i . e .t h e p . d .b e t w e e nt h e p o s i t i v ep o l eo f t h e b a t t e r ya n dt h e u n c h a r g elde f t p l a t ei s
V.
Hencethe chargesINITIALLY
passthrougha potentialdifferenceof V volts.
A s t h e c a p a c i t oar c q u i r e s . n . r r " ,t h e p o t e n t i aol n t h e l e f t p l a t er i s e sf r o m 0 +
Vil
Vz+
Vs
etc.
The potentialdifferencethat the chargepassesthrough will therrefore
DECREASE
from
-------0
V + (V -vr)---+ (V - Vz), +
etc.untilthe pointwherethe batteryvoltageis
e q u a lt o t h e c a p a c i t ovro l t a g ew
, h i c hc a u s e tsh e p . d .b e t w e e nt h e b a t t e r ya n d c a p a c i t otro b e
zero.
Duringthe entirechargingprocess,
the potentialdifferencethat the chargepasses
through,
would havechangedfrom V to zero.Hencethe AVERAGE
potentialdifferencethat the total
c h a r g eQ p a s s etsh r o u g ,hr t
T
-f,
T h ew o r k d o n ei n m o v i n ga c h a r g eQ t h r o u g ha n a v e r a g ep o t e n t i adl i f f e r e n c o
e f Y is givenby
2
W:V 2
T h ew o r k d o n ei n m o v i n gt h e c h a r g ei s e q u a lt o t h e e n e r g ys t o r e di n t h e c a p a c i t o r
S u b s t i t u t i nC
g =
o
|
givesthree usefulequationsfor energystoredin the capacitor;
- ---Q2
w- QV
22C2
Y o um u s tl e a r nt h e s ee q u a t i o n s ! ! ! ! !
67
CVz
C a p a c i t oQr u e s t i o n s
( 1 ) F i n dt h e e q u i v a l e ncta p a c i t a n ci n
e e a c hc a s e :
(i)
r-IHH
471tF
A n s7 . 6 2 p r F
100prF 10;tF
100uF
(ii)
108.25;rF
:f
10uF
47p,F
80prF
]H
20prF
A n s1 1 6 U F
100prF
80prF
Ans.200;rF
( 2 )T h et e r m i n a l sa t A a n d B a r e n o w c o n n e c t e tdo a 2 0 Vs o u r c ei n e a c hc a s e R
. e d r a we a c h
c a p a c i t o(rb )T h e
c i r c u i t o i n c l rJ e t h e p o w e rs u p p l ya, n df i n d ( a ) T h ep . d .a c r o s sE A C H
c h a r g es t o r e di n E A C Hc a p a c i t o ra n d ( c )T h eenergystoredin EACHcapacitor.
68
20v
(1)
V + z: 3 . 2 4 V ; V r c o
: 1 , . 5 2 V ; V r-c I S . Z 4 V
Q + z: Q r c : 0 r o o - I . S 2 x 1 , 0 - 4 C
E+z=2.45x1.0-4J
E r . o o_ 1 , 1 5x 1 0 - 4 J
471tF 1009F 10pF
El,o:1'15x10-3J
100sF
V r c o: 2 0 V ; V t o= 1 , 6 . 5 V ; V +: z 3 . S V
(2)
Q r c o_ 2 x 1 , 0 - 3 c
Q+z=Qrc:1.65x1'0-4C
Erco_ 0.02J
E + z: 2 . 9 x L 0 - 4 J
20v
80uF
E t : 1 . 3 6x 1 0 - 2 J
201tF
Veo: 4V; V2s- 16V; Vlss : 20V
Q e o: Q z o: 3 . 2 x i . 0 - 4 C
Q r o o_ 2 x . 1 , 0 - 3 C
100pF
E e o= 6 . 4 x 1 0 - 4 J
Ezo:2.56x1,0-3J
Erco_Z.0xI0-2J
20v
69
A CAPACITOR.
AND DISCHARGING
CHARGING
it acrossa resistoras shown:
d connecting
A c a p a c i t oirs n o r m a l l yd i s c h a r g eby
C
R
d
e c r e a s ewsi t h t i m e ,a n d h e n c et h e
c
h
a
r
g
e
s
t
o
r
e
d
R
a
c
r
o
s
s
d
i
s
c
h
a
r
g
e
s
,its
A st h e c a p a c i t o r
e i t ht i m e .
t i l l a l s od e c r e a sw
o f t h e c u r r e n ti n t h e c i r c u i w
magnitude
and
of currentdependson the magnitudeof the productof resistance
The rateof decrease
c a p a c i t a n c ie. e
, o n t h e v a l u eo f R C .
T h eq u a n t i t yR Ci s c a l l e dt h e T I M EC O N S T A N Ta n d h a su n i t so f s e c o n d s .
Hence 10 x 1F= lsecond .
=
D u r i n gt h e d i s c h a r g et h, e c u r r e n th a st h e m a x i m u mv a l u el oa t t i m e 0 .
"/nc
o f t h e c u r r e n ta t a n yt i m e t , i s g i v e nb y / = l o €
T h em a g n i t u d e
-ft
S i m i l a r lfyo r c h a r g eQ a t a n Yt i m e t ;
Q - Qse-tl*c
Thevoltagea t a n y t i m e t d u r i n gd i s c h a r g ei s g i v e n b y :
V : Vor-tlnc
following
Thegraphsof eithervoltagev . st i l T € i ,c u r r e n vt . s .t i m e o r c h a r g ev . s .t i m e h a v et h e
shapefor DISCHARGE
"/nc
I : Ioe
-ft
(V,l, or Q)
-/nc
-ft
Q - Qoe
-/nc
V - Voe
-ft
time
70
CHARGtNG
OFA CAPACTTOE
W h e na c a p a c i t oirs b e i n gC H A R G E D
t h e V O L T A Ga
En d C H A R G E
i n c r e a sw
e i t h t i m e u n t i lt h e y
r e a c ha m a x i m u mv a l u e T
. h ec u r r e n th o w e v e rd, e c r e a s ewsi t h t i m e a n d h a st h e s a m es h a p ef o r
discharging.
Voltage
c ur r e n t
charge
time
-fr
I : Ioe "/RC
Q - Q o( 1 - e - t / n c )
V - Vo(I - ,-t /nc)
N o t et h e d i f f e r e n c eisn e q u a t i o n fso r
charging
a n dd i s c h a r g i n T
g .h eV o , Q o
a n d l ev a l u e sr e p r e s e nt th e m a x i m u m
v a l u e si n t h i sc a s e .
time
7t
C O M P A R I S OONFC H A R G I NAGN D D I S C H A R G I N G
c a . s e:s
C o n s i d et rw o s e p a r a t e
t o a v o l t a g eo f 2 0 V a n dt h e n m a d et o
( 1 ) A c a p a c i t ohr a v i n ga c a p a c i t a n coef 2 5 p r iFs c h a r g e d
d i s c h a r gteh r o u g ha 5 0 k Or e s i s t o rD. e t e r m i n teh e t i m e t a k e nf o r t h e p . d .a c r o s tsh e c a p a c i t o r t o
f al l t o 5 V .
a n dt h e p . d .a c r o s ist i s 5 V .l t i s
( 2 )A c a p a c i t ohra v i n ga c a p a c i t a n coef 2 5 ; r Fi s p a r t i a l lcyh a r g e d
t h e n c h a r g e dt h r o u g ha 5 0 k Or e s i s t otro a p . d .o f 2 0 V .D e t e r m i n teh e t i m e t a k e nf o r t h i s
process.
charging
l n t h e f i r s tc a s e
tlnc,
V - Voe
-t
/ 51
v
v
-RC
:1..73s
(-s0000
x
25x
10-t)(r"^)
r"uot
t"% -#
;
l n t h e s e c o n dc a s e ,
tlnc)
V - Vo(I - e
V'
-r,
Vo
/
5t
tn(r-fi)
-
/
7\
\
vo/
-tr
/
I/l
-t
\
Vo/
RC
-t
soooffi;
r : 5 0 0 00 x 2 5 x 1 0 - 6x l n ( 0. 7 5 ) : 0 . 3 6 s
from20Vto 5Vis
for discharging
thatthe time required
Th" resultsindicate
throughthe samepotentialdifference.
totallydifferentfromthat for charging
72
Significance
of the TimeConstant
R e c a lt lh a t f o r D I S C H A R G I N
t hGe,V o l t a g ea t a n yi n s t a n it s g i v e nb y :
V : V o r - t l * r , w h e r eR Ci s t h e T i m eC o n s t a n t
l f w e c o n s i d ear t i m e t w h e r et = R Ct,h e nt h e e q u a t i o nb e c o m e s :
V - V o e "- 1
' o, r v - - V o
ev
Thisequationtellsus that the TimeConstantis the time for whichthe potential
differenceacrossthe capacitorfallsto 1of its initialvalueduringa discharge.
e
11,
T h e u a l u eo f e - 2 . 7 1 8 , h e n c e 0.368
;
LTIBHencewhen t : CR, V : 0.36BVo
A f t e ra p p r o x i m a t e 3
l yt i m e c o n s t a n t st h, e c a p a c i troi s o l m o sct o m p l e t e ldy i s c h a r g e d .
T h i sc a nb e d e m o n s t r a t eadsf o l l o w sw h e nt = 3 C R
V - V o e - = ; V : 0 . 0 5 7 0w h i c h i . sc l o s et o z e r o .
S i m i l a r lfyo r t h e c h a r g i npgr o c e s s
V -Vo(I -e
tlnc);wheftt:3RC,
V -VoI-e-3);
H e n c ea f t e r3 t i m e c o n s t a n ttsh e c a p a c i t oirs a l m o s tf u l l yc h a r g e d .
73
V - 0.9SI/0
SHEET
TUTORIAL
DISCHARGE
CAPACITOR
t o 6 V a n dt h e n m a d et o d i s c h a r gaec r o s sa L 2 0 k Qr e s i s t o r .
( 1 ) A 5 0 p Fc a p a c i t oirs c h a r g e d
C a l c u l a t:e
( i )T h et i m e c o n s t a not f t h e d i s c h a r gcei r c u i t
( i i ) T h et i m e st h a t w o u l dh a v ee l a p s e fdo r t h e p . d .a c r o s tsh e c a p a c i t o r t foa l lt o 3 V a n d 1 V
( i i i )T h ec h a r g et h a t w o u l dh a v ef l o w e dt h r o u g ht h e r e s i s t oar f t e r5 s e c o n d s
( i v )T h ee n e r g yd i s s i p a t ebdy t h e r e s i s t oar f t e r2 s e c o n d s . ( H i n t ; f itnhde i n i t i ael n e r g ya n d
, e n f i n dt h e d i f f e r e n c e )
t h e n t h e e n e r g ya f t e r2 s e c o n d st h
4t.)
t.7 x 10-4C
(Ans:6s;4.1,6s;1'0.75s;
; 4.4 x tO
( 2 ) A 1 0 O 0 p cFa p a c i t oirs c h a r g e dt o 1 2 Va n dt h e n m a d et o d i s c h a r gaec r o s sa L . 5 M O
resistor.
C a l c u l a t:e
( i )T h et i m e c o n s t a not f t h e d i s c h a r gcei r c u i t
f o r t h e p . d .a c r o s st h e c a p a c i t otro f a l lt o 9 V ,6 V
( i i )T h et i m e st h a t w o u l dh a v ee l a p s e d
'
and l-V
( i i i )T h ec h a r g et h a t w o u l dh a v ef l o w e dt h r o u g ht h e r e s i s t oar f t e r8 7 5 s e c o n d s
s n dt h e c u r r e n ti n t h e c i r c u i t
( i v )T h ee n e r g yd i s s i p a t ebdy t h e r e s i s t oar f t e r2 0 0s e c o n d a
at thatinstant
(v)
T h ep . d .a c r o s st h e c a p a c i t oar f t e r3 0 0 s
1,.7xIOZJ;7 x10-6A;9.82V)
3727s;5.3x1-0-3s;
(Ans:1500s;432s;1040s;
t o 8 V a n dt h e n m a d et o d i s c h a r gaec r o s sa 3 0 0 k O
( 3 ) A 2 5 0 0 p rcFa p a c i t oirs c h a r g e d
resistor.
C a l c u l a t:e
( i )T h et i m e c o n s t a not f t h e d i s c h a r gcei r c u i t
f o r t h e p . d .a c r o s st h e c a p a c i t otro f a l lt o O Va n d
( i i )T h et i m e st h a t w o u l dh a v ee l a p s e d
4V
( i i i )T h ec h a r g et h a t w o u l dh a v ef l o w e dt h r o u g ht h e r e s i s t oar f t e r5 0 0 s e c o n d s
( i v )T h ee n e r g yd i s s i p a t ebdy t h e r e s i s t oar f t e r5 0 0s e c o n d s
(750s;2t6s;520s;0.01C;0.06J
)
74
(4) A 1500;rFcapacitoris chargedto 6V andthen madeto discharge
acrossa 800kO
resistor.
C a l c u l a t:e
( i )T h et i m e c o n s t a not f t h e d i s c h a r gcei r c u i t
( i i )T h et i m e t h a t w o u l dh a v ee l a p s e d
f o r t h e e n e r g yo f t h e c a p a c i t otro f a l lt o S O %
of its
i n i t i avl a l u e
(iii)Thetime that would haveelapsedfor the energyof the capacitorto fallto 2O%ofits
i n i t i a vl a l u ea n dt h e c u r r e n ti n t h e c i r c u i a
t t thatinstant.
(iv)Theenergystoredin the capacitorafter its initialchargefallsto 30%ofits initial
value
( v )T h e p . d .a c r o s st h e c a p a c i t owr h e n i t s e n e r g ys t o r e di s 4 5 %o f t h e i n i t i a vl a l u e .
(Ans:1200s; 4I7s; 967s;3.35x10'
A; 2.43xt03t;+.OZV)
( 5 ) A 2 0 p Fc a p a c i t oirs c h a r g e d
t o a p . d .o f 6 V b y a b a t t e r y l.t i s t h e n d i s c o n n e c t ef rdo m t h e
b a t t e r ya n d c o n n e c t e a
d c r o s sa n i n i t i a l l yu n c h a r g e4d0 p r F
capacitor.
(i)
D e t e r m i n e J hpe. d .a c r o s st h e c o m b i n a t i oann dt h e e n e r g ys t o r e d .
(ii)
T h ec o m b i n a t i oonf t h e t w o c a p a c i t o r(sc o n n e c t eidn p a r a l l e li)s t h e n c o n n e c t e d
a c r o s sa 4 5 0 k Or e s i s t oirn o r d e r t o d i s c h a r giet . D e t e r m i n teh e t i m e t h a t m u s t
e l a p s ef o r t h e e n e r g ys t o r e di n t h e c o m b i n a t i otno f a l lt o o n e q u a r t e ro f i t s i n i t i a l
v a l u ea n dt h e c u r r e n t h r o u g ht h e r e s i s t oar t t h a t i n s t a n t .
(Ans: 2V; 1..2x1,0-a
J; 1,8.7s; Z.22x1,0-6
A)
75
Solutionsfor CapacitorDischarge
( 1 ) ( i ) T i m eC o n s t a nrt = C R= 5 0 x L 0 - 6
x 1 , 2 0 x1 0 3 =6 s e c o n d s .
(ii) Use7 - Vor-il^, where Vo : 6V, and"V : 3V
-rt
H e n c e3 - 6 e - t l n cb u t R C : 6 , s o 3 : 6 e - t l u ;
3-t-t
ln= : . ; -0.693 ::
666
3
Z-
,-tlu
;t : 0.693x 6 x- 4.I6s
- 'r//o ; l n 1 , - t
WhenV : IV ;I _ 6e
; t :10.75s
6: 6
( i i i )U s e Q: Q or . - t l n cb u t f i r s t f i . n d "Q s ; Q o : C V s - 5 0 x 1 0 - 6 x 6 : 3 x 1 0 - 4 C
H e n c eQ - 3 x 1 0 - a e - s l o- 1 . 3 x I 0 - 4 C
T h a tv a l u eo f Q r e p r e s e n ttsh e c h a r g er e m a i n i n ign t h e c a p a c i t oar f t e r5 s e c o n d sT.h ec h a r g e
t h a t w o u l dh a v eb e e nl o s tf r o m t h e c a p a c i t oi rs e q u a l t ot h e D I F F E R E NbCeEt w e e nt h e i n i t i a l
c h a r g ea n dt h e f i n a lc h a r g e .
i . e .3 x 1 0 - + 6 - 1 . 3x 1 0 - 4 C: 1 , . 7x I 0 - 4 C
( i v )F i n d . I n i t t a l E n e r g y s t o r e d .t . e . E s - ) t u &
-
1* 50 x 10-6 x 62 9 x r0-4J
A f t e r 2 s e c o n d sf ,i - n dt h e n e w v o ' I t a g et . r . V - V o r - t l n c ; V - 6 e - z l o- 4 . 3 V
1,^1
50x 10-6x4.32:4.6 xI0-4J.
. ThenuseagatnEZCVt:rx
T h ed i f f e r e n c ien t h e t w o v a l u e so f t h e c a l c u l a t eedn e r g i e rse p r e s e n ttsh e e n e r g yd i s s i p a t ebdy
t h e c a p a c i t oi .re . 9 x 1 0 - + 1- 4 . 6 x 1 0 - a - 4 . 4 x t 0 - 4 J
76
( 2 ) ( i ) T i m ec o n s t a n rt = c R= 1 0 0 0x l - 0 ' 6x 1 . 5x 1 0 6 =1 5 0 0s e c o n d s
( i i ) U s e7 - V o r - t l n cw h e r e V o: 1 2 V , a n d V : 9 V
H e n c e9 :
7 2 r - t l a cb u t R C : 1 5 0 0 , s o 9 :
L 2 e - t l r c o o;
*:
,-t/rroo
9-t-t
l n - : - : : i ; - 0 '.228888 : ' r
; t : 0 , 2 8 8 x 1 5 0 0: 4 3 2 s
1,2 15oo
15oo
W h e n V: 6 V ; 6 : 1 2 r - t l r c o o ; l n
; t:1040s
*:#
W h e n V: i , V; 1 : r 2 r - t l r c ;ol o
n+ : # : ;
t : 3727s
( i i i ) U s e Q : Q o r - t l n cb u t f i " r s t f i . n d . Q s ; Q o : C V s - 1 0 0 0 x 1 0 - 6 x 1 2 : 1 , 2 x
H e n c eQ - t 2 x 1 0 - 3 r - u " / r r 0 0 - 6 . 7 x 1 0 - 3 C
10-3C
T h a tv a l u eo f Q r e p r e s e n ttsh e c h a r g er e m a i n i n ign t h e c a p a c i t oar f t e r8 7 5s e c o n d sT.h ec h a r g e
t h a t w o u l dh a v eb e e nl o s tf r o m t h e c a p a c i t oirs e q u a lt o t h e D I F F E R E N C
b eEt w e e nt h e i n i t i a l
c h a r g ea n dt h e f i n a lc h a r g e .
i . e .1 2 x I 0 - 3 C - 6 . 7 x I 0 - 3 C : 5 . 3 x 1 0 - 3 C
( i v )F f . n d I. n t t i a l E n e r g y s t o r e di . . e . E s: ) t U &
-!
x 1 0 0 0 x 1 0 - 6 x 1 . 2 2: 7 2
x I0-3J
Af ter 200 second.s,
f ind"the new voltage t. e.V - Vor-tlnc;V : 1.2r-'oolrs00 I0.SV
1^1
: 2 x 1 0 0 0 x 1 0 - 6 x 1 0 , 5 2: 5 5 x t 0 - 3 J .
T h e n u s ea g a i n E tCVt
T h ed i f f e r e n c ien t h e t w o v a l u e so f t h e c a l c u l a t eedn e r g i e rse p r e s e n ttsh e e n e r g yd i s s i p a t ebdy
t h e c a p a c i t oi .r e . 7 2x 1 0 - : 7 - 5 5 x 1 0 - 3 : 1 . 7 x t 0 - 3 J
T h ec u r r e n ti n t h e c i r c u iat t t h a t i n s t a nits f o u n df r o m I - Y R
r j ' s v- 7 x 1,0-6A
1 . 5x 1 0 b
( i v )U s eV : V o r - t l * , w h e r e V o: 1 , 2 V , a n dt ": 3 0 0 s , R C : 1 5 0 0 s
r sl o o- g . B z V
V : 1,ze-=oo
77
MAGNETIC
FIELDS
co
g n d u c t oor f l e n g t h/ i s p l a c e di n a m a g n e t i c f i e l da,f o r c ei s
W h e n e v ear c u r r e n t - c a r r y i n
e x e r t e do n t h e c o n d u c t opr r o v i d i n tgh a t t h e c u r r e n ti s n o t p a r a l l et lo t h e f i e l d .
T h ef o r c ea c t i n go n t h e c o n d u c t oirs g i v e nb y t h e e q u a t i o n :
F : Bllstn 0
W h e r e0 i s t h e a n g l eb e t w e e nt h e d i r e c t i o no f t h e c u r r e n ta n dt h e d i r e c t i o no f t h e m a g n e t i c
field.
T h e q u a n t i t y/ r e p r e s e n ttsh e l e n g t ho f c o n d u c t otrh a t i s i n t h e m a g n e t i fci e l da n d " 8 " i s c a l l e d
t h e m a g n e t i fcl u x d e n s i t y .
T h ea b o v ee q u a t i o ni n d i c a t etsh a t t h e f o r c ei s m a x i m u mw h e ns i n O= 1 , a n dt h i so c c u r sw h e n0
= 9 0 " , i . et h
. e f o r c ei s m a x i m u mw h e nt h e c u r r e n ta n dt h e d i r e c t i o no f t h e m a g n e t i c f i e ladr ea t
9 0 "t o e a c ho t h e r .
H e n c eF : B I L w h e nt h e c u r r e n ta n dt h e f i e l da r ea t r i g h ta n g l e st o e a c ho t h e r .
F r o mt h i s l a t t e re q u a t i o nw, e c a nd e d u c et h a t B - I , 1 * h " n 0 = 9 0 " )
IL
Definitionof FluxDensity: T h em a g n e t i fcl u xd e n s i t yB ,c a nb e
d e f i n e da st h e f o r c ea c t i n gp e r u n i t
c u r r e n ti n a w i r e o f u n i t l e n g t ha t r i g h t
anglesto the field.It is a vector
T h eu n i t o f m a g n e t i fcl u x d e n s i t yi s t h e T e s l a( T ) .
T h eT e s l ai s d e f i n e da sf o l l o w s :
A Teslais the magneticflux densityof a uniform field that
causesa force of 1N to act on a conductor.1m long and
carryinga currentof 1A,when the conductoris placed
perpendicularto the magneticfield
78
W h e na c u r r e n t c- a r r y i n g
c o n d u c t oirs p l a c e di n a m a g n e t i fci e l d ,t h e d i r e c t i o nof the forceon
the conductoris givenby FLEMING'S
LEFTHANDRULE
R e c a ltlh a t
fhumb
Forefinger
I
iI
I
THRUST
seCond
Itunnrrur
I
V
FIELD
In Fleming'sLeft hand Rule,the current,field and force are alwaysat right anglesto each
other.
Forceon a Charge
W h i l et h e e x p r e s s i oFn= B l Ls i n 0i s u s e dt o d e t e r m i n et h e f o r c ea c t i n go n t h e c o n d u c t o rt ,h e
e q u a t i o nc a nb e m o d i f i e dt o d e t e r m i n et h e f o r c eo n e a c hc h a r g et h a t c o n s t i t u t teh e c u r r e n t .
F o ra c o n d u c t ocr a r r y i n ga c u r r e n t| , l e t t h e T o t a ln u m b e ro f c h a r g etsh a t w o u l dh a v ef l o w e d
a f t e rs o m et i m e b e " n ". '
L e tt h e m a g n i t u d e
o f o n e c h a r g eb e " q " .
Hence the current
nq
t-
t
B u tt h e f o r c eo n t h e c o n d u c t o irs e q u a lt o F r= B I Ls i n 0
Substituttng f or I gi"uesFt:
Bnql x stn?
T h i sF 1r ep r e s e n t tsh e forceon "n" charges
i n t h e c o n d u c t o rh, e n c et h e f o r c eo n o n e c h a r g ei s
givenby :
BqIx stn?
I
d"i"stance
:5f teedv.
rtr2 b u t t h e q u a n t i " t y- t
t
ttrn
HenceF, - Bqv sin?
The force on a chargemovingwith a
speedv througha magneticfield of
flux densityB is givenby
F - Bqv sin9
79
l f t h e c h a r g em o v e sa t r i g h ta n g l e st o t h e f i e l d ,t h e nt h e f o r c ew i l l b e a t r i g h ta n g l e st o b o t ht h e
f i e l da n dt h e v e l o c i t yo f t h e c h a r g e
C o n s i d ear p o s i t i v ec h a r g ee n t e r i n ga u n i f o r mm a g n e t i fci e l da t r i g h ta n g l e st o t h e . f i e l d :
T h ef i e l di s d i r e c t e di n t ot h e p l a n eo f t h e p a p e r . a n db y F l e m i n g l' se f th a n dR u l e ,t h e f o r c ei s
. o w e v e irf F l e m i n g 'lse f t h a n dr u l ei s a p p l i e da t a l m o s te v e r yp o i n ti n
i n i t i a l l yd i r e c t e du p w a r d sH
t h e p a t ho f t h e c h a r g ei,t w i l l b e s e e nt h a t t h e m o t i o no f t h e c h a r g ei s c i r c u l aar s l o n ga st h e
c h a r g er e m a i n si n t h e f i e l d .
T h ec h a r g eu n d e r g o ecsi r c u l a r
m o t i o na n dt h e M a g n e t i cF o r c e=
C e n t r i p e t aFlo r c e .
BQv Hence
BQY :
LEARN
mvz
wherem is
r
t h e m a s so f t h e c h a r g ea n d r i s t h e
r a d i u so f t h e c i r c l e .
mvz
r
80
M A G N E T IF
CI E L DTSU T O R I ASLH E E T
( 1 ) A w i r ec a r r y i n g
a c u r r e n to f 8 A a n d 1 . 5m e t r e si n l e n g t hi s p l a c e di n a f i e l do f f l u xd e n s i t y
0 . 4 5 T. W h a ti s t h e f o r c eo n t h e w i r e i f i t i s p l a c e d
( a ) A t r i g h ta n g l e st o t h e f i e l d
( b ) A t 6 0 ' t ot h e f i e l d
( c ) A t 4 5 = t ot h e f i e l d
( d ) A l o n gt h e f i e l d
( A n s 5: . 4 N ; 4 . 6 8 N
3 .; 8 2 N ; 0()U s eF = B r Ls i n 0 )
( 2 ) A t h i n c o n d u c t o6r m l o n ga n d c a r r y i n g
a c u r r e n to f 0 . 4 5 Ai s p l a c e di n a u n i f o r mm a g n e t i c
f i e l da t r i g h ta n g l e st o t h e f i e l d . O n l yp a r to f t h e c o n d u c t oirs e n v e l o p e d
i n t h e f i e l dw i t h
3 . 6 me x t e n d i nogu t o f t h e r e g i o nc o n t a i n i ntgh e f i e l d .l f t h e f o r c eo n t h e w i r e i s 7 . 2 Nf,i n d
t h e m a g n e t ifcl u xd e n s i t y . ( H i n t O
: n l y2 . 4 mo f t h e w i r e i s s u b j e c t etdo t h e f i e l d .
( A n s 6: . 6 7 T \
( 3 ) A p i e c eo f w i r e 6 5 c m l o n g c a r r y i n ga c u r r e n to f 1 . 2 A i s p l a c e da t a n a n g l e0 t o a u n i f o r m
m a g n e t i cf i e l d o f F l u x d e n s i t y4 . 5 T . l ft h e f o r c eo n t h e w i r e i s 2 . 8 8 N ,c a l c u l a t et h e v a l u eo f 0 .
( A n s :5 5 " )
( a ) W h a t l e n g t ho f c o n d u c t ocr a r r y i n ga c u r r e n to f 3 . 3 4p l a c e da t a n a n g l eo f 7 0 "t o a m a g n e t i c
f i e l do f f l u xd e n s i t y1 . 8 T , w i lcl a u s ea f o r c eo f 1 . 9 5 Nt o b e e x e r t e do n t h e c o n d u c t o r ?
- .
( A n s .3:5 c m )
( 5 ) l f 1 . 5 5 mo f w i r e c a r r y i n g
a c u r r e n to f 3 . 4 Ap l a c e di n a u n i f o r mf i e l do f f l u xd e n s i t y2 . 2 T ,
w h a t m u s tb e t h e a n g l eb e t w e e nt h e c o n d u c t oar n dt h e f i e l di f t h e f o r c eo n t h e c o n d u c t o r
i s5 N ? ( A n s : 2 5 . 5 " ) .
( 6 ) A n e l e c t r o ne n t e r sa u n i f o r mm a g n e t i c f i e lodf f l u x d e n s i t2y . 5 f w i t h a v e l o c i t y o f
3 . 5x L 0 6 m 1
s a t a n a n g l eo f 4 0 " t ot h e f i e l d .C a l c u l a t e
h e f o r c eo n t h e e l e c t r o n .
=
( A n s 9: x 1 0 ' 1 3 N
)
B
q
v
s
i
n
0
l
tF
'(7)Aproton(mass=!.67x1027Kgandcharge=1.6x1-0-1eC)
isfiredataspeedof6.5xl06ms-1
a t r i g h ta n g l e st o a m a g n e t i fci e l do f f l u xd e n s i t y3 x l - 02 T .D e t e r m i n teh e r a d i u so f t h e p a t ho f
t h e p r o t o nw h i l ei t i s i n t h e f i e l d .( H i n t U
: s eF - r y
a n d .F -
B q v ) ( A n2s . 2 6 m )
( 8 ) A n e l e c t r o ni s a c c e l e r a t ef dr o m r e s tt h r o u g ha p . d .o f 3 7 5 0 V l.t e n t e r sa r e g i o nw h e r e
perpendicula
B= 4 x L0-3T
t or i t sv e l o c i t yC. a l c u l a tteh e r a d i u so f t h e p a t hi t w i l l f o l l o w .
31
{ A n s5 . 2 c m ) R e c a i r, ' I r d sosf e l e c t r o ni s 9 . 1 x 1 0 k g a n d c h a r g e= 1 . 6 x l - 0 - 1 e c
F t r s t f i r t d s p e e do f e L e c t r o nf r o m W - Q V :
81
1,
,^r'
1_
t h e nuseBQ
mu2
r
MagneticFluxPatterns
T h em a g n e t i cF i e l dP a t t e r nd u et o :
(1) A longstraightconductor--
C u r r e n itn t o p l a n eo f P a P e r
Thefield linesare Circular
C u r r e n to u t o f p l a n eo f P a P e r
t a n dG r i pR u l e .
T h ed i r e c t i o no f t h e f i e l dl i n e si s g i v e nb y t h e R i g h H
Thisrulestatesthat i f t h e r i g h th a n di s u s e dt o g r i pt h e c o n d u c t o rw, i t h t h e t h u m bp o i n t i n gi n
t h e d i r e c t i o no f t h e c u r r e n tt,h e n t h e d i r e c t i o no f c u r lo f t h e f i n g e r sw i l l givethe directionof the
m a g n e t ifci e l dl i n e s .
(2)Aflat circularcoil:
A L O N GS O L E N O I D
Thernagnsti*
fi*ld is
O n l o o k i n ga t t h e c o i l
f r o m o n e e n d ,i f t h e
ffi= FI"nl
*snceRtr*ted
int*a n*arly
nifsrmfield
n ths *sntsr
c ur r e n ti s
aN t i c l o c k w i s, e
then
a l*ng
t h a t e n d i s a N - p o l ei;f
s*lsnsid.Ths
fieldoutside
is wsakand
dir,erge*t.
i t i s c l o c k w i s ei t, i s a S pole
T h e r i g h th a n dg r i pr u l ec a nb e
a p p l i e dh e r ea g a i nt o g e t t h e
d i r e c t i o no f t h e f i e l d .H o l dt h e c o i l
w i t h t h e r i g h th a n ds o t h a t t h e c u r lo f
t h e f i n g e r sg i v et h e d i r e c t i o no f t h e
C U R R E NtTh;e t h u m bp o i n t i n g
o u t w a r d sg, i v e st h e d i r e c t i o no f t h e
FIELD.
83
FLUXDENSITY:
OF MAGNETIC
CALCULATION
trg_r
a c u r r e n tl ; B = . * t ,
t i r ec a r r y i n g
( 1 ) A t a d i s t a n c e" r " f r o m a l o n gs t r a i g hw
w h e r e$ oi s c a l l e dt h e p e r m e a b i l i t y offr e e s p a c ea n d h a sa v a l u eo f 4 n x 1 0 - ' H m
T h es y m b o H
l i s t h e u n i tc a l l e dt h e H e n r yw h i c hi s a m e a s u r eo f I n d u c t a n c e
n o t r e q u i r e db y s y l l a b u s .
( 2 ) A I t h e c e n t r eo f a f l a t c i r c u l acr o i lo f N t u r n s' 2,gr= *
N I , w h e r e " r " i s t h er a d i u s
ofthecoil.
( 3 ) N e a rt h e c e n t r eo f a l o n gs o l e n o i,d B : V o n I , w h e r e" n " i s t h e n u m b e ro f
turns per unit length
B - P t f o r a l o n gs t r a i g hwt i r e ,w h e r er i st h e
zfir
d i s t a n c fer o m t h e a x i so f t h e w i r e
E r u f f o r t h e c e n t r eo f a f l a t c i r c u l acr o i l ,
2r
w h e r er i s t h e r a d i u so f t h e c o i l
B -
B :
VonI
n e a rt h e c e n t r eo f a l o n gs o l e n o i d
N i s t h e n u m b e ro f t u r n sa n dn i s t h e n u m b e ro f t u r n s
per unit length
84
MAGNETIC
FIELDS
TUTORIAL
SHEET
( 1 ) C a l c u l a t teh e m a g n e t i cf l u x d e n s i t ya t a p o i n t ( a ) 1 2 c m a n d ( b ) 1 . 5 mf r o m a l o n gs t r a i g h t
w i r e c a r r y i n ga c u r r e n to f 5 . 8 A . ( A n s : 9 . 6 11x 0 6 T ;7 . 7 3x t O 7 t )
( 2 1A w i r e c a r r y i n ga c u r r e n to f 8 A p r o d u c e as f i e l do f f l u x d e n s i t y7 . 4 x 1 0 6 Ta t a d i s t a n c ex c m
a w a yf r o m t h e w i r e .D e t e r m i nteh e v a l u eo f x , ( A n s2 1 . 6 c m )
( 3 )W h a t m u s tb e t h e m a g n i t u d e
o f t h e c u r r e n ti n a w i r e i f t h e c u r r e n tp r o d u c e a
s
f i e l do f f l u xd e n s i t y5 . 8x 1 0 6 Ta t a d i s t a n c e
o f 2 2 c mf r o m t h e w i r e . (A n s6 . 3 8 A )
( 4 ) A f l a t c i r c u l a rc o i l h a s 5 0 t u r n s o f w i r e , a n d t h e d i a m e t e ro f t h e c o i l i s 4 4 c m .W h a t i s t h e
magnitude
o f t h e f l u xa t t h e c e n t r ei f 1 . 8 Af l o w st h r o u g ht h e c o i l ? ( A N2S. 5 7x t 0 4 t )
( 5 ) H o w m a n yt u r n so f w i r e a r e t h e r e i n a f l a t c i r c u l acr o i lo f w i r e o f d i a m e t e r6 8 c mi f a c u r r e n t
s m a g n e t i fcl u xd e n s i t yo f 6 x 1 0 4 T ?( A n sI 2 O )
o f 2 . 7A p r o d u c e a
( 6 )W h a t i s t h e m a g n i t u d e
o f t h e c u r r e n ti n a f l a t c i r c u l acr o i lo f w i r e i f i t h a s2 5 0 t u r n so f w i r e ,
i t s d i a m e t e ri s 7 5 c ma n ' dt h e f i e l da t i t s c e n t r ei s 9 x 1 0 4 T ? ( A n2s. 1 5 4 )
( 7 ) A n e l e c t r o no r b i t sa n u c l e u sw i t h a s p e e do f 1 . 8 x L 0 6 m s - ' . T hr e
a d i u so f t h e c i r c u l a o
r rbitis
7 . 8 x i - 01 0 m . F i nt hd e m a g n i t u d oe f t h e f l u xd e n s i t ya t t h e n u c l e u ds u et o t h e e l e c t r o n 'm
s otion..
T h ee l e c t r o n icch a r g ei s 1 . 6x 1 0 1 s C .
( H i n t : F i n dt h e c u r r e n td u e t o t h e e l e c t r o n ' m
s o t i o n ;C u r r e n =
t c h a r g ep e r s e c o n dy; o u r i r u s t
m a n i p u l a t teh e c i r c u m f e r e n cteh,e s p e e da n dt h e c h a r g et o g e t c h a r g ep e r s e c o n d ) .
( A n s0
. .047T)
( 8 )A n a i r - c o r es o l e n o i dh a s1 5 0 0l o o p sa n d i s 8 0 c ml o n gw i t h a d i a m e t e o
r f 3 c m .l f a c u r r e n to f
6 4 i s s e n tt h r o u g hi t . W h a t w i l l b e t h e m a g n i t u d eo f t h e f l u xd e n s i t yi n s i d eo f i t ? ( 0 . 0 1 4 T )
( 9 )A l o n gw i r e c a r r i e sa c u r r e n to f 2 0 A a l o n gt h e a x i so f a s o l e n o i dw, h i c hh a sa i a d i u s0 f 1 O c m .
T h es o l e n o i d
h a sa n i n d e p e n d e nc tu r r e n ta, n dt h e f i e l dd u et o t h e s o l e n o i d
i s 4 x 1 0 - 3 TF.i n dt h e
r e s u l t a nfti e l da t a p o i n t3 m m f r o m t h e s o l e n o i da x i s .( H i n t ;T h e l o n gw i r e p r o d u c e si t s o w n f i e l d
t h a t i s p e r p e n d i c u l at or t h e f i e l d o f t h e s o l e n o i dc; o n s i d e trh e t w o f i e l d sa s v e c t o r st h a t a r e
perpendicula
t or e a c ho t h e r ,a n df i n dt h e r e s u l t a nvt e c t o r . ( A n4s. 2 2x t O 3 l )
Soiution
f o r q u e s t i o n7 : F i n dt h e c urrrreenntti n L r e v o l u ttiiooninn w h i c hc a s eN ( n o o f t u r n s = 1 )
x s?eed
-Q
t
ctrcumf erence
4 n x L 0 - 7 x 1 . . 6 x 1 0 - 1 ex 1 . 8 x 1 0 6
: 0.0477
2 x 7 . 8 x 1 0 - 1 0x 2 x r x 7 . B
x 10-10
I - charge
per
er u
rge p
unit
n i t ttime
ime -9
B5
-
FORCES
MAGNETIC
of Force:
Direction
LEFT
W h e na c h a r g em o v e si n a n e l e c t r i cf i e l d ,t h e d i r e c t i o no f t h e f o r c ei s g i v e nb y F L E M I N G ' S
H A ND R UL E .
Recall: ,Thumb
tl
,
seCond
Forefinger
lt
rl
++
Thrust
C ur r e n t ( c o n v e n t i o|n
)a
Field
T h ed i r e c t i o no f t h e c u r r e n ti s t h e d i r e c t i o no f m o t i o nof P!SI[!VE charge.
C o n s i d et h
r e f o l l o w i n gD i a g r a m s :
xxxxxx'x
E-'
X
XX
XX
c h a r g eh a sa n i n i t i a l
T h ep o s i t i v e
v e l o c i t yt o w a r d st h e r i g h t ,h e n c et h e
c o n v e n t i o n aclu r r e n ti s t o w a r d st h e
l n t h e a b o v ed i a g r a mt,h e e l e c t r o n
h a sa n i n i t i a vl e l o c i t yt o w a r d st h e
r i g h t ,h e n c et h e C O N V E N T I O N A L
C U R R E Ni sTt o w a r d st h e l e f t 'T h ef i e l d
i s d i r e c t e di n t o t h e P a P e ra, n d b Y
F l e m i n g ' Ls e f th a n dR u l e t, h e f o r c ei s
r i g h t .T h ef i e l di s d i r e c t e di n t o t h e
p a p e ra n d b y F l e m i n g 'Ls e f th a n dr u l e ,
t h e f o r c eo n t h e P o s i t i v ec h a r g ei s
initialld
y ownwards.
initialld
y i r e c t e du P w a r d s .
force
A s l o n ga st h e m o v i n ge l e c t r i c h a r g ei s p e r p e n d i c u l taor t h e a p p l i e dm a g n e t i fci e l d ,t h e
o n t h e c h a r g ec a u s e si t t o u n d e r g oC I R C U L AMRO T I O Na, n dt h e f o l l o w i n ge q u a t i o ni s a p p l i e d
BQv -
mvz
r
86
Considerthe casewhen the moving chargeis perpendicular
to an ELECTRIC FIELD.
W h e nt h e e l e c t r i c h a r g ep a s s e s t h r o u ga hn E L E C T RFI ICE L D
t h a t i s p e r p e n d i c u l a r t o t hmeo t i o n
o f t h e c h a r g et,h e p a t ho f t h e c h a r g ei n t h e f i e l di s p A R A B o L l c .
H e n c ew h i l et h e m a g n e t i fci e l dp r o d u c e C
s I R C U L AMRO T I O Nt ,h e e l e c t r i cf i e l dp r o d u c e s
P A R A B O LM
I CO T I O N .
The force experiencedby the chargedue to the ELECTRIC
FIELDis given by
F - Eq where E is the Electri.c Field Strength.
87
C o n s i d et h
r e c a s en o w w h e r et h e c h a r g ep a s s etsh r o u g ha c o m b i n a t i oonf a n E l e c t r iac n d
MagnetiF
c i e l dt h a t a r ep e r p e n d i c u ltaor e a c ho t h e ra n da l s op e r p e n d i c u ltaort h e i n i t i a l
v e l o c i t yo f t h e c h a r g e .
, n dt h e E l e c t r iFc o r c ew i l l b e
T h eM a g n e t i fco r c ew i l l b e d o w n w a r d(sF l e m i n gl' es f th a n dr u l e ) a
u p w a r d s ( A t t r a c t ioofno p p o s i t ec h a r g e s.)
( 1 ) P a t hA i n d i c a t etsh a t t h e u p w a r df o r c ei s g r e a t e tr h a nt h e d o w n w a r df o r c ei . e .
Eq
( 2 ) P a t hB i n d i c a t e s t h a t t h e E l e c t r i cf o r c e i s e q u a l t o t h e M a g n e t i cf o r c e E Q :
( 3 ) P a t h C i n d i c a t e st h a t t h e M a g n e t i cF o r c ei s g r e a t e rt h a n t h e E l e c t r i cF o r c ei . e ,
Bqv > Eq
88
BqV
The Earth's
Magnetic Field
The Earth has a substantialmagneticfield, a fact of somehistoricalimportancebecauseof
the role of the magneticcornpassin explorationof the planet.
Structureof the Field
The Jield lines defining the structure of the magneticfield are similar to thoseof a simple
bar magnet,as illustratedin the following figure.
lnnerYan
A l l e nB e l t
Outer
Allen
i
!
I
't
I
II
I
I
M a g n e t i cF i e l dL i n e s
The Harth's rnasnetis field an* Van Allen nadiation belts
It is ll'ell knorvn that the axis of the magneticfield is tipped with respectto the rotation axis
of the Earth. Thus, true north (definedby the directionto the north rotational pole) does
not coincidervith magneticnorth (definedby the directionto the north magneticpole) and
compassdirectioirsmust be correctedb,vfixed amountsat given points on the surfaceof the
8,.:rc"th
to vield true directions.
89
Van Allen Radiation Belts
A fundamental propcrtr of magneficfields is
t h a t t h e r c r e r t l ' o r c e so n m o v i n g e l e c t r i ea l
c h a r g e s .T h u s . a n t a g n e t i ei i e l d t ; u l t r a p
c h a r g e d p a r t i c l e s: u c l t x - -r i * c t r r ; n s * n d
p r o t o n s a s t h c " \ a r c f o r c e d t o c r e ' eu I e i t
s g * * ; ' . : l ! i ;i .: ;: , : : : , : :b: i t C ka n t l l r - t f t ha i O n gt h e
?,
!
{ield lines.
Earth
*-
'Mirror
Fcints
A s i l l u s t r a t e d i n t h e a d j a c c * t f * i l l u r r .t h t '
.
_1--.-J
:l
I
charged particles are rellectedat "rnirror
p o i n t s " u h e r c t h r l l c l i l l i n c s c { ) m cc l o s e
t
l ,* {agn*
ti cFi + l d
Lin*
t o g e t h e ra n d t h e s p i r a l s t i g l i t r ' ; : .i l n e * f t h e
f i r s t f r u i t s o f e a r l v s p a c ee r p l o r a t i o n n a s t h e
d i s c o v e n i n t h e l a t e 1 9 5 0 st h a t t h e E a r t h i s s u r r o u n d e d b v t r v o r e g i o n s o f p a r t i c u l a r l v h i g h
concentration of charged p*rticles called the l/an Allen rucliation belts.
T h e i n n e r a n d o u t e r V a n A l l e n h c l f s a r e i l l u s t r a t e d i n t h e t o p t i g r r r e .T h e p r i m a r v s o u r c e o f
thesechar.gedparticlesisthe.=,.ii-:':-::::::ig-:::s"lE*:E*=emanatingfromtheSunthatwecallthe'
i d , , r i ; s i ' E " g ' ; - ,r,r"e: , s: ih- .a. l{lss c e i n ; r s u h s e q u e n ts e c t i o n ,t h e c h a r g e d p a r t i c l e s t r a p p e d i n t h e
E a r t h ' s m a g n c t i c f i e l d a r r r e s p c r r s i b l ef n r t h e ; : . ; : ; i . ''' . {' :l 't:i a r t h e r n a n d S o u t h e r n L i g h t s ) .
Origin of the t{agnetic Field
l{agnetic fieldsare producerlhr the motion of electricalcharges.For example,the
magneticlield of a bar magnetresultsfrom the motion of negatively'chargedelectronsin
the magnet.The origin of the Earfh's magneticfield is not completel_-v
understood,but is
rrith tlcctrical currentsproducedby'the couplingof convective
thought to bc irssttciirtcd
effectsand rotation in the spinningliquid metallicouter core of iron and nickel.This
mechanismis termt'd the il-t'ttutn*e.ff'et't.
Rocksthat are lbrmed from the molten statecontain indicatorsof the magneticfield at the
'f
time of their solidification. hc studv of such "magneticfossils"indicatesthat the Earth's
magneticlield re\er-cs ifself everv million vearsor so (the north and south magneticpoles
suir.i'). This is br-rtonc detailof thc magnetictield that is not rvellunderstood.
90
The Earth's
Magnetosphere
The solar wind mentionedaboveis
::
a stream of ionizedgasesthat
,',
blows outward from the Sun at
about 400 km/secondand that
variesin intensitvwith the amount ..,',
of surfaceactil'itv on the Sun. The
Earth's magneticfield shieldsit
from much of the solar wind. When the solar lvind encountersEarth's
magneticfield it is
deflectedlike water around the bow of a ship' as illustratedin the adjacent
image(F{*eer**).
The imaginarv surfaceat which the solar wind is first deflectedis
calledthe bow shock.The
correspondingregionof ipace sitting behindthe borvshockand surroundingthe
Earth is
termed the s;trs*rl<'F*;sgJe{'i'i';
it representsa region of spaceclominatedbv the Earth,s
magneticfield in the sensethat it largely preventsthe solar u,ind from
entering.However,
somehigh energvchargedparticlesfrom the solar wind leak into the
magnetosphere
and
are the sourceof the chargedparticlestrapped in the yan Allen belts.
91
Electricand MagneticFields
Applicationof Mutuallyperpendicular
The MassSpectrometer:
T h i si n s t r u m e nits u s e dt o d e t e r m i n et h e m a s s e os f a t o m sa n d i o n st o a v e r yh i g hd e g r e eo f
accuracy.
I t u s e si o n s i e dg a s ,a sa s o u r c eo f p o s i t i v ei o n s ,a d e f l e c t i o n
s y s t e mc a l l e da V E L O C I TSYE L E C T O R
r h i c hi s u s u a l l a
y s h e e to f p h o t o g r a p hfiicl m .
a n d a d e t e c t ow
l e t a lp l a t e sw i t h m u t u a l l y
T h ev e l o c i t ys e l e c t ocr o n s i s tos f t w o s e p a r a t e pd a r a l l em
p e r p e n d i c u lealre c t r i a
c n d m a g n e t ifci e l d sb e t w e e nt h e m .
W h e ni o n so f a c h a r g eq a n d s p e e dv p a s st h r o u g ht h e f i e l d st,h e M a g n e t i cF o r c eo n e a c h
B tQV;
chargeis givenb.y F !
T h e E l e c t r ifco r c ea c t i n go n e a c hc h a r g ei s g i v e nb VF Z - E q .
l f t h e f i e l d sa r ea d j u s t e ds o t h a t t h e r ei s n o d e f l e c t ino, t h e nE Q :
B tQV
E
:B1'
T h i su n - d e f l e c t ebde a mi s t h e n p a s s e d
t h r o u g ha r e g i o no f a n o t h e rm a g n e t i fci e l da l o n ew h e r e
t h e f l u x d e n s i t yi s 8 2 .T h i sf i e l dc a u s e tsh e b e a mo f i o n st o u n d e r g oc i r c u l am
r o t i o n ,w i t h t h e
r a d i u so f t h e p a t hb e i n gr .
T h i si m p l i e st h a t V
-
Hehce Bzgv:
mvZ
l f w e s u b s t i t u,:t V :
r
_+
t
Ur,
Bze
thenBzg :
q
m
mv
r
mE
rB r'
E
rB 11Bz
T h eq' u a n t i 'tnyat i s c a l l e dt h e c h a r g et o m a s sr a t i o ,a n d o n l yi o n so f a p a r t i c u l acrh a r g et o m a s s
r a t i o ,w i l l b e d e f l e c t etdh r o u g ha p a r t i c u l ar ra d i u sr t o r e a c ht h e d e t e c t o r .
X
B2
q1
X
DETECTOR
ELECTROMAG
NETISM
W h e na s o f t i r o nc o r ei s p l a c e di n s i d ea s o l e n o i d
carrying
a c u r r e n tt,h e M a g n e t i cF l u xD e n s i t y
B o f t h e c o i li s s i g n i f i c a n tilny c r e a s e d
T h et e r m " S o f tl r o n "d o e sn o t r e f e rt o t h e t e x t u r eo r t o u g h n e sosf t h e i r o n ,b u t i n s t e a di t i s a
m e a n so f d e s c r i b i ntgh e a b i l i t yo f t h e i r o nt o b e c o m em a g n e t i z ewd h e na c u r r e n ti s i n t r o d u c e d
AROUND
i t a n dt h e a b i l i t y t ol o s ei t s m a g n e t i s m
w h e nt h e c u r r e n it s r e m o v e d .
A s o f t m a t e r i ails o n e t h a t c a nb e c o m em a g n e t i z eadn d b e d e m a g n e t i z eeda s i l y .
T h i se f f e c to f p l a c i n ga s o f ti r o nc o r ei n a c u r r e n tc a r r y i n gs o l e n o i di s s i m i l a tro t h a t o f t h e
p l a c e m e notf i h e d i e l e c t r iicn a c a p a c i t o r .
T h e ' M A G N E T ' p r o d u ci en dt h e s o f ti r o nb y m e a n so f a n e l e c t r i c u r r e n its r e f e r r e d
t o a sa n
ELECTROMAG
NET.
T h ep r i n c i p loef t h e e l e c t r o m a g niesta p p l i e di n
( i ) E l e c t r i dc o o r l o c k s- W h e nn o c u r r e n ft l o w si n t h e c o i l ,t h e l a t c ho f t h e l o c ki s t h e i n t h e
'locked'position
a n d r e m a i n tsh e r eb y t h e e f f e c o
g n i t . W h e nt h e
t f a s m a l sl p r i n gp u s h i n o
s w i t c hi s p r e s s e da,n d a c u r r e n t f l o w si n t h e c o i l ,t h e s o f ti r o nc o r eb e c o m e sm a g n e t i z eadn d
e x e r t sa f o r c eo f a t t r a c t i o no n t h e l a t c h ,p u l l i n gi t d o w n a n d c a u s i n g t h el o c k t o b e c o m eo p e n .
W h e nt h e c u r r e n ti s s h u to f f , t h e s p r i n gt h e n p u s h e st h e l a t c hb a c ki n t ot h e ' l o c k e dp' o s i t i o n .
latch
93
RELAY:
( i i )S w i t c h e sA
: n e x a m p l eo f a n e l e c t r o m a g n e t i sc w i t c h i s t h e E L E C T R O M A G N E T I C
{
contacts
Connections
t o l o a de . g .
T h er e l a yi s a n .
h i g hv o l t a g e
e l e c t r o m a g n e tsi cw i t c ht h a t
operatesfrom a low voltage
equipment
s u p p l ya n d i s u s e dt o c o n t r o l
Ys u a l l Y
a h i g hv o l t a g es u P P | u
t o i n d u s t r i aml a c h i n e r Y
L o w ' ; o l t a q el n P u t
i i I' i :!a v
m a g n e t i z e da n d a t t r a c t s t h e
W h e n t h e s w i t c h i s c l o s e d ,t h e s o f t i r o n c o r e i n t h e c o i l b e c o m e s
t h e c o n t a c t st h a t c o m p l e t e
m e t a l a r r n a t A . L a L i s i ntgh e e n i i a i B t o b e p u s h e du p a n d c l o s i n g
t h e c i r c u i tf o r t h e h i g hv o i t a g ee q u i p m e n t '
94
FORCES
BETWEEN
C U R R E N T . C A R R Y IC
NO
GN D U C T O R I
T h e f o r c e st o b e c o n s i d e r e da r e t h o s e b e t w e e n: ( a )p a r a i i e {i c n , j u e t o r sa n d i n )p e r n e n d i c u l a r
conductors
I n p a r a l l e lc o n d u c t o r st,h e i " ' :ai i " et w o s i t u a t i o n st o L r ec c n s i d e r e c
( 1 ) P a r a l l ecl o n d u c t c r sc a r r y i n gc u r r e n t si n t h e : ; a r : r ed i r e c t i o n .
al
1e
F i e l dd u e t o B o n i y
a
Y
Fi..rd
c i u et o A o n l y
,,i
rql
\-{
l1
XXX
i
'iI
i
I
V
XXX
i
I
XXX
€-F
C o n s i d e tr h e f o r c e a c t i n go n P . T h e c o n d u c t c ri :
i s p l a c e di n t h e F I E L Do f Q . l f w e u s e t h e R i g h t
C t r : s i c l E n: "c w , t h e i i ; r t . a c i i n g o n Q . .T h e c o n d u c _ t o r
Q : s p l a c e cijn t h e F l L r - Do f P . l f w e u s e t h e R i g h t
H a n dG r i p R u l ef o r Q , i t w i l l b e s e e nt h a t t h e
H a n d G r i p R u l ef o r P , i t w i i l b e s e e nt h a t t h e f i e l c jo n
f i e l d o n t h e l e f t s i d eo f Q w h i c h i s e x p e r i e n c e d
t h e r i g h t s i d e o f P w h i c h i s e x p e r i e n c e db y Q , g o e s
b y P , g o e si n t o t h e p l a n eo f t h e p a p e r .U s i n g
o u t o f t h e p i a n e o f t h r p a p e r .L l s i n gF l e m i n g ' sL e f t
F l e m i n g ' sL e f t H a n d R u l ef o r t h e f c r c e o n P , i t
! - l a n dR u i ei o r t i : e i . ; r r e o n Q , i t w i l l b e s e e nt h a t
w i l l b e s e e nt h a t t h e f o r c e i s d i r e c t e dt o w a r C s
t h e f o r c e i s d i r e c t e dt o w a r d s P .
a.
l f t h e p e r p e n d i c u l adri s t a n c e
b e t w e e nt h e c o n d u c t o riss
Q d u e t o P , i s g i v e n0 v b p
t h e n t h e f i e l de x p e r i e n c e a
dt
l -L onl
7nr
a n dt h e f o r c ee x p e r i e n c ebdy O .i s g i v e nb y F q =B t q l s i nB * t f u k l
{sin g0)
Jo-l'Iq
S i m i l a r l y t h e f c- e e x p e r i e n c e d b y P i s g i v e n b y F p = B I p L s_i n O
I. lsine=11
2nr
f J H E N C U R R E N TA
S R E I N T H E S A M E D I R E C T I O NT, H E R EI S A F O R C EO F A T T R A C T I O N
A C T I N GO N
E A C HC O T { D U C T O RI F. T H E C U R R E N T A
S R E I N O P P O S | T TD I R E C T I O N S
T ,H E R EI S A F O R C EO F
REPULSION
A C T I N GO N E A C HC O N D U C T O R
C o n s i d et rh e f o l l o w i n ga r r a n g e m e notf c u r r e n t - c a r r y icnogn d u c t o r s -
A
X
X
X
X
XX
Y
T h ef i e l dd u e t o C o n i t s l e f t ,i s i n t ot h e p l a n e
, h i l eo n i t s r i g h t ,i t i s o u t o f
o f t h e p a p e rw
t h e p l a n eo f t h e p a p e r . ( R i ghhat n dg r i p
l f w e e x a m i n et h e f i e l dd u e t o A a t t h e
u n d e r s i do
e f A , i t w i l l b e s e e nt h a t b y
t a n dG r i pR u l et,h e
a p p l y i n tgh e R i g h H
f i e l di s d i r e c t e di n t ot h e p a p e r .
Rule)
F l e m i n g 'Lse f th a n dR u l e ,t h e
B ya p p l y i n g
by C at its lower
forceF1experienced
W h e nF l e m i n g l' se f th a n dr u l ei s a p p l i e d,
t h a t p o r t i o no f t h e w i r eA w h i c hi s t o t h e l e f t
e n d i s d i r e c t e dt o w a r d st h e R I G H T .
o f w i r e C ,w i l l e x p e r i e n caef o r c et h a t i s
u p w a r d sw
, h i l et h a t p o r t i o no f w i r e A t h a t i s
H o w e v e ra, t t h e u p p e rs i d eo f A , t h e f i e l d
i s o u t o f t h e p l a n eo f t h e p a p e ra, n dt h e
by C at its upperend,
forceexperienced
w i l l b e F za n d i t i s d i r e c t e dt o w a r d st h e
L E F T . ( F l e m i nLge' fstH a n dR u l e )
T h ew i r eC w i l l e x p e r i e n caen
A N T I C L O C K WM
I SOEME N T .
96
t o t h e r i g h to f C w i l l e x p e r i e n caef o r c et h a t
is downwards.
T h ew i r e A w i l l t h e r e f o r ee x p e r i e n cae
C L O C W I SMEO ME N T .
THEHALLEFFECT
i f e l e c t r o n isn a b e a ma r e d e f l e c t e db y a m a g n e t i fci e l d ,t h e n e l e c t r o n isn a s o l i dC O N D U C T O R
s h o u l da l s ob e d e f l e c t e b
d y a n e x t e r n am
l a g n e t i c f i e lT
dh
. i sw a sd e m o n s t r a t ebdy E d w i nJ .H a l l
i n 1 . 8 7 9w, h o w a st h e n a 2 4 y e a ro l d g r a d u a t ea t J o h nH o p k i n sU n i v e r s i t yT. h ed e f l e c t i o n
of
c h a r g ec a r r i e r si n a c o n d u c t obr y a m a g n e t i fci e l di s k n o w na st h e H A L LE F F E C T .
T h i sH a l lE f f e c te n a b l e os n et o d e t e r m i n e
w h e t h e rt h e c h a r g ec a r r i e r isn a c o n d u c t oar r e
p o s i t i v eo r n e g a t i v e .
C o n s i d ear M E T A L L cI C
o n d u c t oirn f o r m o f a t h i n s t r i po f w i d t h ' w ' a n dc a r r y i n ag c u r r e n t ' l ' a s
s h o w nA
. m a g n e t ifci e l cB
i i s a p p l i e dp e r p e n d i c u ltaort h e c u r r e n at n di n t ot h e p l a n eo f t h e
p ap e r
XXXXXXXX
B
1A
X
X
X
X
.x
X
X
X
t
I
W
I
pV
Letus makethe ASSUMPTION
that the chargecarriersin the metalare positively
charged.- .
lf Flemirig's
Lefthandrule is applied,the forceactingon the chargecarrierswill be upwards,
and all of the positivechargeswill accumulate
on the sideAB, leavingCDwith a negative
charge.
Therewouldthereforebe a potentialdifferencecalledthe HALIVOLTAGE
set up acrossthe
width 'w'and if our assumption
wascorrect,then AB shouldhavea higherpotentialthan CD,
sinceAB shouldbe positively
chargedand CDnegatively
charged.
However,when a voltmeter is connectedacrossthe width, it is found that the side AB is at
the lower potential and is therefore negativelycharged.This impliesthat our ASSUMPTION
.WAS WRONG,and that the chargecarriersin a metallicconductorare negativelycharged
(electrons|.
97
OFTHEHALLEFFECT
APPLICATION
whichis madefrom a
The HallEffectis usedto measuremagneticflux density.A HALLPROBE
h
a n d l ei,s p o w e r e db y a
l
o
n
g
i
n
s
u
l
a
t
i
n
g
t h i n d i s co f s e m i c o n d u c t i nmga t e r i a la n df a s t e n e dt o a
s m a l ld . c .s o u r c et o d r i v ea s m a l lc u r r e n t h r o u g hi t . A m i l l i v o l t m e t eorr a m i c r o v o l t m e t ei sr
c o n n e c t ea
d c r o s st h e l e a d sa, n d w h e nt h e p r o b ei s i n s e r t e di n a m a g n e t i fci e l dt h e H a l lv o l t a g e
bythe millivoltmeter.
ismeasured
T h ep r o b ei s f i r s tc a l i b r a t e bd y p l a c i n gi t i n a f i e l do f K N O W NF L U XD E N S I TsYo, t h a t t h e
m i l l i v o l tp
s e rT e s l ac a nb e o b t a i n e d .W h e ni t i s t h e n p l a c e di n a n u n k n o w nf i e l d ,a n dt h e H a l l
o f B c a nt h e n b e c a l c u l a t e d .
v o l t a g eo b t a i n e dt,h e m a g n i t u d e
FLUX
MAGNETIC
'A'
to a magneticfieldof flux densityB.
Considera loopor a coilof area placedperpendicular
T h ef l u x t h r o u g ht h e c o i li s t h e n u m b e ro f m a g n e t i fci e l dl i n e sp a s s i n tgh r o u g ht h e c o i la n d i s
g i v e nb y t h e f o r m u l a :
0=BA'
The unit of magneticflux is the Weber (Wb)
FLUXLINKAGE:
y ,t h e f l u x
l f a c o i lh a s ' N ' t u r n sa n di t s a r e a ' A ' i sh e l dp e r p e n d i c u l a rat of i e l do f f l u x d e n s i t B
l i n k a g et h r o u g ht h e c o i li s t h e p r o d u c to f t h e f l u xa n dt h e n u m b e ro f t u r n si n t h e c o i l .
H e n c ei n t h i sc a s e ,
o-NBA
l . W e b e r= l T e s l ax L m 2 O R l T e s l a= 1 W b . m - 2
The Weber is the magneticflux or numberof magneticfield linesthat passesthrough a loop
of area 1m2if the loop is at right anglesto a magneticfield of flux density1 Tesla.
98
ELECTROMAG
NETICINDUCTION
W h i l ee a r l ye x p e r i m e n tdse m o n s t r a t etdh a t a c u r r e n tp r o d u c e as m a g n e t i c f i e l dF,a r a d a y w a s
a b l et o d e m o n s t r a tteh e R E V E R S
eE
f f e c ti . e .A m a g n e t i fci e l dc a np r o d u c ea n e l e c t r i c u r r e n t .
l f a b a r m a g n e ti s m o v e di n a n d o u t o f a l o o po f w i r ew h i c hi s c o n n e c t e to
d a galvanometer,
a
c u r r e n it s I N D U C EiD
n t h e c o i lw h e n e v e r t h e
m a g n e its i n m o t i o n .
S
T h ef o l l o w i n go b s e r v a t i o nwse r e m a d e :
( 1 ) A c u r r e n te x i s t si n t h e c i r c u i o
t n l yf o r t h e i n s t a n t w
s h e nt h e r ei s r e l a t i v em o t i o n
b e t w e e nt h e m a g n e ta n dt h e l o o p .T h ec u r r e n g
t o e st o z e r ow h e nt h e m o t i o ns t o p s .
( 2 ) T h eg r e a t e tr h e s p e e do f r e l a t i v em o v e m e n tt,h e g r e a t e ri s t h e m a g n i t u d e
of the
c ur r e n t .
( 3 ) W h e nt h e m o t i o no f t h e m a g n e ti s r e v e r s e dt h
, e d i r e c t i o no f t h e i n d u c e dc u r r e n ti s a l s o
reversed.
i cd u c t i o n
w a sf o r m u l a t e d .
F r o mt h e s eo b s e r v a t i o nFs a
, r a d a y 'Lsa wo f E l e c t r o m a g n e It n
Faraday'sLaw statesthat the MAGNITUDE
of the inducede,m.f.in a coil, is directly
proportionalto the rate of changeof flux linkagein the coil.
99
R e c a l l t h aftl u x a n df l u x l i n k a g er e f e rt o t h e n u m b e ro f m a g n e t i fci e l dl i n e sp a s s i ntgh r o u g ht h e
coil.
Faradaye
' sx p e r i m e n tssh o w e dt h a t t h e n u m b e ro f f i e l dl i n e sp a s s i ntgh r o u g ht h e c o i li s
p r i m a r i l yb y t h e R A T Ea t
unimportanT
t . h ei n d u c e de . m . f .a n d i n d u c e dc u r r e n ta r ed e t e r m i n e d
w h i c ht h a t n u m b e ro f f i e l dl i n ec h a n g e s .
r o v e si n a m a g n e t i c f i e l dI n. t h i sc a s et h e
C o n s i d enr o w a d i f f e r e n ct a s ew h e r ea c o n d u c t om
, n dt h e c o n d u c t om
r o v e sd o w n w a r d w
s itha
f i e l di s d i r e c t e di n t ot h e p l a n eo f t h e p a p e r a
v e l o c i t yv . T h e l e n g t ho f t h e c o n d u c t oirs l .
XXXX
XXXX
-AXX
l"
I
I
XXXI
XX
XX
YV
XX
XX
T h eM A G N I T U DoEf t h e e . m . f .i n d u c e da c r o s st h e e n d so f t h e c o n d u c t oirs g i v e nb y
E -dQ - Bdo.
dt
.
d"A
dt
T h eq u a n t i t y r e p r e s e n ttsh e r a t eo f c h a n g eo f a r e at h a t t h e c o n d u c t ocr u t st h r o u g ha n d h a s
fr
u n i t so f m ' s - t .
F r o mt h e s eu n i t sa l o n ei t c a nb e d e d u c e dt h a t - I e n g t h x u e l o c i . t y
#
HenceE - BIv.
LEARN
100
l n t h e c a s ew h e r ea m a g n e ti s p l u n g e di n a n do u t o f a l o o po f w i r e ,t h e i n d u c e de . m . f .i s s t i l l
g i v e nb y E s i n c e@= B A ,t h e a r e ao f t h e l o o pi s c o n s t a nitn t h i sc a s e b
, u t t h e v a l u eo f
# "nd
B c h a n g e ds e o e n d i n o
g n i t s p o s i t i o na w a yf r o m t h e l o o p .H e n c ef o r t h i s p a r t i c u l acra s e ,
E_A9
dt
l f h o w e v e tr h e c o i lh a s' N ' t u r n st h e nt h e M A G N I T U DoEf t h e e . m . f .i n d u c e di s p r o p o r t i o n at ol
t h e r a t eo f c h a n g eo f F L U XL I N K A GaEn dE - N # .
T h ea b o v ee q u a t i o nh o w e v e ri s i n c o m p l e t a
e st h e D I R E C T I OoN
f t h e i n d u c e de . m . fm u s ta t s ob e
considered.
T h ed i r e c t i o ni s g i v e nb y L e n z ' sL a wa n dt h e r e s u l t i n e
g q u a t i o ni s :
-dQ
E-
dt
LEARN!!
101
LAW
LENZ,S
T h i sl a w d e t e r m i n etsh e d i r e c t i o no f t h e i n d u c e dc u r r e n t '
to
Lenz,sLaw statesthat when an emf is inducedin a conductor,the inducedemf tends
circuit.
opposethe changecausingit, and it doesopposeit when a currentflows in the
p r o d u c ea
T h i sl a w c a nb e r e s t a t e da s : T h e p o l a r i t yo f t h e i n d u c e de . m . f. i s s u c ht h a t i t t e n d st o
w
h
i
c hc a u s e s
f
l
u
x
c u r r e n t h a t w i l l c r e a t ea m a g n e t i fcl u xt o o p p o s et h e c h a n g ei n t h e m a g n e t i c
t h e i n d u c e de . m . f .
The combinationof Lenz'slaw and Faraday'sLawgivesthe all important
equationfor both the magnitudeand directionof the inducede.m.f.
_da
F_
Ll
dt
The negativesignindicatesthat the inducede.m.f.is in the opposite
directionto the changeof flux
n t f l o w si n t h e c i r c u i tt,h e m a g n e t i c
T h i sc a nb e i n t e r P r e t etdo m e a n
f i e l dd u et o i n d u c e dc u r r e n tw i l l o p p o s et h e m o t i o nt h a t c a u s e tsh e e l e c t r o m a g n e ti incd u c t i o n '
m u s tb e
T h ed i r e c t i o no f t h e i n d u c e de . m . f .i s t h e s a m ea st h a t f o r t h e i n d u c e dc u r r e n t b, u t i t
n n l yo c c u r sw h e na c u r r e n tf l o w i n t h e c i r c u i ti n w h i c ht h e c o i l
r e m e m b e r etdh a t t h e o p p o s i t i o o
o r l o o pi s c o n n e c t e dT.h i sc a nb e d e m o n s t r a t eidn t h e a r r a n g e m e nbte l o w .
of thecoil'
T h e m a g n g ti s s e t i n t o o s c i l l a t i osno t h a t i t w i l l b e c o n t i n u o u s lbye m o v i n gi n a n d o u t
i
n
ane.m.f. duced
W h e nt h e s w i t c hi s o p e n ,n o c u r r e n ft l o w si n t h e c i r c u i tb, u t t h e r ew i l l b e
m a g n e ti s
a c r o s st h e l e n g t ho f t h e c o i l .S i n c et h e r ei s n o i n d u c e dc u r r e n tt,h e o s c i l l a t i oonf t h e
n o t h a m p e r e di n a n Yw a y .
sets
W h e nt h e s w i t c hi s c l o s e da, n i n d u c e dc u r r e n t f l o w si n t h e c i r c u i ta, n dt h i s i n d u c e dc u r r e n t
gagnet.
u p a m a g n e t i fci e l dt h a t o p p o s e tsh e m o t i o no f t h e o s c i l l a t i nm
to
T h i so p p o s i t i o tno m o t i o nc a u s e tsh e o s c i l l a t i o nt os d i e d o w n a n dc a u s e tsh e m a g n e
eventuallycometo rest.
1,02
Lenz'sLaw is an exampleof the Principleof Conservation
of Energy.
s i a g r a mw, h e nt h e s w i t c hi s c l o s e da, n dt h e m a g n e ta p p r o a c h et sh e c o i l ,t h e
F r o mt h e p r e v i o u d
i n d u c e dc u r r e n tf l o w si n a d i r e c t i o nt o p r o d u c ea N o r t hp o l ea t t h e t o p e n d o f t h e c o i lt o r e p e l
t h e i n c o m i n gN o r t hp o l eo f t h e m a g n e tT. h ew o r k d o n ei n o v e r c o m i ntgh i sf o r c eo f r e p u l s i o ni,s
c o n v e r t e di n t o e l e c t r i c aeln e r g ya n d a s i g n i f i c a n tsl ym a l l e a
r m o u n to f h e a t .
W h e nt h e N o r t hp o l eo f t h e m a g n e ti s m o v i n gu p w a r d sa w a y f r o mt h e c o i l ,t h e i n d u c e dc u r r e n t
flowsin sucha way to producea Southpoleon the top of the coilto attractthe departingNorth
poleof the magnet,and the work done in overcoming
thisforceof attraction,is convertedto
e l e c t r i c ael n e r g ya n da s m a l l e ar m o u n to f h e a te n e r g y .
103
INDUCTION
OF ELECTROMAGNETIC
APPLICATIONS
( 1 ) T h e A . C .G e n e r a t o r :
W h e n a c o i li s m a d et o r o t a t ei n a m a g n e t i cf i e l d ,a n e . m . f .i s i n d u c e da c r o . stsh e e n d so f
the coil.
T h ed i a g r a ms h o w sa n a l t e r n a t i n g
curreng
t e n e r a t oar l s oc a l l e da n
A L T E R N A T OTRh.i sc a nb e c o n v e r t e d
t o a d . c .g e n e r a t obr y r e p l a c i ntgh e
T INGS
s l i pr i n g sw i t h S P L I R
S l i pr i n g s
T h e c o i l r o t a t e sr n a c i o c k w i s em o t i o n , a n d t h e m a g n e t i cf i e l d i s d i r e c t e df r o m l e f t t o r i g h t .
A t t h e i n s t a n tw h e n t h e c o i l i s i n a h o r i z o n t apl o s i t i o nd u r i n gi t s r o t a t i o n t, h e R A T Eo f c h a n g eo f
f l u x i s m a X i m u m a, n d t h e m a x i m u me . m . f .i s i n d u c e d .
t h e r ei st h e n i a x i n - r unmu m b e ro f f i e l dl i n e sp a s s i n g
W h e nt h e c o i li s i n t h e v e r t i c apl o s i t i o n
O FF L U XC H A N GiESA M l N l M U N 4a, n dt h e i n d u c e de . m . f i. s z e r o .
t h r o u g hi t , b u t t h e R A T E
l n d u c e de . m . f
T i me
1"04
i
(2) TheTransformer:
T h et r a n s f o r m eirs a d e v i c et h a t i s u s e dt o c h a n g et h e m a g n i t u d e
of analternating
voltage
I t c o n s i s tos f t w o a d j a c e nst e p a r a t e
c o i l so f w i r e ,w o u n do n a n i r o nc o r e .
T h ec o i lo f w i r e t h a t i s c o n n e c t e tdo t h e i n p u tc u r r e n ti,s c a l l e dt h e P R I M A RC
YO I La, n d
t h e o u t p u ti s o b t a i n e df r o m t h e S E C O N D A R
CY
OIL.
)
Power
Station
vP/
\
)
W h e na n a l t e r n a t i ncgu r r e n ti s a p p l i e da c r o s st h e p r i m a r yc o i l ,t h e c u r r e n t h a t f l o w s
t h r o u g hi t , w i l l p r o d u c ea n a l t e r n a t i nm
g a g n e t i fci e l d .T h i sa l t e r n a t i nm
g a g n e t i fci e l d
, i l l b e s t r i k i n gt h e c o i l so f t h e s e c o n d a rcyo i l .H e n c et h e s e c o n d a r y
a r o u n dt h e p r i m a r yw
c o i li s e x p o s e tdo a n a l t e r n a t i nogr c h a n g i nm
g a g n e t ifci e l d
'By
F a r a d a y 'Lsa w ,a n a l t e r n a t i neg. m . f .w i l l b e i n d u c e di n t h e s e c o n d a rcyo i l .
For an IDEALtransformer,there are no power losses.
= O U T P UP
H e n c el NP U TP O W E R
TOWER
L e tt h e p r i m a r yv o l t a g e= V pa n d p r i m a r yc u r r e n t= l p
L e tt h e s e c o n d a rvyo l t a g e= V , a n ds e c o n d a rcyu r r e n t= l ,
S i n c ep o w e r - V l , t h e n f o r n o p o w e r l o s s e sV o l o= V r l ,
T h e p r i m a r yv o l t a g ei s p r o p o r t i o n a tl o t h e n u m b e r o f t u r n s o n t h e p r i m a r yc o i l
i.e.
Vp x N, and si"mi"Iarly V, x IV,
v,- n, _',
w
1/,
105
Ip
ER
IM PORTANTREATURESOF THE TRANSFORM
(1) lron Core:
T h em a g n e t i fci e l dc o u p l i n gt h e c i r c u i t s ( p r i m aar ny d s e c o n d a rcyo i l s c) a nb e i n a i r ,b u t i s
sf
t h) e, c o r e ,i n w h i c ht h e f i e l dc a nb e t h o u s a n d o
c aterial(IRON
u s u a l l yi n a f e r r o m a g n e t im
timesgreaterthan it would be in air, makingthe transformerefficientand small.
(2) Laminations
of the core:
W h e na n a l t e r n a t i ncgu r r e n tp a s s etsh r o u g ht h e c o i l s ( w h i cahr ew o u n do n t h e i r o nc o r e ), a n
a l t e r n a t i nm
g a g n e t i c f i e lids s e t u p a r o u n dt h e c o r ew h i c hc a u s e cs u r r e n t sc a l l e dE D D Yc u r r e n t s
to flow in the core.
o f h e a te n e r g yw h i c hw o u l dl e a dt o a d e c r e a s ien t h e
T h e s ee d d yc u r r e n t sc a u s et h e p r o d u c t i o n
energyinput is converted
of the transformersincenow someof the electrical
overallefficiency
to heatenergywhichwould.bewastedenergy.
y u t n o t t o t a l l ye l i m i n a t e d ) b y
o f E D D yc u r r e n t sc a nb e r e d u c e ds i g n i f i c a n t(l b
T h i sp r o d u c t i o n
l a m i n a t i ntgh e c o r e ,i . € .b y m a k i n gi t f r o m t h i n s h e e t so f m e t a lw h i c ha r et h e n a d h e r e d
t o g e t h e rw i t h a t h i n f i l m o f i n s u l a t i nggl u e .T h e s et h i n f i l m so f g l u ep r e v e n t h e c o n d u c t i o on f
e d d yc u r r e n t sf r o m o n e l a m i n a t i o tno t h e o t h e r .T h i sd e s i g ni s m u c hm o r ee f f i c i e ntth a n h a v i n g
s etal.
t h e c o r em a d ef r o m o n e s o l i db l o c ko f c o n t i n u o um
(3) Efficientcore design:
T h ec o r ei s m a d eo f E s h a p e dl a m i n a t i o ntsh a t a r e i n t e r l e a v esdo t h a t t h e m a g n e t i fcl u x d o e s
i s e l i m i n a t e dT.h i sc o r ed e s i g ni s
n o t p a s st h r o u g ht h e a i r a t a l l h e n c em a g n e t i fcl u x l e a k a g e
e x p l a i n ea
d sf o l l o w s :
Shell-CoreTransformers
(4).
T h e m o s tp o p u l a ra n d e f f i c i e nttr a n s f o r m ecro r ei s t h e S = i r i a * = i = a s i l l u s t r a t eidn f i g u r e
sectionsof metal.Thesesections
of I- =:=*i:=i=*L:r:*
As shown,eachlayerof the coreconsists
s .h el a m i n a t i o nasr e i n s u l a t e fdr o m e a c ho t h e ra n d
a r e b u t t e dt o g e t h e rt o f o r m t h e l a m i n a t i o n T
then pressedtogetherto form the core.
106
LAMINATEDCORE
.Fii
E AND
LAMINATIONS
TRANSFORMW
E RI N D I N G S
A s s t a t e da b o v e t, h e t r a n s f o r m ecr o n s i s tos f t w o c o i l sc a l l e dW I N D I N Gw
Sh i c ha r ew r a p p e d
a r o u n da c o r e .T h et r a n s f o r m eor p e r a t e sw h e na s o u r c eo f a c v o l t a g ei s c o n n e c t e tdo o n e o f
t h e w i n d i n g sa n d a l o a dd e v i c ei s c o n n e c t e tdo t h e o t h e r .T h ew i n d i n g t h a it s c o n n e c t e tdo i h e
s o u r c ei s c a l l e dt h e P R I M A RW
Y I N D I N GT.h ew i n d i n gt h a t i s c o n n e c t etdo t h e l o a di s c a l l e dt h e
S E C O N D AW
RY
I N D I N G( N
. o t e I: n t h i sp a r tt h e t e r m s" p r i m a r yw i n d i n g a" n d " p r i m a r ya" r eu s e d
i n t e r c h a n g e a btl hy e
; t e r m : " s e c o n d a rwy i n d i n g "a n d " s e c o n d a r ya"r ea l s ou s e d
i n t e r c hnag e ab l y . )
F i g u r e( 5 )s h o w sa n e x p l o d e dv i e wo f a s h e l l - t y pter a n s f o r m e T
r .h ep r i m a r yi s w o u n di n l a y e r s
d i r e c t l yo n a r e c t a n g u l acra r d b o a r fdo r m
E. t-*rtdt{ATtoH
IHSULATING
PAPER
I LAMINATION
1.07
l n t h e t r a n s f o r m esr h o w ni n t h e c u t a w a yv i e wi n f i g u r e( 6 ) ,t h e p r i m a r yc o n s i s tos f m a n y
l i r e .T h ew i r ei s c o a t e dw i t h v a r n i s hs o t h a te a c ht u r n o f t h e w i n d i n gi s
t u r n so f r e l a t i v e lsym a l w
i n s u l a t e fdr o m e v e r yo t h e rt u r n . ! n a t r a n s f o r m edr e s i g n e fdo r h i g h - v o l t a gaep p l i c a t i o nssh, e e t s
g a t e r i a sl ,u c ha s p a p e ra, r ep l a c e db e t w e e nt h e l a y e r so f w i n d i n g s t op r o v i d e
o f i n s u l a t i nm
a d d i t i o n ai nl s ul a t i o n .
PAPER INSULATIOH
LAMINATED
CORE
LEADS
I
/
PRIMARY
WINDING
SECONDARY WINDING
. he
W h e nt h e p r i m a r yw i n d i n gi s c o m p l e t e lwy o u n d ,i t i s w r a p p e di n i n s u l a t i npga p e ro r c l o t h T
s e c o n d a rwyi n d i n gi s t h e nw o u n do n t o p o f t h e p r i m a r yw i n d i n gA. f t e rt h e s e c o n d a rwyi n d i n gi s
. e x t t, h e
s e c t i o nosf t h e i r o nc o r ea r e .
c o m p l e t e , ,t iot o i s c o v e r e dw i t h i n s u l a t i npga p e r N
i n t oa n da r o u n ct ih e w i n d i n g as ss h o w n .
inserted
y r o u g h ot u t t h r o u g ha h o l ei n t h e e n c l o s u roef t h e
T h el e a d sf r o m t h e w i n d i n g as r en o r m a l l b
t r a n s f o r m e rS. o m e t i m e st e, r m i n a l sr n a yb e p r o v i d e do n t h e e n c l o s u rfeo r c o n n e c t i o nt so t h e
w i n d i n g sT. h ef i g u r es h o w sf o u r l e a d st,w o f r o m t h e p r i m a r ya n dt w o f r o m t h e s e c o n d a r y .
T h e s el e a d sa r et o b e c o n n e c t e tdo t h e s o u r c ea n d l o a d ,r e s p e c t i v e l y .
108
INDUCEE
D M FT U T O R I ASLH E E T ,
( 1 ) A r o d A B S O c m . l o nr go t a t e si n a u n i f o r mm a g n e t i fci e l dw i t h p o i n t
A a c t i n ga st h e p i v o t .
T h er o t a t i o ni s p e r p e n d i c u l a rtthoe f i e l dl i n e sa n dt h e f r e q u e n coyf r o t a t i o ni s
5 r e v o l u t i o npse rs e c o n dl .f t h e m a g n e t ifcl u xd e n s i t yi d O . 3 Td e t e r m i n teh e e m f i n d u c e d
b e t w e e nt h e e n d so f t h e r o d . ( A n:s3 V )
( 2 ) A 5 o c o i lo f 1 0 0 t u r n sa n dd i a m e t e 6
r c m i s p l a c e db e t w e e nt h e p o l e so f a m a g n e ts o t h a t
t h e f l u xi s m a x i m u mt h r o u g hi t sa r e a W
. h e nt h e c o i li s s u d d e n l rye m o v e df r o m t h e f i e l d ,
a c h a r g eo f 1 x l - 0 4 C f l o w st h r o u g ha 5 9 5 Og a l v a n o m e t ecro n n e c t e tdo t h e c o i l .
C o m p u t eB b e t w e e nt h e p o l e so f t h e m a g n e t .
( 0 . 2 1 T ) l HU
i nst e;V = r n = T ,a n d1 -
f t
( 3 ) A f l a t c o i lo f r a d i u s8 c mh a s 1 5 0l o o p so f w i r e .l t i s p l a c e di n a s i n u s o i d a l
m a g n e t ifci e l d
t h a t v a r i e sa t a r a t eo f 1 . 6 T s -i 1
n s u c ha w a y t h a t t h e m a x i m u mf l u xg o e st h r o u g hi t . F i n d
t h e a v e r a g e m f i n d u c e db e t w e e nt h e t e r m i n a los f t h e c o i l .
(ANS;4.82v)
( 4 ) A c o p p e rd i s co f l d c m r a d i u si s r o t a t i n ga t 2 0 r e v i s a b o ui t sa x i sa n d
w i t h i t sp l a n e
perpendicula
t or a u n i f o r mf i e l dw i t h B =0 . 6 T W
. h a t i s t h e p o t e n t i adl i f f e r e n c b
eetween
t h e c e n t r ea n dt h e r i m o f t h e d i s c .
r r- e
( A n s: 0 . 3 8 V )f H i n t ;I r e v o l u t i o n
= a r e ao f' O"I S C = T t l)" , d ' A:
--v
d,
ttO. Ol
1 ATee Of diSC]
( 5 ) H o wm u c hc h a r g ew i l lf l o wt h r o u g ha 2 0 0 0g a l v a n o m e t ce or n n e c t e d
t o a 4 0 0 0c i r c u t a r
c o i lo f 1 0 0 0t u r n sw o u n do n a w o o d e ns t i c k2 c mi n d i a m e t e
r , i f a m a g n e t i fci e l dB =
0 . 0 1 1 - 3pTa r a l l et lo t h e a x i so f t h e s t i c ki s d e c r e a s esdu d d e n l y t o
zero.
( A n s 5: . 9 p C )
(6)A solenoi6
d 0 c ml o n gh a s5 O 0 C
t u r n sa n d i s w o u n do n a n I R O Nr o d o f 0 . 7 5 c mr a d i u s .
F i n dt h e f l u xd e n s i t ya n df l u x t h r o u g ht h e s o l e n o i w
d h e nt h e c u r r e n itn i t i s 3 4 . T h e
r e l a t i v ep e r m e a b i l i toyf i r o n i s 3 0 0 .
( b )T h ef l u xi s r e d u c e d
t o a v a l u eo f l m W b i n a t i m e o f 0 . 0 5 0 s. F i n dt h e i n d u c e de m f i n
t h es o l e n o i d .
( A n s: 9 . 4 2 ;T1 . 7 m W b
; 67V)
109
for previoustutorial
Solutionguidelines
o n e .H e n c ef o r e a c hr e v o l u t i o n
L . T h es h a p et h a t t h e r o d c u t so u t a s i t r o t a t e si s a c i r c u l a r
t h e r a d i u sr i s t h e l e n g t ho f t h e
t h e a r e ao f f i e t dl i n e st h a t a r ec u t i s e q u a lt o n r 2 w h e r e
rod.
o f t h e i n d u c e de m f ,i t i s n e c e s s a r y tdoe t e r m i n et h e r a t e
e agnitude
ln determiningthm
of Area)'
at whichthe flux is cut whichin this caseis B x (rateof change
w h i c hw o u l db e e q u a lt o t h e
T h e r a t eo f c h a n g eo f a r e ai s t h e a r e ac u t o u t p e r s e c o n d ,
o f r e v o l u t i o npse r
p r o d u c to f t h e a r e ao f t h e c i r c l e ( i . e1.r e v o l u t i o na)n dt h e n u m b e r
second.'
is equalto 600 o, the radiusis 0.03m,N= 100Q= l- x 10-4c'
2. Thetotal resistance
e x p e r i e n c easc h a g ei n
T h e m o v e m e not f t h e c o i lo u t o f t h e f i e l dm e a n st h a t t h e c o i l
causea
s n e m ft o b e
t h e n u m b e ro f m a g n e t i c f i e lldi n e sc u t t i n gi t , a n dt h i sc h a n g e
L 'asw )
i n d u c e di n t h e c o i l ( F a r a d a Y
a n d h e n c ea c u r r e n tw i l l
T h ei n d u c e de m f i s a p p l i e da c r o s sa t o t a l r e s i s t a n coef 6 0 0 0
f l o w t h r o u g ht h i s r e s i s t a n c e '
m o v e d ,h e n c ei t i s
w e d o n o t k n o wt h e v a l u eo f t h e t i m e f o r w h i c ht h e c o i lw a s
n e c e s s a r y tsoe t u p a n e q u a t i o nw h e r et h e t i m e c a nb e e l i m i n a t e d '
F r o m O h m ' sL a w , ,:
NBAIt
V I n d u c e d " e m:f
,4\
tr/
l
n: T
- g'e
A r s o c u r r e n t r - T o t attme
'.|'charg
t
Q)
(1) and (2)
Thecurrent ts the sametnboth equattonsso we c&nequate
NBAI. n
v
lt
:
R
t,
NBA
;-Q
110
henceBcanbefound
3 . T h ei n d u c e e
dm
. f i . sg t v e n b yE _ -
#:
-rO#
wttere#: r.6Ts-l
4 . T h eh i n t i s a l r e a d yg i v e ni n t h e q u e s t i o n
5. Applythe samemethodas in question2. Thewood doesNOTaffectthe Flux
6. ln usingthe formulafor Fluxdensityincludethe relativeperrrreability
of iron, so the
f o r m u l an o w b e c o m e B
s -
V oV r n I
lJT
2
lVl.of,ufe
I I M P O R T A NETQ U A T I O N S
(1)A sinusoidal voltage or current i.srepresented by i x - xssinat
(2)Ther . m.s.ualueof an a.c.ts Ir.^.r.:
'ff
(3)Gainof an inverttngamplifter ts gtvenby:Gatn+
L
( 4 ) G a t n o f I N O N - I N V E R T I N Ga m p l i f t e r t s g t v e n b y G a i n - t *
Rr
&
$ ) f h e G a t n o f a c o m b t n a t i o no f C a s c a d e dA m p l i f t e r s i s A A r A z A 3. . .
"Everythingis determined... by forces over which we have no
control.lt is determinedfor the insecfsas well as for the star.
Human beings, vegetables,or cosmic dusf - we all dance t.oa
mysteriousfune, intonedin the distanceby an invisiblepiper."
- Albert Einstein
112
A L T E R N A T I NC
GU R R E N T
A n a l t e r n a t i ncgu r r e n it s o n e i n w h i c ht h e d i r e c t i o cn h a n g eps e r i o d i c a l l y .
s i nu s o i d Ia
s qu ar e w a v e a . c .
rrnllroo
o r c ur r e n t
VALUE
o f t h e c u r r e n ot r v o l t a g ew, h i c hi s
I n b o t hc a s e s t, h e r ei s a m a x i m u mv a l u eo r P E A K
u s u a l l yw r i t t e na s I n . . uo ,I V r u ^ .
T h ef r e q u e n coy f t h e a . c .i s t h e n u m b e ro f c y c l e p
s e rs e c o n da n di s m e a s u r eidn H e r t z( H z ) .
v a l u eo f a n a . c .i st h a tf r a c t i o no f t h e p e a kv a l u et h a t w o u l dd i s s i p a t e
T h eR O O TM E A NS Q U A R E
h e a ta t t h e s a m er a t ea sa s t e a d yd . c .t h r o u g ha g i v e nr e s i s t o r .
H e n c ef o r a n yr e s i s t o r ' R t' ,h e c u r r e n dt i s s i p a t ebdy a s t e a d yd . c .i s g i v e nb y P - l z d . r .aRn dt h e
e q u i v a l e npto w e rd i s s i p a t ebdy a n a l t e r n a t i ncgu r r e n t ,i s P - I ? . ^ . r R .
r h e nt h e S I N U S O I D A
aL
. c .p a s s etsh r o u g hi t , i s t h e
T h ep o w e rd i s s i p a t ebdy t h e r e s i s t ow
A V E R AE
GP O W E R .
AVERAGE POWER=
MAXiMUM POWER
t2p
+
( S I N U S O I D cAaLs e )
T h i sa v e r a g ep o v / e i , a l s oe q u a l t o I i ^ r R .
,.)
HencellrnrR :
lmK
-------\
t--
z
#-,
Ir^,
113
:lm
<-
,tz
LEARN
S i m i l a rV
l yr * r : ' r ,
+-
LEARN.
Another useful f ormula is i Vr^-I--n
, I r ^ , -jry'+
i O-
V^I^
t:
AveragePower
In the caseof the squarewave from the abovegraph,the magnitudeof the currentand
voltageis constant,and the power dissipatedis constant. Hencefor the squarewave shown
above,the r.ms.valuesof both voltageand currentwill be the sameas the steadymaximum
value.
Sincean alternating
voltageor currentcanbe represented
by a sinewave,the voltageor
currentat any instantcanbe expressed
as :
V:Vmsinat
and I - I^sinat,
where a is the AN GULAR FREQUEN CY of the pd. or current.
The angularfrequencyu = 2nf, where'f is the frequencyof the currentor voltage.
F o re x a m p l et h e m a i n ss u p p l yi n T r i n i d a dh a sa f r e q u e n c o
y f 6 0 H z .H e n c et h e a n g u l a r
frequencyu,ris givenby ul = 2n x 60 = 1,20nrad.s-1.
( R e c a tl h
l a t f o r t h e s i n ew a v e ,1 c y c l er e p r e s e n t2sn r a d i a n s ) .
W h e na l t e r n a t i ncgu r r e n t sa n dv o l t a g e a
s r e m e a s u r e dt h, e i n s t r u m e n tm
s easure
the r.m.s.
v a l u e sr a t h e rt h a nt h e p e a kv a l u e s .
1.1.4
Advantagesof transmissionof currentsusingalternatingcurrentsand highvoltages:
(1) Alternatingcurrentsareeasierand cheaperto producethan directcurrents.The
alternating
currentis producedby rotatinga coilin a magneticfield and the mechanical
energyfor rotation comesfrom steamwhich is producedfrom the heatingof water with
naturalgasbeingthe fuel.
Directcurrentsare producedfrom chemicalreactionsin a batterywhich is relatively
expensiveto manufacture.
(2) The alternating
voltagescan be increased
or decreased
as requiredwith minimum
energywastageby meansof a transformer,whereasthe direct voltagehasto be altered
by meansof resistors
whichhavesignificant
energydissipation.
(3) Thereis significantlandpollutionwith the disposalof batterieswhereaswith a.c.
generationthere is the issueof fossilfuel emissions.
(4) When alternating€u rrentsare transmittedat high voltages,there is significantlyless
powerlossesin the cablesas comparedto low voltagetransmission.
Thisis explainedasfollows:The currentpassing
throughthe cablesfor a particular
street or district, is determinedby the power requirementof that district.lf a district
hasan averagepowerrequirementofsay 2MW, and the resistance
ofthe overhead''
cablesis 0.20.
Whenthe a.c.is transmittedat 240V,the currentthroughthe cableswill be:
P ower
Cut'rent - Voltag e
2,000,000
.
= 8333/, and the power lossesin the cableswould be l2R,whichwould give
I ='#
a
a value of 1.39x 106W,which is in fact almostthe power requiredby the district.
Howeverif highvoltagesare usede.g.24000V,the currentin the cableswill be
2'000'000
= 83,4and the oowerlossin the cablesis 832xo.2= 1378W.
f 24000
'
The highvoltagescanbe steppeddown to 110Vor 220Vby meansof a transformeron reaching
the district
115
QUESTIONS
( 1 ) I n t h e f o l l o w i n gw a v e f o r m sr e p r e s e n t i nagn a l t e r n a t i ncgu r r e n tt,h e p e a kv a l u eo f t h e
a . c .i s 5 A .C a l c u l a t eh e r . m . sc. u r r e n it n e a c hc a s e .
c ur r en t
(a)
currenI
A N S: 3 . 5 4 A ,3 . 5 4 A 5
, A.
(c)
( 2 ) A v o l t m e t e r e a d s1 1 0 Vw h e n i t i s c o n n e c t e d
a c r o s st h e t e r m i n a l so f a s i n u s o i d aplo w e r
W
r
i
t
e
d
o w n t h e e q u a t i o nf o r t h e i n s t a n t a n e o u s
s u p p l yt h a t h a s a f r e q u e n c yo f 6 0 H 2 .
v o l t a g ep r o v i d e db y t h e s o u r c e .
( 3 ) A s i n u s o i d aal l t e r n a t i n gc u r r e n ti n a 4 0 O r e s i s t o rp, r o d u c e sh e a t a t a r a t e o f 7 5 0 W .
D e t e r m i n teh e m a x i m u mv a l u e so f v o l t a g ea n d c u r r e n t.
1.16
(4) Determine
t h e r . m s .v a l u eo f t h e c u r r e n ti n ( i )a n dt h e
r . m . sv. o l t a g ei n ( i i )
V
A N S :4 4 , 1 1 0 V
( 5 ) W h e na n a . c .s u p p l y . o1f 2 0 V , . *i s, c o n n e c t e tdo a c i r c u i t
i n w h i c ht h e r e is a fuse in series
w i t h a l o a d ,t h e f u s eb r e a k st h e c i r c u i tw h e nt h e c u r r e n jtu s t e x c e e d gs A
r . m . s .l f t h e a . c .s u p p l y
i s r e p l a c e db y a 8 0 V d . c .s u p p l y w
, h a t m u s t b e t h e m a x i m u mc u r r e n tp a s s i n through
g
the fuse
j u s t b e f o r ei t b r e a k st h e c i r c u i t ?
( 6 ) T h es i n u s o i d a
l l t a g eV r a p p l i e da c r o s sa r e s i s t o r ' R ' p r o d u chees a ta t
vo
a m e a nr r t u p . ' l f
t h e s i n u s o i d aal. c .i s n o w r e p l a c e dw i t h s q u a r ew a v ea . c .a s s h o w ni n ( i i )
with the same
p e a kv a l u ea s b e f o r e w
, h a t i s t h e n e w r a t eo f h e a tp r o d u c t i o n ?
v-
IT7
Solutions
( 1 ) I n t h e c a s eo f t h e s i n u s o i d aglr a p h t, h e r . m . s c. u r r e n ti s g i v e nb y :
r-*--'^y-:-3.s4A
rrms-
O-O-rrr
( b )T h eg r a p hi n ( b ) i s a l s oc y c l i cw i t h e a c hc y c l eh a v i n ga c o n t i n u o u5s A r e g i o nf o l l o w e db y a
zeroregion
l f t h i s c u r r e n tw a sa l l o w e dt o p a s st h r o u g ha r e s i s t o rt ,h e p o w e rd i s s i p a t ebdy t h e r e s i s t o r f o r
t h e c o n s t a nct u r r e n to f 5 A w o u l dh a v eb e e nP = l 2 R= 2 5 R .
r o u l dh a v eb e e n0 .
D u r i n gt h e i n t e r v aol f z e r oc u r r e n tt,h e p o w e rd i s s i p a t ebdy t h e r e s i s t ow
IzR + 0
The average power dtsstpated -
Z
25R
:
z
B yd e f i n i t i o nt ,h i sw o u l dh a v eb e e nt h e p o w e rd i s s i p a t ebdy t h e r . m . s c. u r r e n t
H e n cIe? ^ r R: ' +
t
i l , m s :+ - 3 . 5 4 A
12
( c )l n t h i s p a r t ,e a c hc y c l eo f t h e c u r r e n th a sa c o n s t a npt o s i t i v e5 A . f o l l o w ebdy a c o n s t a n t
negative5A.
r d u r i n gt h e p o s i t i v eh a l fc y c l ei s
T h ep o w e rd i s s i p a t ebdy a r e s i s t oR
Pt:
I z R - 5 2 R- 2 5 R
r is
D u r i n gt h e n e g a t i v eh a l fc y c l et,h e p o w e rd i s s i p a t ebdy a r e s i s t oR
P z : 1 2 R- ( - 5 ) 2 R - 2 5 R
Average power dtssipated - Pt * Pz :
25R + 25R
T
T h i si s t h e p o w e rd i s s i p a t ebdy t h e r . m . s c. u r r e n tb y d e f i n i t i o n ,
Hence I?*rR : 25Ri Ir^s : 5A
118
:
50R
Z
: 25R
(2)
V-
V^orsinl)t : V*o* stnZrcft
We first needto find the valueof V.r".
T h e 1 L O Vr e c o r d e db y t h e v o l t m e t e ri s i n f a c t h e r . m . s v. o l t a g eA. l l a . c .m e a s u r i nign s t r u m e n t s
a r e d e s i g n e tdo g i v et h e r . m . s v. a l u e s .
Recallvro,
,t :
.
v^o*
fi;
h, e n c e
V ^ o ,- \n x 110V : 1.55.6V
T he g eneral equati"onbecomesV -
155.6 sinZrcx 60t
V : 155.6sin 12Ont
( 3 )T h eh e a td i s s i p a t ebdy t h e r e s i s t oirs o b t a i n e df r o m t h e f o r m u l a
P - I?^rR;
500 : I?^, x 40 i lrms :
I*o*:lr^rx12:3.54
x ..12=
500
:3.54A
40
54.
V*o*: I^o, x R - 5 x 4 0 : 200v
( a ) F o rt h e p o s i t i v eh a l fc y c l et,h e m a x i m u mc u r r e n ti s 8 A . I f t h e c u r r e n tw a sa c o m p l e t es i n e
wave,the powerdissipated
by a resistorR would havebeen
12D
P :'^7"
z
wave the power d.i"sstaptedwouldbe
P; Howaver f or a HALF si"ne
IhorR
B yd e f i n i t i o nt ,h e p o w e rd i s s i p a t ebdy t h e r . m . s c. u r r e n tw o u l db e e q u a lt o t h i s l a t t e rv a l u e ,
I?.*.r.R:
IlnorR
; hence Ir.^.r. :
119
I?,o* - I*o*
B
:
4
2
t:
44
r for a
F o rt h e v o l t a g eg r a p h t, h e p o w e rt h a t w o u l dh a v eb e e nd i s s i p a t etdh r o u g ha r e s i s t oR
F U L Ls i n eg r a p hw o u l dh a v eb e e n :
Io --
v"1n'
:zR
is
H e n c ef o r a H A L Fs i n eg r a p ht h e p o w e r t h a t w i l l b e d i s s i p a t e d
, - v'?''n'
I
4R
Vl,n,
:P:
Bydefinltiort,f
L { e n cver ^ r . - v r y ' - 2 :
22
V|o,
*
- 1,i,ov
. ence
q u e s t i o nT.h ef u s ew i l l b r e a kw h e n e v etrh e c u r r e n et x c e e d8sA r . m . s H
( 5 )T h i si s a T R I C K
t h e s u p p l yv o l t a g ea, c u r r e n to f 8 A r . m . s i.s r e q u i r e dt o m e l tt h e f u s ew i r e .
R E G A R D L EoSf S
a
s g i v e nb y
( 6 )I n t h e c a s eo f t h e s i n u s o i d a l v o l t a tghee, p o w e rd i s s i p a t ei d
'Dl
Vior
zR
F o rt h e s { u a r ew a v e ,t h e v o l t a g eh a sa C O N S T A NMTA G N I T U DoEf V . . " , a n d i n t h i sc a s et,h e
p o w e rd i s s i p a t eids g i v e nb Y
'D2 - _
V
v t2
nQx
R
Hence P2:2P1
124
(Ans.2P)
SEMICONDUCTORS
T h ee l e c t r o n cs o n t a i n e d
i n s o l i d sa, r ec o n t a i n e d
i n v a r i o u sb a n d so f e n e r g yl e v e l s .
T h eh i g h e sbt a n di s c a l l e dt h e C O N D U C T I OBN
A N Da n d i n m e t a l st,h e r ea r ef r e ee l e c t r o ntsh a t
m o v ew i t h i nt h e c o n d u c t i obna n d .
I n i n s u l a t o r es l, e c t r o nasr eh e l di n a l o w e rb a n dc a l l e dt h e v a l e n c e
b a n d ,a n dt h e r ei s a
r e l a t i v e llya r g eg a pb e t w e e nt h e v a l e n c eb a n da n dt h e v a c a n ct o n d u c t i o b
na n d .
H e n c ef o r i n s u l a t o rtsh e e l e c t r o n m
s u s tb e s u p p l i e d
w i t h a r e l a t i v e llya r g ea m o u n to f e n e r g y t o
j u m p i n t ot h e c o n d u c t i obna n di n o r d e rt o p a r t i c i p a ti en e l e c t r i c ac lo n d u c t i o n .
I n s e m i c o n d u c t osr su c ha s s i l i c o nt,h e e n e r g yl e v e ld i f f e r e n c b
e e t w e e nt h e v a l e n c eb a n da n d
t h e c o n d u c t i obna n di s m u c hs m a l l etrh a nt h a tf o r i n s u l a t o r as n
, dh e n c et h e ya r ea b l et o
c o n d u cw
t h e no n l ya s m a l lp o t e n t i adl i f f e r e n c ies a p p l i e da c r o s st h e c r y s t a l .
Conductiob
nand
P a r t l yf i l l e dw i t h c h a r g e
Conductiob
nand
E m p t yo f c h a r g ec a r r i e r s
Conductiob
nand
A l m o s te m p t y { c o n t a i nos n e
o r t w o c h a r g ec a r r i e r s
c ar r ie r s
V a l e n c eb a n d ( f i l l e d )
V a l e n c eb a n d ( f i l l e d )
E n e r g yb a n d sf o r M E T A L
E n e r g yb a n d sf o r I N S U L A T O R
E n e r g yb a n d sf o r
SEMICONDUCTOR
T h eg a pb e t w e e nt h e v a l e n c e
b a n da n dt h e c o n d u c t i obna n di s c a l l e dt h e F O R B I D D E
GN
Ap.
1.21
Conductionin Semi-conductors:
l n t h e c r y s t aol f s e m i c o n d u c t i nmga t e r i a li,f a n e l e c t r o ne s c a p efsr o m i t s p o s i t i o ni n t h e l a t t i c e ,
a v a c a n c yi s l e f t b e h i n da, n d s i n c et h i s p o s i t i o nw a si n i t i a l l yn e u t r a lt,h e l o s so f t h e e l e c t r o nw i l l
l e a v ei t w i t h a p o s i t i v ec h a r g ea n d i s c a l l e da H O L E .
T h i sh o l ea c t sa s a v a c a n c fyo r a n yo t h e re l e c t r o na n dw h e n i t i s f i l l e d ,a n o t h e rh o l ew i l l b e
createdwherethe other electronwould haveleft.
s o v ei n o n e d i r e c t i o nh, o l e sm o v ei n
A l o n ga n ys t r a i g hlti n e ,i t w i l l b e s e e nt h a t w h i l ee l e c t r o n m
t h e o p p o s i t ed i r e c t i o n .
CARRIER.
CHARGE
as a POSITIVE
The holecanthereforebe considered
l n t h i sc a s et h e n u m b e ro f h o l e sw i l l b e e q u a tl o t h e n u m b e ro f m o b i l ee l e c t r o nasn ds i n c ea l lo f
s o n o t c o m ef r o m a n ye x t e r n asl o u r c et,h e s e m i c o n d u c t oi srs a i dt o b e
t h e c h a r g ec a r r i e r d
INTRINSIC.
s i l l e s c a p ef r o m t h e i r
o f a s e m i c o n d u c t oi sr i n c r e a s e dm, o r ee l e c t r o nw
W h e nt h e t e m p e r a t u r e
n i l l i n c r e a s ae n d r e s i s t a n cwei l l
s i t e sa n d m o r eh o l e sw i l l b e c r e a t e dh, e n c e t h ec o n d u c t i o w
decrease.
. h ea m o u n t
e st h e i rc o n d u c t i o ins t o o l i m i t e dT
I n t r i n s i sc e m i c o n d u c t oar sr e o f l i t t l ei m p o r t a n c a
o f c o n d u c t i o tna k i n gp l a c ec a nb e i n c r e a s egdr e a t l yb y i n c r e a s i ntgh e n u m b e ro f a v a i l a b l he o l e s '
or electrons.
T h i sc a nb e d o n eb y a d d i n ga s m a l la m o u n to f a n o t h e rm a t e r i atlh a t c a ns u p p l ya d d i t i o n a l
a n dt h e m a t e r i atlh a t i s a d d e di s c a l l e da n
h o l e so r e l e c t r o n sT.h i si s r e f e r r e dt o. a s D O P I N G
IMPURITY.
e m i c o n d u c t ba rn dt h e r e
t hr a t i s p r o d u c e db y d o p i n gi s c a l l e da n E X T R I N SsI C
T h es e m i c o n d u c t o
aretwo types,i.e.n-typeand p-type.
122
The n-typesemiconductor:
The 'n' refersto NEGATIVE
type.When a siliconcrystalis dopedwith a penta-valent
element
s u c ha s p h o s p h o r o uosr a n t i m o n yt,h e i m p u r i t ya t o m r e p l a c eosn e o f t h e s i l i c o na t o m si n t h e
c r y s t a lS. i l i c o nh a sa v a l e n c eo f f o u r a n d s h a r e sf o u r e l e c t r o nw
s i t h n e i g h b o r i nagt o m s .W h e n
t h e p h o s p h o r o uast o m r e p l a c eas s i l i c o na t o m ,f o u r o f i t sv a l e n c ee l e c t r o nw
s i l l b e u s e di n
c o v a l e nbt o n d i n gT. h ef i f t h e l e c t r o nw i l l b e a b l et o d r i f t i n t h e c o n d u c t i o b
n a n da n dt a k ep a r ti n
c o n d u c t i o na,n dt h e i m p u r i t ya d d e di s c a l l e da D O N O Ra s i t d o n a t e se l e c t r o nfso r c o n d u c t i o n .
The majority chargecarriersin the n-type materialare electrons.
T h ed o p e ds e m i c o n d u c t iosrh o w e v e sr t i l le l e c t r i c a lnl ye u t r aal s n e i t h e r t h es i l i c o n o r t h e
d o n o ra t o mh a da n yn e t c h a r g eos n t h e m .
The p-typesemiconductor:
The 'p' refersto POSITIVE
type and the majoritychargecarriersare holes.
T h i si s m a d eb y d o p i n gs i l i c o n
w i t h a t r i - v a l e ni tm p u r i t ys u c ha s B o r o no r A l u m i n i u m .
A s i l i c o na t o m i s r e p l a c e db y t h e t r i v a l e n at t o ma n dt h r e eo f t h e v a c a n c i el se f t b y t h e m i g r a t e d
s i l i c o nw i l l e db e f i l l e db y t h e t h r e ev a l e n c ee l e c t r o n isn t h e i m p u r i t yT. h ef o u r t hv a c a n c w
y i l J.
t h e na c L e p a
t n e l e c t r o nf r o m t h e v a l e n c eb a n d ,t h u sc r e a t i n ga h o l ei n t h e v a l e n c eb a n d .T h e
v a l e n c eb a n dw i l l t h e n b e a b l et o c o n d u c w
t i t h t h e p r e s e n c oe f h o l e si n i t .
T h e i m p u r i t yi n t h i s c a s ei s c a l l e da n A C C E P T O R .
Material
N u m b e rd e n s i t yo f
Resistivity
C h a r g eC a r r i e r s
(r-')
(o.m)
Metal
9 x 1028
2 x 10' 8
Intrinsic
Semiconductor
3x103
Extri nsic
S e m i c o n d u c t o| r1 x 1 0 " t o 1 x 1 O 2 a 5 x 1 0 8
L23
LEARNTHE NUMBERS
T H EP . NJ U N C T I O N
A p - nj u n c t i o ni s a s e m i c o n d u c t corry s t at lh a t i s d o p e di n s u c ha w a yt h a t o n e r e g i o ni s n - t y p e
a n dt h e a d j a c e nrte g i o ni s p - t y p e .
T h i sc a nb e v i s u a l i z eadsa b l o c ko f n - t y p em a t e r i abl e i n gp r e s s e d
a g a i n sat b l o c ko f p - t y p e
material.
C
p
C
Fo
o
on
o
C
clo
<tO
n
-+t+-
O
O
At the interface
o f t h e p - nj u n c t i o nt h e r ei s a n i n i t i a d
l i f f u s i o no f e l e c t r o nfsr o m t h e n - t y p e
, n d a d i f f u s i o no f h o l e sf r o m t h e p - t y p et o t h e n - t y p e .
t o t h e p - t y p ea
T h em o t i o no f b o t ht h e H o l e sa n dt h e e l e c t r o npsr o d u c ea D I F F U S I OCNU R R E Nl 6T; 6 t h ai ts
c o n v e n t i o n a ldl yi r e c t e df r o m p t o n , i . e .c o n v e n t i o n ac lu r r e n ti s t h e f l o w o f p o s i t i v ec h a r g e .
W h e na n e l e c t r o nf r o m t h e n - s i d ec r o s s etsh e j u n c t i o na n d e n t e r st h e p - s i . d ei ,t f i x e si t s e l f
i n t oa v a c a n st i t ea n dp r o d u c eas n e g a t i v ceh a r g eo n t h e p - s i d en e a r t h ej u n c t i o n .
S i m i l a r l yw,h e na h o l ec r o s s etsh e j u n c t i o na n de n t e r st h e n - s i d ei ,t w i l l p r o d u c ea p o s i t i v e
c h a r g eo n t h e n - s i d en e a rt h e j u n c t i o n .
T h ed i f f u s i o no f c h a r g ec a r r i e r cs a u s e tsh e b u i l d -u p o f a n e t p o s i t i v ec h a r g ei n a n a r r o w
s t r i po n t h e n - s i d ec l o s et o t h e j u n c t i o na, n d a n a r r o ws t r i po f n e g a t i v e
c h a r g eo n t h e p - s i d e
c l o s et o t h e j u n c t i o n
T h e r e g i o no n e i t h e rs i d eo f t h e j u n c t i o nb e c o m e a
s l m o s tf r e bo f m a j o r i t yc h a r g ec a r r i e r s
a n d i s c a l l e dt h e D E P L E T I OLN
AYER
o r B A R R I ELRA Y E R .
T h ee x i s t e n coef t h i sd e p l e t i o nl a y e rp r e v e n t sa n yf u r t h e rd i f f u s i o no f c h a r g ec a r r i e r s .
r)
a)
.)
n
C
124
T h e r ei s a s m a l lv o l t a g ea c r o s st h e d e p l e t i o nl a y e rc a l l e dt h e j u n c t i o np . d .o r b a r r i e rv o l t a g e .
The depletionlaye.ris approximately1Fm thick.
S i n c et h e d i f f u s i o nc u r r e n ti s t h e f l o w o f h o l e s i,t i sj u s t b a l a n c e b
d y t h e f l o w o f e l e c t r o n isn
theopposite
direction
a n dt h e c u r r e n dt u et o t h e m o t i o no f t h e e l e c t r o niss c a l l e dt h e D R I F T
C U R R E NlTa , i r ,
T h ed r i f t c u r r e n ti s a l w a y si n t h e o p p o s i t ed i r e c t i o nt o t h e d i f f u s i o nc u r r e n t .
Recallthat diffusioncurrentis directedfrom p-typeto n-typewhile DRIFTcurrentis directed
from n-typeto p-type.
BIAS
W h e na b a t t e r yi s c o n n e c t e a
d c r o s st h e p - nj u n c t i o ns u c ht h a t t h e p o s i t i v ep o l eo f t h e b a t t e r y
c o n n e c t tso t h e p - s i d eo f t h e j u n c t i o na n dt h e n e g a t i v e
p o l eo f t h e b a t t e r yc o n n e c t tso t h e n s i d eo f t h e j u n c t i o nt,h e p - nj u n c t i o ni s s a i dt o b e F O R W A R B
Dl A s E D .
l f h o w e v e tr h e p o s i t i v eO o ' "o f t h e b a t t e r yw a sc o n n e c t e tdo t h e n - s i d eo f t h e p - nj u n c t i o n ,
and
the negative
p o l ec o n n e c t e tdo t h e p - s i d et,h e nt h e j u n c t i o nw o u l dh a v eb e e nR E V E R S E
BIASED.
F o r w a r dB i a s
R e v e r s eB i a s
I n t h e f o r w a r db i a sc o n n e c t i o nt h, e
I n t h e r e v e r s eb i a sc o n n e c t i o nt ,h e n -
d e p l e t i o nl a y e ri s o v e r c o m e
,becomes
t y p e i s m a d em o r e p o s i t i v ea n d t h e p -
n a r r o w e r a n d h o l e sf r o m t h e p - t y p e
t y p e m o r e n e g a t i v eT. h i sc a u s e sa n
w i l l f l o w t o t h e n - t y p e ,a n d e l e c t r o n s
i n c r e a s ien t h e w i d t ho f t h e d e p l e t i o n
f l o w i n gi n t h e o p p o s i t ed i r e c t i o nT. h e
l a y e r ,c a u s i n ga v e r y h i g hr e s i s t a n c e
c u r r e n ti s i n t h e o r d e ro f m i l l a m p e r e s .
s o t h a t t h e c u r r e n ft l o w i n gt h r o u g h
t h e j u n c t i o ni s a p p r o x i m a t e zl ye r o .
. p - n j u n c t i o nt h e r e f o r ea c t s a s a r e c t i f i e r ,
a s i t a l l o w sc u r r e n tt o f l o w i n o n e d i r e c t i o no n l y .
1,25
for a p-n iunction.
The current - voltage characteristic
Current/mA
Forward
lv
T h ej u n c t i o ns t a r t st o c o n d u c itn t h e f o r w a r dd i r e c t i o na t o r a r o u n d0 . 6 V ,a n dt h i sp . d .i s c a l l e d
t h e s a t u r a t i o vn o l t a g e .
illiam
nection
DIODE:
OFTHEP.NJUNCTION
APPLICATIONS
- h e n a n a . c .i s a p p l i e dt o t h e d i o d e ,t h e j u n c t i o nc o n d u c t si n t h e
( 1 ) A s a h a l f w a v e r e c t i f i e rW
forwarddirection,but preventscurrentfrom flowingin the reversedirection.
(2) As a full-waverectifier.
l n t h i sc a s ef o u r d i o d e sa r ec o n n e c t e tdo f o r m a b r i d g e
W h e n X i s p o s i t i v ec, u r r e n tf l o w st o B
t o R t o C a n dt h e n t o Y .
W h e nY i s p o s i t i v et,h e c u r r e n tf l o w s
t o D t o R t o A a n dt h e n t o X .
I n b o t h h a l fc y c l e st,h e c u r r e n ti n R i s
i n t h e s a m ed i r e c t i o n .
126
(3) The manufactureof the light-emittingdiode(LED)
W h e na n e l e c t r o nf r o m t h e c o n d u c t i o b
n a n df a l l si n t o a h o l et h e v a l e n c eb a n d ,t h e r ei s a
l o s so f e n e r g yo f t h e e l e c t r o nw
s h i c hi s d i s s i p a t eads h e a t .T h e r em u s tb e a l a r g e
n u m b e ro f e l e c t r o n - h o lceo m b i n a t i o nt so c a u s et h e e m i s s i o o
n f l i g h t .T h i si s a c h i e v e d
by
a p p l y i n ga s t r o n gf o r w a r db i a so n a h e a v i l yd o p e dp - nj u n c t i o n
(a) The photo diode:
W h e na c u r r e n ti s p a s s e d
t h r o u g ha p - nj u n c t i o nl i g h ti s e m i t t e d .l t f o l l o w st h e n t h a t i f
l i g h ti s ' i n c i d e notn a p - nj u n c t i o na, c u r r e n tc a nb e p r o d u c e di n t h e c i r c u i ct o n t a i n i n g t h e
p - nj u n c t i o n .
T h i sa p p l i c a t i oins c o m m o ni n t h e o p e r a t i o no f a T V r e m o t ec o n t r o l .
W h e nt h e r e m o t ec o n t r o li s c l i c k e da, l i g h te m i t t i n gd i o d ee m i t si n f r a r e dl i g h ti n a
particularsequenc,e.
The receiving
deviceon the TV detectsthe infrared lightand
p r o d u c ee
s l e c t r i c asli g n a ltso c h a n g et h e T V s e t t i n g s .
1.27
THETRANSISTOR
W h i l et h e p - nj u n c t i o ni s a t w o t e r m i n a dl e v i c ea n di s u s e dp r i m a r i l ya s a r e c t i f i e rt ,h e
s a na
l yo r ea p p l i c a t i o nt h
i sR
a t h r e et e r m i n a dl e v i c ea n d h a ss i g n i f i c a n tm
TRANSISTO
g n di s a n e s s e n t i a l
, itchina
s i n g l ep - nj u n c t i o na, s i s i t u s e df o r s i g n aal m p l i f i c a t i o sn w
EIDR C U I T S ( l C ' s )
c o m p o n e ni tn a l l I N T E G R A TC
T h eJ U N C T I OTNR A N S I S T Oi sRm a d eu p o f t w o p - nj u n c t i o n sc o n n e c t e idn s e r i e s, b a c k
to back.
T h ea r r a n g e m e no tf t h e j u n c t i o n cs a ng i v ee i t h e ra P N Po r a n N P Nt r a n s i s t o r .
A n n p nt r a n s i s t o r
A p n pt r a n s i s t o r
emitter
CIRCUIS
TYMBOLS:
NPN
PNP
I
C
-\
(
I
c
r28
T h ed i r e c t i o no f t h e a r r o w h e a do n
n d i c a t ew
s h e t h e rt h e
t h e E M I T T EiR
t r a n s i s t oi rs n p no r P n P T
. h ea r r o w
of conventional
p o i n t si n t h e d i r e c t i o n
c u r r e n ti . e .f r o m p t o n .
O P E R A T I O N AAL M P L IF IE R S
T h eo p e r a t i o n aalm p l i f i e r ( a b b r e v i a o
t epd- a m p )i s a d e v i c et h a t c a np e r f o r mm p t h e m a t i c a l
o p e r a t i o nssu c ha sa d d i t i o nm
, u l t i p l i c a t i oann di n t e g r a t i o n .
I t h a so n e O U T P U a
Tn dt w o I N P U T Sa,n d h a sa s e p a r a t e
s e to f c o n n e c t i o nt sh a t a r e u s e dt o
p o w e ri t . T h ep o t e n t i a l d i f f e r e ntchea t i s u s e dt o s u p p l yp o w e r t oi t i s c a l l e dt h e S U p p L y
VOLTAG
E.
T h eS U P P LVYO L T A Gm
E u s tn o t b e c o n f u s e d
w i t h t h e I N P U TV O L T A G E
a s, t h e l a t t e ri s t h e o n e
t h a t i s s e n tf o r A M P L I F I C A T I tOhN
r o u g ht h e o p - a m p w
, h i l et h e s u p p l yv o l t a g ei s u s e dt o p r o v i d e
e l e c t r i c aeln e r g yf o r t h e o p - a m pt o b e c o m ef u n c t i o n a l .
T h eo p - a m ph a st w o i n p u tt e r m i n a l m
s a r k e da s ( + ) a n d( - )
T h e( + )t e r m i n ails c a l l e dt h e N O NI N V E R T I NTG
E R M I N AaLn d R E T A I Nt hSe s i g no f t h e v o l t a g e
a p p l i e dt o i t . i . e +
. v er e m a i n sa s+ v ea f t e ra m p l i f i c a t i oann d s i m i l a r l -yv e r e m a i n sa s- v e a f t e r
a m p l iifc a t i o n
T h e( - )t e r m i n a il s c a l l e dt h e I N V E R T I NTGE R M I N AaLn d i t c o n v e r t s+ v et o - v e a f t e r
a m p l i f i c a t i oann d- v e i s c o n v e r t e d
t o + v ea f t e ra m p l i f i c a t i o n .
T h et e r m i n a l s+ V ,a n d- V , a r et h e S U P P L Y v o l t atgeer m i n a l sa n dt h e s em u s tn o t b e c o n f u s e d
w i t h t h e l NP U Tt e r m i n a l s .
T h eS U P P L Y t e r m i n a rl set h e c o n n e c t i o nf so r w h i c ht h e o p - a m pr e c e i v eas p o t e n t i adl i f f e r e n c e
t o t r i g g e ri t i n t oa c t i o nT
. h es u p p l yv o l t a g e a
s r et y p i c a l l 6
y V o r 9 V ,a n dt h e s ev o l t a g e a
sre
NEVEa
Rm p l i f i e ad r e n yt i m e .
1,29
T h i ss u p p l yv o l t a g ec a nb e c o n s i d e r etdo b e s i m i l a tro t h e v o l t a g ew h i c hm u s tb e a p p l i e da c r o s s
a m u s i ca m p l i f i e irn o r d e rt o g e t i t t o w o r k .E g .A m u s i c i a h
n a st o p l u gi n a n a m p l i f i etro t h e
L 2 0 Vm a i n si n o r d e rt o g e t h i se l e c t r i cg u i t a rt o o p e r a t eW
. h i l et h e s o u n df r o m t h e g u i t a ri s
convertedto an electrical
signaland amplified,the t20V from the mainsis not amplified
Bythe law of conservation
of energy,the maximumoutputvoltageVocanneverexceedthe
s u p p l yv o l t a g eV , .T h i ss i m p l ym e a n st h a t i f t h e m a x i m u mi n p u te n e r g y ( p ecro u l o m b i)s V , t h e n
t h e m a x i m u mo u t p u ti s a l s oV , T h ec o n d i t i o nf o r w h i c ht h e O u t p u tV o l t a g e= V , i s c a l l e d
5-AIUBAIJSN.
The OP-AMPis ialled a differentialamplifierbecause
the output is directlyproportional
to the
D I F F E R E NiC
nE
v o l t a g eb e t w e e nt h e t w o i n p u t s .
F r o mt h e a b o v ed i a g r a r nV, r i s a p p l i e dt o t h e N O N- I N V E R T I NTGE R M I N AaLn dV 2t o t h e
I N V E R T I NTG
ERMINAL.
C o n s i d et rh e f o l l o w i n gs i t u a t i o n:s
( 1 ) V 1 > V ;zT h i sm e a n st h a t t h e r ei s a g r e a t e vr o l t a g ea t t h e n o n - i n v e r t i nt ge r m i n atl h a nt h a t
a t t h e i n v e r t i n gt e r m i n a la n d s i n c et h e a m p l i f i c a t i o ins t h e s a m ef o r e a c h i n p u t ,w e
w o u l d e x p e c ta N E TP O S I T I VoEu t p u tv o l t a g et;h a t i s V s i s p o s i t i v e0 1 s i m p l yw h e n t h e
t w o i n d i v i d u aol u t p u t sa r ea d d e dt h e n e t r e s u l tw i l l b e p o s i t i v e .
(2) Vz >Vr j Vowould be NEGATIVE
as the output from V2 has greatermagnitudeand is
n e g a t i v ea s c o m p a r e dt o t h e s m a l l e rp o s i t i v eo u t p u tf r o m V l . W h e nt h e t w o i n d i v i d u a l
o u t p u t sa r e a d d e d t, h e n e t r e s u l tw i l l b e n e g a t i v e .
( 3 ) V r = Y z, T h e o u t p u tf r o m V 1w o u l d h a v et h e s a m em a g n i t u d e
a s t h a t f o r V 2b u t t h e y w i l l
h a v eo p p o s i t es i g n sT. h i sw o u l dr e s u l ti n t h e n e t o u t p u tb e i n gz e r oi . e .V 6= I
130
A M P L I F I EGRA I N :
T h ep u r p o s eo f a n a m p . l i f i ei sr t o i n c r e a steh e m a g n i t u d o
e f t h e i n p u ts i g n aal n d a s a r e s u l t h e
o u t p u ts h o u l da l w a y sb e a n i n t e g r am
l u l t i p l eo f t h e i n p u t .
T h er a t i oo f t h e o u t p u tv o l t a g et o t h e i n p u tv o l t a g ei s c a l l e dt h e G A I No f t h e a n a p l i f i e r .
G A I N:
VOUt
Vtn
S i n c eG A I Ni s a r a t i oo f v o l t a g e si t, h a sn o u n i t s ,a n d i s u s u a l l yr e p r e s e n t ebdy t h e s y m b o "l A "
H e n c ei f t h e v o l t a g e V
s l a n d V 2 d r €a p p l i e dt o t h e i n p u tt e r m i n a l st ,h e n e t i n p u tv o l t a g ew i l l b e (
V r V z) a n d i f V o i s t h e o u t p u tv o l t a g et,h e n
Vo:
A(Vr-Vz)
PRoIEBTIES-QE"T"H"r_"QP-""AMP"",""
T h ep r o p e r t i eos f a t h e o r e t i c aild e a lo p - a m pa r ec o n s i d e r efdo r t h e p u r p o s eo f d e r i v i n g t h e
relevantequationsfor it. Howevera REALOP-AMPhasproperties
that areverycloseto but not
i d e n t i c awl i t h t h o s eo f t h e i d e a lo p a m p .
PROPERTtESOF THE B,E*A!=,_OJ:AMP:
(1) A very high open loop gain of approximately
10sat low frequencies
but it decreases
as
f r e q u e n c yo f t h e i n p u ts i g n a il n c r e a s e s( R
e
m
e
m
b
e
r
t
h
a
t
.
t h e i n p u ts i g n adl o e sn o t h a v e
t o b e a c o n s t a nvt o l t a g ei,t c a nb e s i n u s o i d aaln d h e n c ei t w i l l h a v ea f r e q u e n c y ) .
T h et e r m " o p e nl o o p "m e a n st h a t t h e r ei s n o e l e c t r i c aclo n n e c t i ofnr o m t h e o u t p u tb a c k
t o t h e i n p u t .C o n s i d et rh e f o l l o w i n gd i a g r a m s :
In Diagraffi
A, there is no
connection
from the output back
D i a g rm
a A
t o t h e i n p u t ,o r t h e r ei s n o l o o p
f r o m o u t p u tt o i n p u t ,a n dt h i s
c i r c u i ti s c a l l e da n O P E NL O O P
C I R CIUT .
1.31
I n d i a g r a mB t h e r ei s a l o o pf r o m
t h e o u t p u tt o t h e i n p u ta n dt h i s
D i a g r a mB
c i r c u i ti s c a l l e da C L O S ELDO O P
IRCUIT
C I R C U IoTr a F E E D B A C K
( 2 ) A v e r y h i g h i n p u t r e s i s t a n c e ( i m p e d a n cr ie=)1 0 1 2 Oa,n d t h e r e f o r ed r a w s a m i n u t e
e x t e r n a l l ay t
c u r r e n tf r o m t h e s u p p l y . T h iisn p u t r e s i s t a n cies n o t o n e t h a t i s c o n n e c t e d
t h e i n p u t sb
, u t i n s t e a di t i s a r e s i s t a n ct eh a t i s m a n u f a c t u r ei dn t ot h e o p - a m p .
r o =1 0 0 0 ; w h i c hm e a n st h a t i t s o u t p u tc a nb e
(3) A very low output resistance(impedance)
t r a n s f e r r e dw i t h o n l y l i t t l e l o s so f p o w e rt o l o a d sg r e a t e rt h a n a f e w k i l o h m s (. R e c a l l
t h a t p o w e rl o s s= l 2 R )
Ri
i n p u ti m p e d a n c e
Ro output impedance
1.32
P R O P E R T IoEf tSh e I D E A LO PA M P
( 1 ) T h ei n p u ti m p e d a n cies l N F l N l T E
, n c et h e i n p u tc u r r e n it s Z E R O .
he
( 2 ) T h e i n p u t sa r e b o t h a t t h e s a m ep o t e n t i a lT. h i sc a n b e e x p l a i n e d
a s f o l l o w s :F o ra v e r y
h i g ho p e nl o o pg a i n ,t h e g a i n4 6 =o o
R e c a lA o - b -
Vi
"o
hence Vi _
vo-vo
Ag
co
- o
B u t V i= ( V z - V r ) = 0
H e n c eV z = V r
( 3 ) Z e r oo u t p u tl m p e d a n c e .
l d e aO
l PA m p
R e aO
l PA m p
( 1 ) I n f i n i t ei n p u ti m p e d a n c e F i n i t ei n p u t i m p e d a n c e1 0 1 2 0
( 2 )Z e r oo u t p u ti m p e d a n c e L o wo u t p u t i m p e d a n c -e 1 0 0 0
( 3 ) l n f i n i t eo p e n l o o p g a i n
F i n i t eo p e nl o o pg a i n- 1 0 '
133
CS.
R CHARACTERISTI
TRANSFE
vs.INPUTVOLTAGE.
is simplya graphof OUTPUTVOLTAGE
CHARACTERISTIC
TheTRANSFER
Saturation
Vrlt-tV
R e c a lt lh a t t h e m a x i m u mo u t
v o l t a g ei s e q u a lt o t h e s u P P l Y
v o l t a g eV , a n dw h e nt h i sh a P p e n s ,
t h e o p a m p i s s a i dt o b e
SATURATED
Saturation
in
i n V o l t s t, h e i n p u tv o l t a g ei s m e a s u r e d
Notethat whilethe cutPutvoltagei s m e a s u r e d
.mlgrgyg.!!r,
134
l f t h e i n p u tv o l t a g ei s S I N U S O I D At hLe n t h e g r a p h so f I n p u tv s .t i m e a n d o u t p u tv s .t i m e d r a w n
s e p a r a t e lwy i l l b e a sf o l l o w s :
CLIPPING
INPUT/microvolts
Output/volts
time
T h eo u t p u ts i g t r ai ls n o t a n e x a c tr e p l i c ao f t h e i n p u tb e c a u s teh e o p a m p b e c o m e s a t u r a t e d . s o
t h a t t h e m a x i m u mv a l u eo f V si s e q u a lt o V , w h i c hi s a c t u a l l yl e s st h a nA o V i a n tdh e p e a k so f t h e
output graphare saidto be C[PP"E_,D_*
In mostcases,
the amplifier
reaches
saturation
almostinstantaneously
sothata sinewaveinputwould
givea SQUARE
WAVEOUTPUT
l n p u t / mi c r o v o l t s
Output/Volts
135
SPECIAL
CASEWITHTWO INPUTS
C o n s i d eT
r h e c a s ew h e r e t h e r e i s a n i n p u t t o b o t h t h e i n v e r t i n a
g n dn o n - i n v e r t i nt egr m i n a l s :
t
Vort
I
T h e i n p u t st o t h e i n v e r t i n ga n d n o n i n v e r t i ntge r m i n a l sa r ea ss h o w n :
Output/V
T h ei n i t i a pl o s i t i v ep a r to f t h e s q u a r ew a v eo u t p u ta r i s e sf r o m t h e f i r s tp a r to f t h e i n p u tv o l t a g e O
sA
w h e r et h e i n p u tt o t h e + v et e r m i n a il s g r e a t e trh a nt h a t o f t h e - v e t e r m i n a l , r e s u l t iinnga + v eo u t p u t .
A f t e rt h a t ,t h e i n p u to n t h e - v e t e r m i n a bl e c o m e g
s r e a t e trh a nt h a t o n t h e + v et e r m i n a l ( A B sCo) t h e
o u t p u ti s- v e i . e .i n v e r t e dF. r o mC t o D t h e o u t p u tw i l l a l w a y sb e p o s i t i v eb e c a u s fei r s t l yt h e
rragnitude
o f t h e i n p u tv o l t a g eo n t h e + v et e r m i n ails g r e a t e trh a nt h a t o n t h e - v e t e r m i n a l .
S e c o n d l yw,h e nt h e m a g n i t u d o
e f t h e i n p u tv o l t a g eo n t h e - v e t e r m i n a il s g r e a t e trh a nt h a t o n t h e
+ v et e r m i n a lt,h e o u t p ; r ti st s t i l lp o s i t i v eb e c a u s n
e o w t h e i n p u to n t h e i n v e r t i n tge r m i n ails - v e s o t h e
o , ! i . :,,i w i l l b e i n v e r t e dt o a + v ev a l u e .
136
F E E D B A CAKN DT H EI N V E R T I NAGM P L I F I E R
F e e d b a ci kn a n o p a m p i s a c h i e v e bd y m a k i n ga n e l e c t r i c ac lo n n e c t i ofnr o mt h e o u t p u t t oo n e
o f t h e i n p u t s, u s u a l lvyi aa r e s i s t ocra l l e dt h e F E E D B A C
RK
E S I S T ORRr
F o ra n I N V E R T I NaG
m p l i f i e rt h
, e o u t p u ti s c o n n e c t etdo ' t h ei n v e r t i n ign p u to f t h e a m p l i f i e r v i a
R i3 ss h o w n :
T h ef e e d b a ccku r r e n it s l i a n dt h e i n p u tc u r r e n its l ; n .T h eo p a m p h a sa h i g hi m p e d a n caen d
t h u sd r a w sa m i n u t eo r n e g l i g i b lceu r r e n t .
l f K i r c h o f f 'csu r r e n tl a w i s a p p l i e da t t h e j u n c t i o nP , t h e nl ; p+ 1 1
= 0 , o r l i n= - l r .T h i sa l s oi n d i c a t e s
t h a t l 1 dp n d l i a r ei n A N T I P H A SwEh i c hi st r u ef o r t h e I N V E R T I NoG
pamp.
T h ef o l l o w i n g
t w o I M P O R T A NATS S U M P T I OaNrS
em a d e :
( 1 ) T h ef e e d b a ccku r r e n lt 1i s e q u a tl o t h e i n p u tc u r r e n lt i n d, u et o t h e i m p e d a n coef t h e o p
a m p b e i n gs o h i g ht h a t i t d r a w sn e g l i g i b lceu r r e n t .
( 2 ) T h ep o t e n t i aal t t h e p o i n tP i s s o s m a l l t h a itt i s a p p r o x i m a t et od z e r o ,a n dt h e p o i n tp i s
t h e r e f o r ec a l l e dt h e V I R T U AELA R T H .
A point is saidto be at VIRTUAL
EARTH
when the potentialat that point is so small
that it can Deapproximatedto earth potentiali.e.zeropotential.
1.37
e f t h e v e r yl a r g eG A I No f t h e o p a m p .
T h ev i r t u a e
l a r t hc o n d i t i o na r i s e sb e c a u s o
e . g .l f g a i n A = 1 0 s , t f r e n f: 1 0 s
a n di f V o = V , = g V t h e nV p : # =
T h ef o l l o w i n gd e r i v a t i o fno r G a i ni s v e r yi m p o r t a n t :
S i n c eV p =0 , V i n= l i n R iann dV o= l r R r
B u t l i n- - l 1( a ss h o w np r e v i o u s l y )
Gain_*:(*,*l :(_ f^y^)
G a i n o f I N V E R T I NoGp q m p - - X
138
O
T H EN O N . I N V E R T I N
AG
MPLIFIER
1
I
vi
I
t__T h e i n p u tv o l t a g eViii s; a p p l i e dt o t h e n o n -i n v e r t i n g
t e r m i n a (l + ) .T h i sp r o d u c e as n
o u t p u tV ot h a t i s i n p
1 hr a s ew i t h t h e i n p u t .
Negativefeedbackis obtair
n1 (ed( b y f e e d i n gb a c ka f r a c t i o no f t h e o u t p u tt o t h e
I N V E R T I Ni nGp u t ( - ) .
ThevoltageacrossRi_is
equalto VnjO=Vn.The potentialat the non-inverting
input is Vn.Fora
largeopen loopgain.Vn=Vi.soin the derivations,
we useVninsteadof Vi .because
Vo_g_an
be
expressed
in termsof a resistance.
ie. Vo_lB;whereasV; hasno e.xpression
in termsof a
resistance
O.u r a i m i s t o g e t a n e x p r e s s i ofno r t h e G A I Ni n t e r m so f t h e r e s i s t a n c et hsa t a r e
a v a i l a b lien t h e c i r c u i t .
Thevoltageacross(R6 Ri)= Vo
S i n c eR r a n dR id r e i n s e r i e st h e s a m ec u r r e n ft l o w st h r o u g ht h e m t o t h e e a r t h ;
V p = l R i a n d V o = l ( R t+ R i )
vo
Rf+Ri
vp
R,
Rr
: 1 * J - G A I N f o r N O N I N V E R T I N GO P A M P
rti
G a i n o f I N V E R T I N Go p q m p -
139
-
"t
_^
SN D B A N D W I D T H
G A I N . F R E Q U E NGCRYA P HA
g r a p hi s p l o t t e d .
i pe t w e e nG a i na n df r e q u e n c ya,l o g a r i t h m
y r e l a t i o n s hb
I n o r d e r t od i s p l a a
T h i sl o gg r a p hi s u s e df o r t w o r e a s o n s :
( 1 ) l t a l l o w ss m a l l enr u m e r i c avla l u e st o b e u s e d
( 2 ) l t g i v e sa l i n e a vr a r i a t i o n
b e t w e e nt h e p a r a m e t e rosn t h e x a n dy a x e s .
'Gain
T h i s P o r t i o ni s t h e r e g i o no f n o
f e e d b a c ka n d t h e g a i n h e r e i s c a l l e c
t h e O P E NL O O PG A I N .
10s
l-04
an d w i d t h
103
102
bandwidth
.10
C
1,
i.0
1,oz l-03 !04
10s 106 1,06 1,01 108
Frequency/Hz
T h eh o r i z o n t al iln e sC Da n d E Fr e p r e s e n t t hgea i nw h e nf e e d b a ciks p r e s e n ti ,. e .c l o s e dl o o p
s e g a i nw h e nn o f e e d b a ciks p r e s e n ti ,. e .o p e nl o o pg a i n .
g a i nT
. h es l a n t e dl i n eA B r e p r e s e n t h
A t C D ,t h e r ei s m o r ef e e d b a ctkh a na t E F .
T h ed i a g r a ms h o w st h a t t h e c l o s e dl o o pg a i ni s a l w a y sl e s st h a nt h e o p e nl o o pg a i nb u t i t i s
m o r ed e s i r a b lbee c a u s e :
( 1 ) T h eg a i ni s p r e d i c t a b l e .
( 2 ) S t a b i l i t yi s g r e a t e rw h i c h m e a n st h a t t h e g a i n i s c o n s t a n to v e r a w i d e r r a n g eo f
frequencies.
n f g a i nw i t h f r e q u e n c y
i ' , r U n l i k et h e o p e nl c o pg a i nt h e r ei s n o v a r i a t i o o
I4A
E.ANDIUDI.II;
Thedifferencebetweenthe upperand rowerfrequencies
for whichthe gainislonstant,is
calledthe BANDWIDTH
of the amplifierfor a particulargain.
Forexample,on rookingat the previousgraph,for a gainof r.o,
the Bandwidthis (108-1)Hz,
and for a gainof 10OO
the bandwidthis (101t)Uz.
ln orderto havegreaterbandwidth,the negativefeedbackmust greater,
be
but this causesthe
gainto drop.To compensate
for this rowergaintwo or moreop ampscanbe coupredor
typicallyreferredto as CASCADED
C A S C A D EADM P L I F I E R S
A1
A2
W h e nt h e t w o o p a m p sa r ec a s c a d eads s h o w na b o v e t, h e r e s u l t a ngt a i n =
A 41xA2
S i m i l a r lfyo r 3 o p a m p st h a t a r ec a s c a d e dt h, e r e s u l t a ngt a i nA = A r
1,4L
x 4 2x 4 3
OFTHEOPAMP
APPLICATIONS
(1) As a comparator-lt comparesthe two inputsand its outputdependson the differenceof
thetwo inputs.
t r a s a c o n v e r t eor f s i n e
T h ec o m p a r a t ocr a nt h e n b e u s e da s a s w i t c hi n a n a l a r mc i r c u i o
w a v es i g n a l tso s q u a r ew a v es i g n a l s .
a m p l i f i e rT. h i si s u s e di n
( 2 ) A s a S U M M I N GA M P L I F I EwRh e n i t i s u s e da sa n i n v e r t i n g
m i x i n gb o a r d si n m u s i ca n da l s oa sa d i g i t atl o a n a l o g u ce o n v e r t e r .
the voltagefolloweris usedas a bufferor matchingamplifier
FOLLOWER(3) As a VOLTAGE
a n dw i l l b e e x p l a i n e sdh o r t l Y .
THECOMPARATOR
r form a
T h et h e r m i s t o a
r n d t h e f i x e dr e s i s t o R
p o t e n t i adl i v i d e rw h i c hf i x e st h e v o l t a g ea t t h e
( a ) A s a s w i t c hi n a n a l a r mc i r c u i t :
n o n - i n v e r t i ni n
gp u t .
A s t h e t e m p e r a t u r e ' r i s et sh,e v o l t a g ea t t h e n o n i n v e r t i n gi n p u ti n c r e a s e sa,st h e r e s i s t a n coef t h d
t h e r m i s t o rr i s e s ( a s s u m ian gp o s i t i v ec o e f f i c i e n t
t y p et h e r m i s t o r ) .
T h ev o l t a g ea t t h e i n v e r t i n gi n p u ti s s e t b y t h e
potentiometer.
T h e p o t e n t i o m e t eirs s e tt h a t t h e v o i t a g ea t t h e
i n p u ti s t h e s a m ea st h a t a t t h e n o n inverting
i n v e r t i n gi n p u t .T h ed i f f e r e n c eb e t w e e nt h e t w o
v o l t a g e sw i l l t h e n b e z e r o ,a n dt h e l a m po r
buzzew
r ill beoff.
A s s o o na st h e t e m p e r a t u r ei n c r e a s e st h, e
a t t h e i n p u tw i l l b e g r e a t e r
v o l t a g ed i f f e r e n c e
r illbe
, n dt h e l a m po r b u z z ew
t h a n z e r oa
a c t i v a t e dC. o m p a r a t o rasr e v e r ys e n s i t i v e
s w i t c h e sa n d t h e t e m p e r a t u r en e e d st o c h a n g e
b y o n l ya s m a l la m o u n tt o t r i g g e ro n t h e a l a r m .
r42
T H ES U M M I N GA M P L I F I EORRA D D E R
W h e nt h e o p a m p i s u s e da s a m u l t i - i n p uitn v e r t i n ga m p l i f i e ri,t c a nb e u s e dt o a d d a n u m b e ro f
v o l t a g e sb, o t h a . c .a n d d . c .
C i r c u i t s u c ha st h e s ea r e u s e di n t h e m u s i ci n d u s t r y t oc o m b i n eo u t p u t sf r o m m i c r o p h o n e s ,
electricguitars,keyboards,
voiceetc.
A f o u r i n p u ts . u m m i nagm p l i f i e irs s h o w nb e l o w :
RF
v1
ll
1I
vo
t
j
v4
+
W h e nK i r c h o f f ' sc u r r e n tl a w i s a p p l i e dt o t h e j u n c t i o na t P ,t h e t o t a lc u r r e n te n t e r i n gP i s e q u a l
t o t h e t o t a lc u r r e n tl e a v i n g .
Hence:
It*lz*ls*l+=l
I43
- + ( t h e riesa m i n u ss i g ns i n c et h e
T h ec u r r e n t h r o u g ht h e f e e d b a c kr e s i s t oirs g i v e nb y I a m p l i f i eirs i n v e r tni g )
1r:;:
V.
t(1
V,
;lz -i;
-vo
Rf
Is:
V,
Io:
&,
V4
Rn
- + V"" + \
_ V',^ + V?
R1
R2
R3
Ra
T h eG E N E R AELQ U A T I O fNo r t h e s u m m i n ga m p l i f i eirs t h e r e f o r e :
vo:
(3,,
Rr
* JV"
RzL
.frvs*P-r-)
a
A M P L I F I Ei sRt h e c o n v e r s i oonf d i g i t asl i g n a ltso
O n ei m p o r t a nat P P l i c a t i oonf t h e S U M M I N G
a n a l o g usei g n a l sT.h i st Y P eo f c o n v e r t eirs u s u a l l ya b b r e v i a t eadsa D / A c o n v e r t e r .
t o e l e c t r i c apl u l s e st h a t h a v eo n l y t w o s t a t e si . e .1 ( h i g h )a n d
l n d i g i t a ,t ' r n . l s , d a t ai s c o n v e r t e d
O( l o w ) .
l yv e ra r a n g eo f v a l u e s
t o a v o l t a g et h a t v a r i e sc o n t i n u o u s o
I n a n a l o g u se i g n a l sd,a t ai s c h a n g e d
i . e .a w a v e f o r mo f t h e v o l t a g er e p r e s e n ttsh e d a t a '
ANALOGUE
144
Analoguesignalsare subjectedto noiseand distortion,whiledigitalsignalsare extremelyclean
and undistorted.
However,there are manyapplications
whichcanonly operateon an analogue
basise.g.thosewhichinvolvethe humansensessuchas hearingand seeing.
The humanear candetectcontinuousvariationsin the intensityand pitchof sound;if it were to
operateon a digitalbasisthen would eitherdetector not detectsoundbut it wouldnot be able
to distinguish
changesin intensityand pitch.
The sametype of reasoning
is usedfor the humaneyetd detectchangesin lightintensityand
changesin colour.
H e n c ef o r r a d i oa n dt e l e v i s i obnr o a d c a sitt,w o u l db e l o g i c a l t oc o n v e rat n a n a l o g usei g n aflr o m
t h e r a d i oa n n o u n c et ro a d i g i t asl i g n a l t h at th e n l e a v e tsh e t r a n s m i t t e ar ,n dt h e n b a c kt o
a n a l o g uaet t h e r a d i or e c e i v e T
r .h er e a s o nf o r u s i n gt h e d i g i t asl i g n a l f otrr a n s m i s s i oi snt h a t i t
i s f r e ef r o m n o i s ea n d d i s t o r t i o na n dw o u l dt h e r e f o r eg i v ea b e t t e rq u a l i t yo f s o u n da t t h e
receiver.
l n p r a c t i c eD, / A c o n v e r t e ri sn c o r p o r a taeS U M M I N G
A M P L I F I E. R
T h ei n p L lst i g n a las r et y p i c a l l5y V f o r a b i n a r yi n p u to f l a n d0 V f o r a b i n a r yi n p u to f 0 .
1.45
T h ef o l l o w i n ge x a m p l es h o w st h e c o n v e r s i oonf d i g i t atl o a n a l o g u e :
A m a x i m u mp . d .o f 5 V i s a p p l i e dt o e a c hi n p u t .
1ko
2kC)
VD
VC
1
VB
16kO
vo
t
I
VA
J
F r o m t h e g e n e r a lf o r m u l a f o r t h e o u t p u t o f t h e s u m m i n ga m p l i f i e r ,
Vo: -
Rr
Rr
f"Vr- nU,
Rr
-iV3-
Rr
rur*
T h eo u t p u tv o l t a g ed u e t o t h e 2 k Or e s i s t oar l o n e- V p
-lkfi
Vo:
Zkn
x5:-2.5V
- 1ko
Vc:4kA x5:-1.25V
-Iko
x5: -0.625V
B/.O
-1k0"
Ve:16k'x5:-0'31'3V
Vs:
r46
T h e m i n i m u mv o l t a g eo u t p u tt h a t i s n o n z e r o i s c a l l e dt h e L E A SsTl G N l F l c A NBTl r ( L S Ba) n d
in
t h e a b o v ee x a m p l et,h e L S Bi s e q u a lt o 0 . 3 1 3 V .
T h ec a l c u l a t i o nssh o wt h a t w h i l et h e i n p u ti s o n l y0 o r 5 V ,t h e o u t p u tr a n g e sf r o m 0 t h r o u g h
a
s e r i e so f v o l t a g e u
s p t o a m a x i m u mo f - ( 2 . 5 +j . . 2 5 + 0 . 6 2 5 + 0 . 3=1 -34). 6 g g v .
A s e r i e so f i n t e r m e d i a toeu t p u t sc a nb e o b t a i n e df r o m t h e f o l l o w i n gb i n a r yd i g i t s :
0001i.e only outputfrom Va
0011- Va+Ve- -(0.625+0.3j.3)= -0.938V
0 1 0 1 = V c * V R = - ( j , . 2 5+ 0 . 3 1 3 )= _ 1 . 5 6 3 V
1 0 1 1- V p+ V s + V=a- ( 0 . 3 L 3+ 0 . 6 2 5 + 2 . 5=)- 3 . 4 3 g V
= -0,.25+ 0.625+ 0.313)= _2.1ggv
0111=Vc*Ve*VR
=Vo*Vc+Vs+V4
=- (0.313+ 0.625+ 1.25+ 2.5)=-4.6ggV
1J1,1.
0 1 0 0 = V c =- 1 , . 2 5 V
Theabovecalculations
showthat therecan be a seriesof differentbinarydigitsto givea wide
rangeof outputvoltagesand in this way,the inputdigitalstatesof "on and off" can be
converted
t o a n A N A L O G USEI G N A L .
r47
FOLLOWER
THEVOLTAGE
T h i si s a s i m p l e vr e r s i o no f t h e n o n - i n v e r t i nagm p l i f i e r .
Rt=0
OV
t o e a r t h ,w h i l ea v o l t a g ei s a p p l i e dt o t h e n o n - i n v e r t i ntge r m i n a l ,
t e r m i n a li s c o n n e c t e d
T h ei n v e r t i r l g
h e n c et h e a m p l i f i eirs N O NI N V E R T I N G .
R e c a lt lh a t f o r t h e n o n - i n v e r t i nagm p l i f i e rt ,h e G A I Ni s g i v e nb y :
Rr
A_L+J 'R,
t o l e a v ea n o p e nc i r c u i t .
R,1i s m a d ez e r o ,a n dt h e R ii s r e m O v e d
I n t h e v o L T A G EF O L L O W E R
t m a k i n gR ii n f i n i t e .
T h i si s e q u i v a l e nt o
T h ec i r c u i tf o r t h e v o l t a g ef o l l o w e rw i l l t h e r e f o r eb e a ss h o w nb e l o w :
148
S i n c et h e g a i nA : ' 1 + y , t h e n i n t h i s
R1'
n
case,A=1*"-1.
oa)
T h eg a i no f t h e v o i t a g ef o l l o w e r = 1 .
T h ev o l t a g ef o l l o w e ri s n o t u s e da s a v o l t a g ea m p l i f i e rb, u t i n s t e a di t i s u s e da s a V O L T A G E
BUFFER.
T h ev o l t a g eb u f f e ra c t st o i s o l a t ea n i n p u ts i g n a l f r o ma l o a d ,w h i l ea l l o w i n gt h e v o l t a g eo u t p u t
t o b e i d e n t i c at lo t h e i n p u ti n m a g n i t u d e
a n d i n p o l a r i t yi.. e .V o= V i n
T h ev o l t a g ef o l l o w e rh a sa h i g hi n p u ti m p e d a n cseo t h a t i t d r a w so n l ya m i n u t ec u r r e n ta, n da t
t h e s a m et i m e ,i t h a sa l o w o u t p u ti m p e d a n cteh u sp r o d u c i n a
g s u f f i c i e n t l ya r g ec u r r e n ta t t h e
o u t p u t .( R e c atl hl a t 1 _
f
r o t h a t w h e n R i s s m a l l I, w o u l db e s u f f i c i e n t l ya r g e . )
C o n s i d et rh e f o l l o w i n gc i r c u i t :
A v o l t m e t e ri s c o n n e c t e d
t o m e a s u r ea p . d .o f 1 V
a c r o s sa 1 M O r e s i s t o rA. t y p i c a vl o l t m e t e rh a sa
r e s i s t a n coef 1 0 0 k Oa n d i n t h i s c i r c u i tt,h e v o l t m e t e r
w i l l d r a wm o s to f t h e c u r r e n a
t n dg i v ea n i n a c c u r a t e .
reading.
T h ev o l t a g ef o l l o w e rc a n b e u s e da ss h o w nb e l o w .
tv
1 M O\
1MO
l"Mo
T h eo p a m p i n p u td r a w sf a r l e s sc u r r e n t h a n t h e v o l t m e t e r b
, utthe OUTPUT
p r o v i d e se n o u g hc u r r e n t
f o r t h e v o l t m e t e rt o o p e r a t e .
T h ei n p u tc u r r e n ti s a b o u t1 p A ( 1 01 2 A ;w, h i l et h e o u t p u ti s a f e w m i l l i a m p(st 0 3 n ) .
T h i sc i r c u i ti s a l s ou s e f u fl o r d e t e r m i n i n g
t h e c u r r e n t st h r o u g hv e r y h i g hr e s i s t a n c eis. e
, .v e r ys m a l l
c u r r e r r tw
s h e n a v e i ' \ s. e n s i t i v a
e m m e t e ri s n o t a v a i l a b l e .
I n t h i sc a s et h e p . d .i s m e a s u r e a
d s b e f o r ea, n dt h e c u r r e n ti s c a l c u l a t efdr o m r - v
1,49
R
ON OP.AMPS
TUTORIAL
(1)
t h e l a m pi s r a t e da t
r nt h e a b o v ec i r c u i tt,h e o p - a m ph a sa n o p e n r o o pg a i no f 1 o 2 a n d
s 0 m A ,t h e l a m pw i l l b u r n o u t .
2 V ,4 0 m A .l f t h e c u r r e n te x c e e d 4
op-amp is
D e t e r m i n et h e v a l u e o f t h e p r o t e c t i v er e s i s t o rR , w h e n t h e
saturated
C a l c u l a tteh e p o t e n t i aal t t h e n o n - i n v e r t i nign p u t
ii.
e f t h e r h e o s t a t h a t w i l l a l l o w t h e p o t e n t i a la t t h e
t h e r e s i s t a n co
Calculate
iii.
i n v e r t i n gi n p u tt o a t t a i na v a l u eo f 2 ' 3 8 V '
j u s t c a u s e st h e
w h a t m u s t b e t h e p o t e n t i a/ sl a t t h e i n v e r t i n gt e r m i n a lt h a t
iv.
o p - a m Pt o r e a c hs a t u r a t i o n'
of the rheostatbe adjristed
To what valuesof resistancemust the resistance
v.
same initial
s o t h a t t h e o p - a m p j u s t r e a c h e ss a t u r a t i o n ? ( A s s u mt hee
conditiona
s t t h e n o n - i n v e r t i ntge r m i n a l )
Or 3 3 7 k Q )
( R= 2 5 0 (;)2 . 4 V ; 3 2 9 k O2; . 2 8 Vo r 2 . 5 2 Y ; 3 2 3 k o
i.
150
(2)
l n t h e a b o v ec i r c u i tt,h 6 o p - a m ph a sa n o p e n l o o pg a i no f 1 0 a a n dt h e l a m pi s r a t e d a t 1 , . 2 V ,
8 0 m A .l f t h e c u r r e n te x c e e d 8
s 0 m A ,t h e l a m pw i l l b u r no u t .
i.
ii.
iii.
'
iv.
v.
D e t e r m i n et h e v a l u eo f t h e p r o t e c t i v er e s i s t o R
r ,w h e n t h e o p - a m pi s s a t u r a t e d
C a l c u l a tteh e p o t e n t i aal t t h e n o n - i n v e r t i nign p u t
Calculate
the resistanco
e f t h e r h e o s t a t h a t w i l l a l l o w t h e p o t e n t i a la t t h e i n v e r t i n g .
i n p u tt o a t t a i na v a l u eo f 0 . 8 V .
What must be the potentiad
l i f f e r e n c ea c r o s st h e i n p u t t e r m i n a l st h a t j u s t c a u s e st h e
o p - a m pt o r e a c hs a t u r a t i o n .
T h e r e s i s t a n coef t h e r h e o s t a it s s e t i n i t i a l l ya t 2 0 0 k Qa n d t h e n a t 1 8 O O k OD.e t e r m i n ea t
w h i c hs e t t i n go f t h e r h e o s t a w
t i l l t h e l a m po p e r a t ea n d e x p l a i nh o w y o u a r r i v e da t y o u r
conclusion.
( 2 2 . 5 0; I . 2 Y; 1 0 3 6 k O; 3 0 0 U V; 2 0 0 k 4 )
151
(3)
( i ) F i n dt h e p o t e n t i aal t t h e n o n i n v e r t i nign p u t .
( i i )D e t e r m i n teh e v a l u eo f t h e r e s i s t a n coef t h e r h e o s t aitf t h e o u t p u to f t h e o p a m pi sz e r o .
( i i i ) T h el a m pi s r a t e da t 3 V ,1 . . 2 Ad,e t e r m i n e t h er e s i s t a n coef t h e p r o t e c t i v e
r e si s t or .
( 1 , 8 V5; 4 M O ;5 O )
1.s2
(4)
I n t h e a b o v em u l t i s t a gaem p l i f i e rt ,h e i n p u tv o l t a g ei s 1 5 m V .D e t e r m i n teh e g a i no f t h e c i r c u i t
a n dt h e o u t p u tv o l t a g e .
A n s: 2 0 0 ; 3 V
a
(s)
500ko
T h e i n p u tv o l t a g ei s V r =2 0 p t VD
. e t e r m i n et h e o v e r a lgl a i na n dt h e o u t p u tv o l t a g e .
( 1 . 5 6x 1 0 " ; 0 . 3 1 2 V )
153
(6)
76kO
200ko
600ko
( i ) l ft h e g a i n sf o r t h e c a s c a d e ad m p l i f i e r a
s r e A z =4- 0 a n d 4 3 - - 8 0 , a n dt h e o v e r a l gl a i no f t h e s y s t e mi s
s r , R za n d R 3.
t h e v a l u e so f t h e r e s i s t o rR
6 . 4 x 1 0 4 d, e t e r m i n e
( i i )l f V r = 3 5 F V ,d e t e r m i n et h e o u t p u tv o l t a g e .
A n s : 4 k O ; 5 k O; 7 . 5 k O; 2 . 2 4 V
(7)
40ko
1s0ko
500ko
s r e A r = -8 0 , a n dA r = -2 0 , a n dt h e o v e r a l gl a i no f t h e
l f t h e g a i n sf o r t h e c a s c a d e ad m p l i f i e r a
s r ,R za n d R 3 .
a m p l i f i e ri s 3 0 0 0 ,d e t e r m i n et h e v a l u e so f t h e r e s i s t o r R
(s00o; 80kct 2sko)
1.54
(8)
2MO
lf V1=2mV,Vz= 10mV,and V3=6mV,
d e t e r m i n et h e o u t p u tv o l t a g e .
-141mV
1
(e)
220kO
V z =1 2 m V
D e t e r m i n et h e o u t p u tv o l t a g ef o r t h e a b o v ec i r c u i t .
( 0 . 2- 2 s v )
155
500ko
Solutionsfor tutorial
(1)
i . W h e nt h e o p a m p i s s a t u r a t e dt h, e o u t p u tv o l t a g ew o u l db e e q u a tl o t h e s u p p l y
v o l t a g ew h i c hi s 1 2 V .I f t h e l a m pi s r a t e da t 2 V t h e nt h e p . d .a c r o s tsh e l a m pi s 2 V ,s o
t h a t t h e p . d .a c r o s tsh e r e s i s t owr o u l db e 1 0 V( l . e '1 2 V- 2 V = 1 0 V ) '
T h er e s i s t oar n dt h e l a m pa r e i n s e r i e s o t h a t i f 4 0 m Ah a st o f l o w t h r o u g ht h e l a m p ,
t h e n 4 0 m Aw i l l h a v et o f l o w t h r o u g ht h e r e s i s t o rS. of o r t h e r e s i s t o rV, = 1 0 Va n d | =
v10
:250Q
0 . 0 4 4. R 1
0.04
i i .T h e b r a n c ho f r e s i s t o rcso n n e c t etdo t h e n o n i n v e r t i n ign p u t h a sa B 0 k Or e s i s t o r
, h et o t a l r e s i s t a n cies 2 0 0 k Oi n t h a t b r a n c ha, n dt h e p o t e n t i a l
s e r i e sw i t h a 1 2 0 k OT
differenceacrossthat branchis 12v-(-r2v)=12V+ r2Y = 2 4 V .
T h ec u r r e n t t h e b r a n c h1 T h e o . d a c r o s st h e
V
R7
#
:-'
24
zxros'
x B 0x 1 0 3: 9 . 6 V
o f p o t e n t i aal c r o s s
l I F F E R E Na
CcEr o s st h e 8 0 k O ,i . et h e d i f f e r e n c e
T h i si s t h e p o t e n t i aD
a c r o s sb o t h e n d s
t h a t r e s i s t o rl.f t h e t o p e n d i s a t a p o t e n t i aol f L 2 Va n d t h e d i f f e r e n c e
o f t h e r e s i s t oirs 9 . 6 V t, h e nt h e p o t e n t i aal t t h e l o w e re n d i s 1 2 V9- . 6 V = 2 . 4 Y
156
S o l u t i o nf o r q u e s t i o n1 c o n t i n u e d ;
i i i .A t t h e i n v e r t i n gi n p u t ,t h e b r a n c ho f r e s i s t o rcso n t ai n a 2 2 O k Q
r e s i s t o irn s e r i e sw i t h a
r h e o s t a tl.f t h e p o t e n t i aal t t h e i n v e r t i n gi n p u ti s 2 . 3 8 V t, h e n t h e p o t e n t i adl i f f e r e n c e
a c r o s st h e
2 2 O k Or e s i s t o irs ( 1 2 V -2 . 3 8 V ) =9 . 6 2 V
T h e c u r r e n t h r o u g ht h e 2 2 0 k Or e s i s t o irs 1 , -
#
-
#
T h ep . d .a c r o s st h e r h e o s t a=t 2 . 3 8 V - ( - 1 2 V
1 4) . 3 8 V
T h ec u r r e n t h r o u g ht h e r h e o s t aw
t i l l a l s ob e t h e s a m ea st h a t c a l c u l a t eadb o v es i n c et h e t w o
14'38
resistorsare in series.I,' = ? - - s s3-,
Rz=329ko.
R2
R2
220x103
i v .T h ep o t e n t i aal t t h e n o n - i n v e r t i nt e
g r m i n a il s f i x e da t 2 . 4 V T
. h eo p a m p r e a c h e s a t u r a t i o n
w h e nt h e o u t p u tv o l t a g ei s l - 2 V .
vin=Ynf- # - a.rzv
; hence
R e c a l lt h a t t h e g a i n o f a n o p a m p i s g i v e n b V A - l s t t
vLn
T h ed i f f e r e n c be e t w e e nt h e p o t e n t i a last t h e i n v e r t i n a
g n d n o n -i n v e r t i p tge r m i n a l m
s u s tb e
0 . 1 . 2 Vs,o t h a t i f V , = 2 . 4 i t h e nV , c a nb e e i t he r 2 . 2 8 Vo r 2 . 5 2 V .
v ' T h i sp a r tr e q u i r e tsh a t y o u f i n dt h e v a l u e so f t h e r e s i s t a n ct e
h a t w o u l dg i v ep o t e n t i a losf 2 . 2 g V
a n d2 . 5 2 Va t t h e i n v e r t i n tge r m i n a l .
C o n s i d efri r s ta p o t e n t i aol f 2 . 2 8 V T
l I F F E R E NaCcEr o s st h e 2 2 O k Ar e s i s t o irs
. h e p o t e n t i aD
1 ' 2 V- 2 . 2 8 V =9 . 7 2 V . T h ec u r r e n t h r o u g ht h e 2 2 0 k Oa n d h e n c et h r o u g ht h e b r a n c hi s g i v e nb y
T h e p . d . a c r o s st h e r h e o s t a t
v.
9.72
Rr
220xI03
2.28v-(-1Zv)=1.4.28v
T h i sc u r r e n t i s t h e s a m e t h r o u g h t h e r h e o s t a t L e . 3 R2
1 4 - 2-8 - s ' 7 2
n
R z = 323kA
R2
,To" ro= ;
U s e t h e s a m e m e t h o d t o g e t t h e r e s i s t a n c ew h e n t h e p o t e n t i a li s 2 . 5 2 V .
Q u e s t i o n2 a n d 3 c a n b e s o l v e ds i m i l a r l yt o q u e s t i o n1
I
I
I
I
157
Solutionfor question4
I n t h e c i r c u i t t h ea m p l i f i e rasr e c a s c a d eadn dt h e o v e r a lgl a i ni s g i v e nb y :
A : ArAzAz
i n c et h e r ei s a s i g n atlo t h e n o n
T h ef i r s ta m p l i f i e rt(o t h e e x t r e m el e f t )i s N O NI N V E R T I NsG
l h i l et h e i n v e r t i n tge r m i n a il s c o n n e c t etdo t h e g r o u n d .
i n v e r t i n tge r m i n aw
Thegain f or thenon-
tnuerttng amp is gtvenby At - t *
Rr
&
30c)
Ar:1+200:2.5
s r e i n v e r t i n gs i n c et h e y h a v en o s i g n aal t t h e n o n i n v e r t i n g
T h e s e c o n da n dt h i r d a m p l i f i e r a
t e r m i n aw
l h i l et h e r ei s a s i g n aal p p l i e dt o t h e i n v e r t i n tge r m i n a l .
_Rr
TheGainf or aninverttngop - emp ts gtvenby A - t
-800
-20
' Az=i--4
andA=-
J
40
- -20
T h e o u e r a l l G o . t nA : 2 . 5 x ( - 4 ) x ( - 2 0 ) = 2 0 0 .
The OutputVoltageVs- Gatn x Vn - 200 x 0'0tSV : 3V
Solutionfor question5
T h ef i r s to p - a m pi s n o n i n v e r t i n gw h i l et h e o t h e rt w o a r e i n v e r t i n g .
T h eg a i no f t h e f i r s to p - a m pi sA t : 1 *
L*
ff:
#
-'U
- -40
T h eg a i no f t h e s e c o n do p - a m pi s A z : - +
-
T h eg a i no f t h e t h i r do p - a m pi s , 4 3- + -
: 6 0 0- - 1 5
#
- 1 ' ' 5 6x 1 0 4
T h e o v e r agl la i n i s g i v e bn y A - A 1 A , A 3 - ( 2 6 ) x ( - 4 0 ) x ( - 1 5 )
The output voltageVo= AV,n=1'56 x 10ax20 x 10-6=0'312V
158
Solutionfor question5
Overall Gatn A = A1A2A3 ; henceA, : -
A
6.4 x l_04
AzAs
40 x B0
T h ef i r s ta m p l i f i e ri s n o n i n v e r t i n gs i n c et h e i n v e r t i n gt e r m i n a li s c o n n e c t e d
t o 0 V w h i l et h e n o n
i n v e r t i n gt e r m i n a li s c o n n e c t e d
to analternating
supply.
Thegainisgib
ve
n , , : 2 0- 1 *
vA
; h e n c e R r :r y -
N-t*T
T h es e c o n da m p l i f i e ri s i n v e r t i n ga n d i t s g a i ni s g i v e nb v A z - - 4 0 :
4ke
a# :
-200kQ
henceR2: Skf)
T h et h i r d a m p l i f i e ri s i n v e r t i n ga n dA z = - - ! f -J
-600ko
: -80:
vvt
R1
R3
R E: 7 . S k e
( i i )T h eo u t p u tv o l t a g eV o= A V t n : 6 . 4 x 1 0 a x 3 5 x 1 0 - 6 - 2 . 2 4 V
Solutionfor question7
Allof the amplifiera
s reinverting,
rA1- 2 -
3000
- -1,.875
-80 x -20
A
AA=
_Rr
40k0"
R,
R1
-80
Az : -I'875
.
A." R=i -20
-Rt
i
Rr : 5000
-15Okcr
Ri
_R,
hence
R2
; Rz: BOkf)
-500k0
R3
159
; R s: 2 5 k A
R2
D I G I T A LE L E C T R O N I C S
L o g i cg a t e s :
( 1 ) T h e N O Tg a t e :
T h i sg a t e h a s o n e i n p u t a n d o n e o u t p u t ;
I N P U T OUTPUT
0
I
1
0
( 2 ) T h eO Rg a t e :
I N P U T OUTPUT
A B
c
0 0
0
0 1
I
1, 0
1
I
I
t
T h i sg a t eh a st w o i n p u t sa n do n e o u t p u t ;
A
C.
B
T h eo u t p u ti s h i g h( 1 )w h e ne i t h e ro r b o t hi n p u t sa r eh i g h .
(3) TheNOR gate:
-;
.
'
T h i sg a t e c a n b e c o n s i d e r e dt o b e t h e N O T o f a n O R g a t e , i . e . N O R= N O T O R
I N P U T OUTPUT
A B
C
1
0 0
0 I
0
1 0
0
1 I
0
A
I
B
( a ) T h eA N Dg a t e :
T h eo u t p u ti s h i g ho n l yw h e n b o t h i n p u t sa r e h i g h .
160
I N P U T OUTPUT
A B
C
0 0
0
0 T
0
I
0
0
1 1
1
( 5 ) T h e N A N Dg a t e :
T h i si s e q u i v a l e nt to a n A N Dg a t ef o l l o w e dby a NOTgate.
I N P U T OUTPUT
A B
c
0 0
1
0 I
1.
t 0
t
1 I
0
A
B
( 6 ) T h e E X C L U S I V E -gO
aR
t e( E X - o R )
T h i si s s o m e w h ast i m i l a r t oa n O Rg a t e ,b u t i t E X C L U D E
t hSe c a s eo f t h e b o t h i n p u t sb e i n gh i g h .
R e c a lt lh a t i n t h e c a s eo f t h e O Rg a t e ,t h e o u t p u ti s h i g hw h e n b o t h i n p u t sa r e h i g h ,h o w e v e r
g a t e ,t h e o u t p u ti s l o w w h e n b o t h i n p u t sa r e h i g h ,b u t i n a l l o t h e rc a s e si,t
withthe EX-OR
b e h a v e s i m i l a tro t h e O Rg a t e .
I N P U T OUTPUT
A
\\------.'
\\
t
-B--l___-/
/-c
A
B
c
0
0
0
0
I
I
1
0
1
I
1
0
A n o t h e re a s yw a y o f r e m e m b e r i n g
t h e t r u t h t a b l ei s t h a t t h e o u t p u ti s
h i g ho n l yw h e nt h e i n p u t sa r e N O T
thesame.
( 7 1 T h e E X C L U S I VNEo-R g a t e( E X - N O R )
I N P U T OUTPUT
--l)
A
-B
\\----\
lL----/
>-.
A
B
c
0
0
T
0
t
0
1
0
0
t
1.
1
I n ' i t i sg a t et h e o u t p u ti s h i g ho n l yw h e n b o t h i n p u t sa r ee q u a l ,h e n c ei t a c t so p p o s i t et o t h e E X O Rg a t e .
t6r
GATE
DIAGRAM
FUNCTION
T h eo u t p u ti s o p p o s i t e
of the input
NOT
OR
T h eo u t p u ti s L o n l y
w h e n e i t h e ro n e o f t h e
i n p u t si s 1 O Rb o t h
i nputsare l -
T h eo u t p u ti s t h e
opposi teof the ORgate.
N O Rm e a n sN O TO R
NOR
The output i s 1-onl y
w h e n b o t h i n p u t sa r e 1
AND
NAND
T h eo u t p u ti s t h e
o p p o s i t eo f t h e A N D
gate.
N A N Dm e a n sN O T A N D
EX-OR
EX-NOR
INPUTS
T h eo u t p u ti s 1 o n l y
w h e no n e o f t h e i n p u t s
a r e1 b u t n o t w h e nb o t h
i n p u t sa r e 1
T h eo u t p u ti s t h e
opposi teof the E X -OR
gate.
r62
OUTPUT
1.
0
0
I
0
0
0
0
1.
L
1.
0
1.
L
t
1.
0
0
T
0
0
I
0
0
I
L
0
0
0
0
1
0
0
1
0
0
T
1.
L
0
0
0
L
T
L
T
0
L
I
T
0
0
0
0
0
L
L
1.
0
T
1.
1.
0
0
0
1
0
T
0
t
0
0
L
1
1.
t
C o m b i n a t i o nosf l o g i cg a t e s
A L Ll o g i cg a t e sc a nb e m a d ef r o m o n l yN A N Dg a t e so r o n l yN o Rg a t e s ( R E M E M I
B E RT H t S ) .
T h e m a i nb e n e f i ti s t h a t t h e s u b s t i t u t i odno e sn o t r e d u c et h e a m o u n t g a t e s
of
butinstead
r e d u c e tsh e n u m b e ro f l cC' 'ssr e q u i r e d . ( lm
c e a n sl n t e g r a t e cdi r c u i ta n d i s n o r m a l l yc a l l e da
"chip")
I n o r d e rf o r a n yg a t et o b e a b a s i cb u i l d i n gb l o c kf o r o t h e rg a t e s i,t m u s t
h a v et h e I N V E R T I N G
c a p a b i l i t yh,e n c eA N Dg a t e sa n d o R g a t e sc a n n o tb e u s e dt o b u i l do t h e r g a t e s .
T h e r ea r es o m eb a s i cd e s i g n tsh a t y o u m u s tl e a r ni n o r d e rt o m a k eg a t e s
f r o m N A N Do n l yo r
N O Ro n l y .
( 1 ) F i r s t l ya, N o T g a t ec a nb e m a d ef r o m a s i n g l ei n p u tN A N D
o r a s i n g l ei n p u tN o Rg a t e ;
NAND
Input Output
0,0
1
t,r
0
NOR
Input Output
0,0
I
r,t
0
In both cases,the output is the oppositeof the input,henceit is a NOTgate.
( 2 ) s e c o n d l ya, n A N Dg a t ec a nb e m a d eb y c o m b i n i n ga N A N Dt o
163
a N O Tg a t e :
A
B
C
c
D
0
0
1.
1.
0
0
1
1
1.
0
1
0
1
T
0
1
I
0
0
T
( 3 ) T h eO Rg a t ei s f o r m e db y a d d i n ga N O Tg a t et o e a c hi n p u to f t h e N A N Dg a t e :
A
B
c
D
E
0
0
1.
1,
0
0
1,
I
0
1
T
0
0
t
1
T
1
0
0
!
( a ) T h e N O Rg a t ec a nb e . m a d ef r o m i n v e r t i n g
t h e O Rg a t e .T h i si s d o n eb y a d d i n ga N O T
g a t ea t t h e o u t p u tf r o m t h e p r e v i o u O
s Rg a t e .
T h et r u t h t a b l ef o r t h e o u t p u to f t h i sc o m b i n a t i o ins t h e N O To f t h a t f o r t h e p r e v i o u tsa b l e .
164
B U I L D I NW
GI T HN O RG A T E S
I n o r d e rt o b u i l da n yg a t e sf r o m N O Rg a t e s i,t i s i m p o r t a ntto p e r f o r mt h e f o l l o w i n gs e q u e n c oe f
actions:
( 1 ) D e s i g nt h e c i r c u i w
t i t h N A N Dg a t e so n l y .
( 2 ) R e p l a c e a c hN A N Dg a t ew i t h a N O Rg a t e
( 3 ) A d d N O Tg a t e st o a l l t h e i n p u t sa n do u t p u t s . ( R e m e m bt h
ea
r t t h e N O Tg a t ei s m a d ef r o m
a N O Rg. a t e ) .
( 4 ) R e m o v ea n y p a i r so f N O Tg a t e si n s e r i e sw i t h e a c ho t h e rs i n c et h e y c a n c eol u t e a c h
other.
E X A MP L E :
( 1 ) s h o wt h a t t h e f o l f o w i n gc i r c u i ti s e q u i v a l e nt to a N o Rg a t e :
F i r s t l yr,e p l a c ea l l N A N Dg a t e sb y N O Rg a t e s :
T i i e i t i n s e r t N O Tg a t e sa t A , B ,a n d C .
165
B yc a n c e l l i nogu t t h e p a i r so f N O Tg a t e st h a t a r e i n s e r i e sw i t h e a c ho t h e r ,t h e e n d r e s u l ti sj u s t
t h e l o n eN O Rg a t eX .
E x a m p l 2e :
D e s i g na n A N Dg a t ef r o m u r i n gN O Rg a t e so n l y .
Step1-;Designthe ANDfrom the NANDgatesi.e.
A NANDgatefollowedby a NOTgate
g i v e sa n A N Dg a t e .i . e N O To f N A N D
=AND
S t e p2 ; R e p l a cteh e N A N Dg a t e sw i t h N O Rg a t e s :
166
Step3; lnsertNOTgatesat A, B, and C.
T h e N O Tg a t e s4 a n d 5 a r e i n s e r i e sa n d w i l l c a n c eol u t e a c ho t h e r ,a n dt h e r e s u l t i n gc i r c u i tw i l l b e :
Tw o successi ve
N OTgatesi n ser ies
w i l l a l w a y sc a n c eol u t e a c ho t h e r
= (posit ive)
as (negati ve)(negati ve)
T h ea b o v ec i r c u i tr e p r e s e n tasn A N Dg a t e
O n l o o k i n ga t t h e i n p u t sA a n d B , i t i s s e e nt h a t t h e
O U T P U Ti s i d e n t i c atlo t h a t o f a n A N Dg a t e
A
0
0
1
I
B
0
1
0
t
NOT A
1,
T
0
0
NOT B
1.
0
t
0
1.67
NOT A NOR NOT B
0
0
0
1
The Exclusive-OR
Gate :
Desired Gate
NOR Construction
TruthTable
lnput A Input B OutputQ
000
011
101
110
The ExclusiveNORgate:
Desired Gate
NOR Construction
t=DrTruthTable
Input A Input B Output Q
0
01
0
1
0
1
0
0
1
1
I
168
B u i l d i n gW i t h N A N DG a t e s
The ORgate:
Desired Gate
NAND Construction
:-TFTruthTable
Input A Input B Outpute
000
011
l0l
111
The NORgate :
A NoR gateis simply an oR gatewith an invertedoutput:
Desired Gate
NAND Construction
:IrTruthTable
Input A Input B OutputQ
0
01
0
l0
I
00
I
10
169
TheExclusive
ORgate
A n X O Rg a t ei s c o n s t r u c t esdi m i l a r l tyo a n O Rg a t e ,e x c e p w
t i t h a n a d d i t i o n aNl A N Dg a t e
i n s e r t e ds u c ht h a t i f b o t h i n p u t sa r e h i g h ,t h e i n p u t st o t h e f i n a lN A N Dg a t ew i l l a l s ob e h i g h ,
a n dt h e o u t p u tw i l l b e l o w .T h i se f f e c t i v e lrye p r e s e n t sh e f o r m u l a :" N A N D ( A
N A N D( A N A N D
B ) )N A N D( B N A N D( A N A N DB ) ) " .
Desired Gate
NAND Construction
lflTruthTable
Input A Input B OutputQ
000
011
101
110
The X- NORgate :
An XNOR gate is simply an XOR gate with an inverted output:
Desired Gate
NAND Construction
lilrTruthTable
Input A Input B OutputQ
001
010
100
111
170
NOTES
171
B IN A R YA D D E R S
T h e r ea r et w o t y p e so f B I N A R A
Y D D E RiS. e .t h e H A L FA D D E R
a n dt h e F U L LA D D E R .
T h e H A L FA D D E R
i s u s e dw h e nt w o o n e b i t n u m b e r sa r ea d d e d T
. h ew o r d " b i t " m e a n sb i n a r y
digit.
T h eo n e b i t n u m b e r sc a no n l yh a v et h e v a l u e so f 0 a n d L .
T h e r ea r ef o u r d i s t i n cst i t u a t i o ntso c o n s i d ew
r i t h t h e H A L FA D D E RT.h e ya r e :
carry
I n ( d ) ,t h e s u mo f L + L = 1 0 ,a n dt h e 0 i s c a l l e dt h e S U Ma n dt h e 1 i s c a l l e dt h e C A R R y .
T h ev a l u eo f 1 0 o n t h e D E C I M A sL c a l ei s 2 . T h i si s e x p l a i n eadsf o l l o w s :
t0
The0isequalto0x20=0
\
T h e1 i s e q u a tl o 1 x 2 r = 2
T h eH A L FA D D E R
i s m a d ef r o m a n E X - O Rg a t ea n da n A N DG A T E :
A
sum
A
0
0
B CARRY S U M
0
0
0
I
0
t
I
0
0
1
1. 1.
1.
0
carry
( T h eE X - O R
g a t eg i v e :l h e s u ma n dt h e A N Dg a t eg i v e st h e c a r r y )
172
T h e H A L FA D D E Rc a na l s ob e b u i l tf r o m t w o N O R sa n d a n A N Dg a t e( L E A R N
T H | S! ! ! ! ! ! ! ! )
SUM
T h ec i r c u i tg i v e st h e
f o l l o w i n gr e s u l t s :
CARRY S U M
A
B
c
D
E
0
0
1,
0
0
0
1' 0
0
1
I
0
0
0
1
I
I
0
T
0
0+0=00
0+l-=01
1+0=01
1+1=10
173
T H EF U L LA D D E R
The FULLADDERaddsthree bits at a time.
C o n s i d et rh e f o l l o w i n gb i n a r ya d d i t i o n :
I t + 1 . L ;t h i s i s r e w r i t t e ni n t h e f o l l o w i n gf o r m a tf o r s i m p l i f i c a t i o n :
t1
0t
L
OL
I
1.1.0
bitsfirst,i.e.the digitson the extremeright;
Consider
the leastsignificant
, e n c et h e m i d d l ec o l u m ng i v e s
1 + 1 = l - 0 .T h e 1 i n t h i s r e s u l ti s c a r r i e dt o t h e n e x tc o l u m n h
j
1
1
( 3 b i t sh e r e ,s o t h a t t h r e eI N P U T aSr e n e e d e d ) A
i s r e q u i r e df o r t h i s
. F U L LA D D E R
L + 1+1
part.
T h e r e f o r e1 1 - + 1 1
, , =1 1 0
174
T h e F U L LA D D E R
i s m a d ef r o m t w o H A L FA D D E RaSn d a n O Rg a t e .( H A ra n d H A 2a r e H A L F
ADDERS)
sum
T h et r u t h t a b l ef o r t h e F U L LA D D E R
i s g i v e nb e l o w :
INPUTS
OUTPUTS
A
B
c
0
0
0
0
0
0
0
1
t
0
0
T
0
1
0
1
0
0
t
0
0
1
1
0
t
T
0
T
0
1
1
0
0
1
0
T
I
I
1.
1,
SUM
CARRY
{l*iriLis is ittr.'ilr-r LtL-nl
i*spir"atiixi.ni*etv-i:ine p*r dcltt [']u-rs!lit'ation.
T'homas .{. Edison. 1*irt;r-';"lE
-ffi:;id:jr', jVr.:
175
CIRCUITS
BISTABLE
A BISTABLE
circuithastwo stablestatesof L or 0 .
. order
A c i r c u i ti s c o n s i d e r etdo b e s t a b l ei f i t s o u t p u ts t a y sf i x e de v e na f t e r t h ei n p u tc h a n g e sI n
f o r t h i sc i r c u i t o m a i n t a i ni t s o u t p u tw h i l et h e r ei s a c h a n g ei n i t s i n p u t , i t m u s ti n c o r p o r a t e
in its design.
sometype of LATCH
i s s i m p l ya p a r to f t h e c i r c u i t h a t " l o c k s "o r " s e t s "t h e o u t p u t ,s o i t w i l l n o t c h a n g e
T h eL A T C H
w h e nt h e r ei s s o m ec h a n g ei n t h e i n p u ts i g n a l .
c a nb e u n d e r s t o o idn t h e f o l l o w i n gc i r c u i t :
o f o p e r a t i o no f t h e L A T C H
T h ep r i n c i p l e
A L AT C HC IR C U IT
output
input
, n d h e n c et h e i n p u tt o B i s h i g h .
W h e nt h e i n p u tA i s h i g ht h e o u t p u ta t C i s h i g h a
g
o
e
s
l
o
w
,
t
h
e
i
n
p
u
t
a
t
B
d
o e sn o t c h a n g ea n d h e n c et h e o u t p u ta t C i s
to
W h e nt h e . i n p utto A
s t i l lh i g h .
t o u l db e t o a s s u m et h a t w h e nt h e i n p u ta t A w a sc u t
A s i m p l ew a y o f u n d e r s t a n d i nt hgi sc i r c u i w
o f f i . e .s e n tf r o m L t o 0 , t h e r ew a sa d e l a y e dr e a c t i o ni n t h e o u t p u t( e g .a m i c r o s e c o n d. )T h i s
d e l a yi s s u f f i c i e nt to a l l o wt h e o u t p u ts i g n atlo f l o w t o B s o t h a t t h e i n p u tt o t h e O Rg a t ew o u l d
b e l a n d0 , h e n c ei t s o u t p u tw o u l db e 1 , a n dt h i so u t p u tw i l l b e c o n t i n u o u s l y f ebda c k t oB s o
t h e O Rg a t ew i l l h a v ea l w a y sa n o u t p u to f 1
T h ea b o v ec i r c u i th a so n e s t a b l es t a t e , b u t f o rp r a c t i c apl u r p o s e ss,o m ec i r c u i t sa r e d e s i g n e d
ircuits.
w i t h t w o s t a b l es t a t e sa n dt h e s ea r ec a l l e dB I S T A B Lc E
A B I S T A B Lc E
i r c u i tc a nb e m a d ef r o m t w o N O Ro r t w o N A N Dg a t e sa, n d s u c ha c i r c u i ti s c a l l e d
a n S - RB I S T A B L E .
The "S" refersto SETandthe "R" refersto RESET.
176
C o n s i d et rh e B I S T A B Lc E
i r c u i t h a t i s m a d ef r o m N O Rg a t e s :
a
a
Q'
Q'
OPERATION:
C A S E1 :
T h e o u t p u to f a N O Rg a t ei s L o n l y i f b o t h i n p u t sa r e 0 . H e n c ei f S o r R i s L , t h e o u t p u to f t h e p a r t i c u l a r
g a t em u s t b e 0 .
F o re x a m p l ei,f S = 1 a n d R = 0 , t h e o u t p u to f G A T E1 i s 0 i . e .Q = 6 .
T h i so u t p u tQ i s f e d t o G A T E2 , s o t h a t n o w G A T E2 h a st w o i n p u t st h a t a r e0 , a n d i t so u t p u tw i l l b e 1 .
i.e. Q' -1,.
T h i s Q ' i s n o w f e d i n t o G A T E1 , s o t h a t G A T E1 n o w h a st w o i n p u t st h a t a r e 1 ( S = 1f r o m b e f o r e) , s ot h a t
Q r e m a i n sa t 0 .
I n s u m m a r v . w h esn= 1 a n d R = 0 . t h e c i r c u i t i s s t a b l e w i et h= 0 a n d e ' = 1 .
CASE
2
'lf S is nowresetto0 sothatS = R =0,
GATE
l stillhasan inputof l from q, sothattheoutputat eis still0.
ThisQ =0 isfed intoGATE2 sothat bothinputsto thisgate=0 andthe outputCf remainsat 1.
CASE3 :
R i s n o w r e s e t o l l e a v i n gs = 0 . T h e i n p u t st o G A T E2 a r e n o w l a n d0 ( e w a s 0 i n t h e p r e v i o u cs a s e ) ,
h e n c eQ ' = 0 a n d t h i s b e c o m e st h e s e c o n di n p u tt o G A T E1 . T h i sG A T E1 t h e r e h a st w o i n p u t so f 0 , s o
t h a t i t so u t p u t
Q = l w h i c hi s f e d b a c kt o G A T E2 t o e n s u r et h a t Q ' r e m a i n sa t O .
CASE4:
R r ss w i t c h e dt o 0 w i t h S = 0 , Q i s s t i l la t L s o g a t e2 h a si n p u t so f 0 a n d 1 s o t h e o u t p u tQ ' i s 0 .
T h eo u t p u t sa r et h e r e f o : €U n c h a n g eads R i s s w i t c h e dt o 0 .
177
i s 'SET"by settingS = 1 and R=0.
The BISTABLE
i s 'RESET"by settingS =0 and R=1.
The BISTABLE
circuit
TheTruth table for the BISTABLE
(uslNGNoRGATEs)
S
R
t
T h et r u t h t a b l es h o w st h a t w h e n S i s
"set" at 1-, with R beingkePtat 0, the
o u t p u t sQ a n d Q ' a r e s t a b l e ( u n c h a n g e d )
I
w h e n S i s c h a n g e dt o 1 .
a a'
1 0 0
0 0 0
0 L I
0 0 1
0
H o w e v e tr h e o u t p u t sc h a n g ew h e n R i s
" R e s e t "t o 1 , a n dt h e s eo u t P u t sr e m a i n
0
s t a b l e( u n c h a n g e dw)h e n R i s c h a n g e d
to 0.
T h e o u t p u t sc a n b e c h a n g e da g a i nb Y
r e s e t t i n gt o t h e o r i g i n asl t a t ew i t h S =1
a n d R =0 .
t78
BistablecircuitusingNANDgates
S
R
a
0
0
Q'
T h et r u t h t a b l es h o w st h a t w h e n S i s
"set" at 1-, with R beingkept at 0, the
1
0 0 0 1
0 I 1, 0
0 0 1 0
L
o u t p u t sQ a n d Q ' a r e s t a b l e ( u n c h a n g e d )
w h e n S i s c h a n g e dt o 1 .
H o w e v e trh e o u t p u t sc h a n g ew h e n R i s
" R e s e t "t o 1 - a
, n dt h e s eo u t p u t sr e m a i n
s t a b l e( u n c h a n g e dw)h e n R i s c h a n g e d
to 0.
T h eo u t p u t sc a n b e c h a n g e da g a i nb y
resettingto the originalstatewith S=1
a n d R =0 .
179
F L I PF L O P S
F l i pF o p sa r e b i s t a b l es t o r a g ed e v i c e tsh a t a r e c a p a b l eo f s t o r i n go n e b i t .
T h ed i f f e r e n c b
e e t w e e na L A T C H
a n da F L I P - F L O
i sPt h e m e t h o dt h a t i s u s e dt o c h a n g et h e i r
states.
L a t c h eas r ec o n s i d e r etdo b e " L e v eTl r i g g e r e d i".,et h e y c h a n g es t a t ed e p e n d i n o
gn t h e v o l t a g e
l e v ea
l pplied.
F l i p - F l o pasr es a i dt o b e " E d g eT r i g g e r e d " , i .teh.e o u t p u td e p e n d so n t h e T R A N S I T I Oo N
f the
gd g e ) .
s i g n awl h i c hc a nb e L O Wt o H I G H ( r i s i n
eg
d g e )o r H I G Ht o L O W( f a l l i n e
T h et e r m s" r i s i n ge d g e "a n d " f a l l i n ge d g e "r e f e r t ot h e s q u a r ew a v es i g n atlh a t i s a p p l i e dt o t h e
deviceto causeit to changeits state.
T h i ss q u a r ew a v es i g n ails c a l l e da c l o c kp u l s e .
Rising
edge
F a l l i n ge d g e
T h ed i a g r a . m
s h o w st h a t f o r t h e c l o c kp u l s et h e " R l S l N G
E D G Ei's a t t h e l e f t s i d eo f t h e
GD G Ei"s a t t h e r i g h ts i d eo f
r : e c t a n g u lparro j e c t i o n(sa r r o w sp o i n tu p w a r d sw
) h i l et h e " F A L L I N E
t h e r e c t a n g u l aprr o j e c t i o n(sa r r o w sp o i n td o w n w a r d s ) .
180
ACTION
OFFLIP
FLOPS
T y p i c aFl L I P - F L OaPrSe r i s i n ge d g et r i g g e r e di ,. e .t h e o u t p u tc h a n g eas t t h e r i s i n ge d g eo f e a c h
c l o c kp u l s e .
T h eT - t y p eF L I P - F L O P :
The "T" in T-typestandsfor the word TOGGLEt h i s m e a n st h a t t h e o u t p u t sQ a n d Q ' t o g g l eo r
e x c h a n g set a t e sa t e a c hr i s i n ge d g e .
T
a
Q'
T h ef l i p - f l o ph ' a st h e f o l l o w i n gd i a g r a m
T h ee x p l a n a t i o n
f o r t h e s ea r r o w s
t o t h e r i g h ti s
g i v e nb e l o w
0
0
1
1
I
0
0
I
0
1
0
I
0
0
I
1,
L
0
+
t
+
t
T h eT r u t ht a b l ea n dt i m i n gd i a g r a masr eg i v e na s
follows:
I
J-O
I n t h e t r u t h t a b l et h e r ea r es o m ea r r o w st o t h e l e f t o f i t . T h ef i r s ta r r o wi n d i c a t etsh a t t h e c l o c k
p
. u l s ei s o n 0 b u t m o v e su p t o 1 . T h i si s a l s os h o w no n t h e c l o c kp u l s et i m i n gd i a g r a mT. h e
s e c o n da r r o wf r o m t h e t r u t h t a b l e ,i n d i c a t etsh a t t h e i n p u tl e v e li s 1 a n dt h e r e f o r em o v e sd o w n
t o 0 e t c .T h i si s n o t s h o w no n t h e t i m i n gd i a g r a mb e c a u s w
e e a r e u s i n gt h e t i m i n gd i a g r a mt o
s h o wo n l yt h e R I S I N E
GD G E Sa,st h e f l i pf l o p i s o n l ya c t i v a t eodn t h e r i s i n ge d g e s .
T h et r u t h t a b l ea l s os h o w st h a t t h e o u t p u t sQ a n d Q ' t o g g l ea t e a c hr i s i n ge d g eo f a c l o c kp u l s e .
F o re x a m p l ea, t t h e f i r s tc l o c kl e v e l, Q = 0 a n d Q ' - 1 . A t t h e s e c o n dp u l s eo f L t h e i n p u tw o u l d
h a v eR I S E N
f r o m C t o 1 , i . e .i t i s a R I S I N G
e d g e ,a n dt h e o u t p u t sw o u l dt o g g l es o t h a t Q = 1 a n d
( j" 1.
181
t o g g l ea s
W h e nt h e c l o c kl e v e lf a l l sb a c kt o Ot h i s i s a f a l l i n ge d g ea n dt h e o u t p u t sw o u l dN O T
n e x tr i s i n g
t h e y r e s p o n do n l yt o r i s i n ge d g e s ,h e n c et h e o u t p u t sr e m a i nu n c h a n g e dA.t t h e
states.
edge,therewould againbe togglingi.e.the Q and Q' would exchange
T h et i m i n gd i a g r a mfso r i n p u ta n do u t p u t sa r es h o w nb e l o w
a
Q'
l & , a n do u t p u t sa r es h o w nh e r e .T h e
lettersA, B, C, D, E,and F are not part of the diagrambut they have
t oints.
b e e ni n s e r t e da s l a b e l sf o r s o m ei m p o r t a n p
A t A t h e i n p u ti s a t a r i s i n ge d g ea n d Q w o u l ds w i t c ht o 1 w h i l eQ '
wouldswitchto 0.
A t B , t h e r ei s a n o t h e r i s i n ge d g ea t t h e i n p u t ,s o Q a n d Q ' w o u l d
n o w t o g g l e. i . e .Q w o u l dg o t o 0 a n d Q ' w o u l dg o t o 1 '
A t C , t h e r i s i n ge d g eo n t h e i n p u tw o u l da g a i nc a u s et o g g l i n g
Q to go to l- and Q' to 0.
,causing
A t o B , B t o C ,C t o D , D t o E , a n d E t o F ,h a v eo n l y
T h es p a c e s f r o m
F A L L I NeGd g e so n t h e i n p u tt i m i n gd i a g r a ma n dt h e r e f o r ed u r i n g
to both Q and Q'.
thereare no changes
theseintervals,
1,82
COUNTERS
A THREE
BITbinarycountercan be madefrom eitherthree fallingedgetriggeredtogglingFLlpFLOPsor three risingedgetriggeredtogglingFLlp-FLOpS.
T h ea r r a n g e m e nf ot r t h e t h r e ef a l l i n ge d g eT - F L I PF-L O P iSs a sf o l l o w s :
Qz(m.s.b.)
Most SignificantBit
Q'
Q r ( 1 .bs .)
LeastSignificantBit
Clock
pulses
A
I t i s i m p o r t a nt o n o t et h e s i g n i f i c a n coef t h e s y m b o l fso r R I S I N G
EDGE
TRIGGERE
Dd F A L L I N G
an
EDGE
TRIGGERED.
The FALLING
EDGE
TRIGGERED
is denotedby
1pr-
T h eR I S I N E
GD G E
T R I G G E R iEsD
d e n o t e db y +
I t i s a n a r r o ww i t h a b u b b l ec l o s e
to thetip
T h e r ei s n o b u b b l e
183
T h et i m i n gd i a g r a masn dt h e c o r r e s p o n d i tnrgu t ht a b l ea r eg i v e nb e l o w :
C l o c kp u l s e s
a o( l . s . b . )
Q, (m.s.b.)
Remembe
: rL s . bm e a n L
s E A SSTI G N I F I C ABNI TT
m . s . bm e a n M
s OSS
TIGNIFICA
BN
I TT
W h e nt h e c l o c kp u l s ei s 0 t h e t o t a l o u t p u ti s 0 .
W h e nt h e c l o c kp u l s e= I , t h e o u t p u ti s 0 0 1 =1 , =2 0( o n l yt h e f i r s tf l i p - f l o pi s o n )
W h e nt h e c l o c kp u l s e= 2 , t h e o u t p u t= 0 1 0= 0 + 2 r + O =2 ( t h es e c o n df l i p - f l o pi s o n a n dt h e
first is off)
W h e nt h e . c l o c p
k u l s e= T , t h e o u t p u t= ! ! L = 2 2+ 2 1+ 2 0= 7 e t c . ( A ltlh r e ef l i p - f l o p a
s r eo n )
W h e nt h e c l o c kp u l s ei s 5 , t h e t i m i n gd i a g r a ms h o w st h a t t h e 3 ' of l i p f l o p i s o n , t h e s e c o n di s o f f
a n dt h e f i r s ti s o n t o g i v e 2 2+ 0 + 2 0= 4 + 1 = 5
Outputs
Numbeo
r f Qz=22=4 Qt= 2'= 2
C l o c kP u l s e m . s b. .
0
0
0
I
0
0
2
0
I
3
0
1
4
1,
c
5
1,
0
6
1,
1,
7
1,
T
1.84
Q o = 2 u= 1
l.s.b.
0
1
0
I
0
I
0
1,
MODULE
3
Atomic and NuclearPhysics
I M P O R T A NFTO R M U L A E :
(1) Photon Energy E - hf
(2)Work functi.onenergy e:
ltfo
( 3 ) E t n s t e t n 's P l t o t o e l e c t r ie
c q u a t i " o n: h f : ,
*;mv2
OR ltf - E + eV,
(4)Attenuatton of X ra"ys:I : Io€GFx)
(S)Energy LeuelTransi"tton AE - hf : Ez - Er
$ ) f h e d . eB r o g l i . eW a u e t e n g t l tl , : L
p
(7)Massnumber-A*Z
(B)Einstei"n'sMassEnergy Relatton L,E = Lmc2
(e) Acti.utty
A =#:
-AN
(I})Number of nuclei present N : Noe-At
( 1 1 )H a t fL i' fe r r , : Y
/2
(1 2 )A c ti u i .ty A = Ao (;)"
A
w here n i s ttte number of hal f ti " ues
185
PARTIC
EFFECT
ThCPHOTOELECTRIC
ROMAG
TE NATUREOF E
RA
, , p a r t i c u l aN
t ea t u r e "m e a n sp a r t i c l ep r o p e r t i eis. e .l i g h ta sw e l l a sa l l o t h e r t y p e so f
T h et e r m
X -rays,radiowavesetc,) behaveas particlesin
radiation(eg.microwaves,
electromagnetic
g a v ep r o p e r t i e s .
a d d i t i o nt o e x h i b i t i n w
CE CoTf l i g h t .
T h i sp a r t i c u l a tnea t u r ei s d e m o n s t r a t eidn t h e P H O T O E L E C TERFI F
Thisphotoelectriceffectcanbestbe explainedasfollows:When lightof a suitablyhigh
frequencyfallson a metalsurface,electronsare emittedfrom the metalsurface.
t h e i re m i s s i o cn a nb e d e m o n s t r a t ebdy
d r ec a l l e dP H O T O E L E C T RaOnNdS
sm.ittea
T h ee l e c t r o n e
meansof a GoldLeafElectroscope.
u . v .l i g h t
lll-e
lila
Polished
Zinc
plate
+++
l
E x p ln
aa t i o n :
is givena negativechargeandthis causesits leavesto divergedue
A Goldleafelectroscope
nf charges.
the repulsioo
A sheetof polishedzincis placedat the top of the capof the ele.ctroscope.
is
U l t r a v i o l el it g h tf o r a m e r c u r yv a p o u rl a m pi s t h e n i l l u m i n a t e od n t h e z i n cp l a t ea n d i t
to
beginto collapse'
observedthat the leavesof the electroscope
c h a r g ew h i c hw o u l dh a v eo c c u r r e d
e f t h e l e a v e si s d u et o t h e l o s so f n e g a t i v e
T h i sc o l l a p s o
were emittedfrom the zincsurface.
rvhenphotoelectrons
p l a t ep l a c e do n
W h e nt h e e l e c t r o s c o pwea s i n i t i a l l yg i v e na p o s i t i v ec h a r g ea, n dt h e nt h e z i n c
t h e c o p ,t h e r ew a sn o c h a n g ei n t h e d i v e r g e n coef t h e l e a v e sw h e nt h e z i n cw a si l l u m i n a t e d
were
with uv " Thismeansrhattherewas no changeof positivecharge,and photoelectrons
attractionof the
asthey were heldin placeby electrostatic
to leavethe electroscope
ur.lilr:l€
positivechargeson the electroscope.
186
Onesignificant
otherobservation
with the caseof the negatively
chargedelectroscope,
was
t h a t w h e n a s h e e to f g l a s sw a s i n s e r t e db e t w e e nt h e u v . s o u r caen dt h e z i n cp l a t e ,t h e r ew a sn o
collapso
ef the leaves..
T h i so b s e r v a t i ocna nb e e x p l a i n e a
d sf o l l o w s :
.
T h eg l a s sc a u s e tsh e R E F R A C T I o
OfNl i g h t .T h eu v l i g h th a sa v e r ys m a l w
l a v e l e n g t ah n d i s t o t a l l y
i n t e r n a l l rye f l e c t e d
s o i t i s u n a b l et o r e a c ht h e z i n cp l a t e .
T h eo t h e rl o n g e rw a v e l e n g t hwsi l l p a s st h r o u g hb u t d o n o t h a v es u f f i c i e net n e r g yt o s u p p l yt o
the electronsto causethem to escapefrom the metalsurface.As a result,the pHOTOELECTRIC
EFFECT
doesnot occur.
L a w so f P h o t o e l e c t r iEcm i s s i o n :
( 1 ) T h e p h o t o c u r r e nw
t h i c h i s t h e n u m b e ro f p h o t o e l e c t r o npse r s e c o n d i, s p r o p o r t i o n a l
t o t h e I N T E N S I ToY
f t h e i n c i d e n rt a d i a t i o n .
( 2 ) T h e m a x i m u m K i n e t i cE n e r g yo f t h e p h o t o e l e c t r o niss d i r e c t l yp r o p o r t i o n atlo t h e
F R E Q U E N Co Y
f t h e i n c i d e n tr a d i a t i o na, n d i s i n d e p e n d e not f t h e i n t e n s i t yo f t h e
i n c i d e n tr a d i a t i o n (. t h e p h o t o e l e c t r o nasr e e m i t t e d w i t h a r a n g eo f k i n e t i ce n e r g i e s
f r o m z e r ou p t o a m a x i m u mv a l u e .
( 3 ) F o r a n y p a r t i c u l am
r e t a l ,t h e r e i s a m i n i m u mf r e q u e n c yo f r a d i a t i o nt h a t w i l l c a u s e
p h o t o e m i s s i oT
nh
. i sm i n i m u mf r e q u e n c yi s c a l l e dt h e T H R E S H O LFD
REeUENCY
,d
an
b e l o wt h i sf r e q u e n c yn, o p h o t o e m i s s i ot na k e sp l a c e .
A c c o r d i ntgo W a v eT h e o r yt,h e i n t e n s i t yo f r a d i a t i o ni s t h e p o w e rp e r u n i t a r e a ,s o a g r e a t e r
l n t e n s i t yi m p l i e sa g r e a t e ra m o u n to f e n e r g yi n c i d e not n t h e m e t a ls u r f a c el .t w o u l db e
r e a s o n a b lteo a s s u m et h e n t h a t w h e nt h e i n t e n s i t yi s i n c r e a s e dt h, e p h o t o e l e c t r o nssh o u l d
a c q u i r ea g r e a t e ra m o u n to f K i n e t i cE n e r g ya st h e y l e a v et h e m e t a ls u r f a c e .
However, this is not observedand the fact that the kineticenergyof the photoelectrons
is
i n d e p e n d e notf i n t e n s i t y , ( L a2w) i s a c a s eo f t h e F A I L U RoEF T H EW A V ET H E o R yo F L t G H T .
Alsoaccording
to WAVETHEORY,
the energytransportedby a waveis continuousand for
f r e q u e n c i ebse l o wt h e t h r e s h o l df r e q u e n c yt h
, e e l e c t r o ns h o u l db e a b l et o a c q u i r es u f f i c i e n t
energyaftera periodof time to be ableto escapefrom the metalsurface.Thisis not observed
a n d i s t h e r e f o r ea n o t h e rc a s eo f t h e F A I L U RoEF T H EW A V ET H E o R yo F L I G H T .
T h eo b s e r v a t i o ni n
s the photoelectre
i cf f e c tc a no n l yb e e x p l a i n e b
d y a s s u m i ntgh a t t h e
r a d i a t i o nh a sa P A . : , T I C U L A
NTAET U R E
T .h a ti s ,e l e c t r o m a g n e triacd i a t i o nb e h a v e sl i k ea s t r e a m
of iiiscreteparticles.
1.87
THEQUANTUMTHEORY
A L B E RE
T I N S T E IdNe v i s e da t h e o r yt h a t w o u l da c c u r a t e leyx p l a i nt h e P h o t o e l e c t rE
i cf f e c t :
T H E Q U A N T U MT H E O R Y .
T h i st h e o r yi m p l i e st h a t a l l e l e c t r o m a g n e rt a
i cd i a t i o ins m a d eu p o f i n d i v i d u auln i t sc a l l e d
PHOTONS.
A PHOTONis a basicunit of electromagnetic
radiationthat behavesas a discreteparticle.
( L E A R)N
E a c hp h o t o nh a sa f i x e da m o u n to f e n e r g yc a l l e da Q U A N T I . J o
Mf e n e r g y .
l f t h e i n c i d e net n e r g yh a sa f r e q u e n c y /t,h e nt h e q u a n t u mo f e n e r g yi n a p h o t o ni s g i v e nb y :
E:hf
W h e r eh i s a c o n s t a nct a l l e dt h e P l a n c k 'cso n s t a n(t h =6 . 6 3x 1 O - 3 a J s )
A l l e l e c t r o m a g n e triac d i a t i o nh a v ea s p e e dc = 3 x 1 0 8 m s1 i n a v a c u u ma, n dt h e w a v e l e n g tohf
( f r o mv : f 7 ) .
a n ye . m .r a d i a t i o ins g i v e nA VA :
|
ht
.
H e n c et h e e n e r g yo f a s i n g l ep h o t o nc a na l s ob e w r i t t e na stE- ' 1'
E_
hc
1
-hf
!-EARNI
i cf f e c ta sf o l l o w s :
T h ep h o t o nm o d e lc a ne x p l a i nt h e r e s u l t so f t h e p h o t o e l e c t r e
T h ei n t e n s i t yo f t h e i n c i d e nrt a d i a t i o ni s a m e a s u r eo f t h e n u m b e ro f i n c i d e npt h o t o . npse r
s e c o n da, n dw h e nt h e s ep h o t o n ss t r i k et h e m e t a ls u r f a c et ,h e r ei s a d i r e c t r a n s f e o
r f energy
f r o m e a c hp h o t o nt o a p a r t i c u l aer l e c t r o nH
. e n c ea g r e a t e ri n t e n s i t yo f i n c i d e nlti g h tm e a n sa
g r e a t e rn u m b e ro f p h o t o n sp e r s e c o n ds t r i k i n gt h e m e t a ls u r f a c ea, n dt h i sw o u l dm e a na
g f e a t e rn u m b e ro f p h o t o e l e c t r o nesm i t t e dp e r s e c o n da so b s e r v e d .
W h e nt h e f r e q u e n c o
y f t h e i n c i d e nlti g h ti s i n c r e a s e dt h, e e n e r g yo f e a c hp h o t o ni s i n c r e a s e(dE
= h f ) ,c a u s i n tgh e p h o t o n st o n o w t r a n s f e a
r g r e a t e ra m o u n to f e n e r g yt o t h e e l e c t r o n sw, h i c h
i n t u r n c a u s e tsh o s ep h o t o e l e c t r o nt os n o w h a v ea g r e a t e rK Ew h e nt h e y l e a v et h e m e t a l
surface.
\ l / h e nt h e i n c i d e nrt a r J i a t i oi n
s b e l o wt h e T H R E S H O L
FD
REQUENC
t hYe, e n e r g yo f t h e p h o t o ni s
'low;:r,ij
t h e e n e r g yt r a n s f e r r etdo e a c he l e c t r o ni n t h e m e t a li s t o o l o w f o r i t t o e s c a p ef r o m t h e
188
m e t a ls u r f a c eT. h ee n e r g yi s n o t c o n t i n u o uasn dt h e e l e c t r o ncsa n n o ta b s o r bm o r e
t h a no n e
q u a n t u ma t a n yt i m e ,a n da r et h e r e f o r eu n a b l et o e s c a p teh e m e t a l .
PROPERTIES
OFTHEMETALSURFACE
F o re a c hp a r t i c u l am
r e t a l ,t h e r ei s a m i n i m u mf r e q u e n c o
y f l i g h tt h a t c a nb e r a d i a t e do n i t s
s u r f a c et o c a u s ep h o t o e m i s s i o n
T h i sm i n i m u mf r e q u e n c o
y f e l e c t r o m a g n e triacd i a t i o n
t h a t i s r e q u i r e dt o c a u s ep h o t o e l e c t r o n
e m i s s i o ins c a l l e dt h e T H R E S H O F
LR
D E e U E N COYRC U TO F FF R E e U E N C
f oYr t h e m e t a l .
R e c a ltlh a t c = ' f l h e n c e l - : . l f t h e t h r e s h o l d
f r e q u e n c yi s d e n o t e db y f o ,t h e n t h e
I
c o r r e s p o n d i nwga v e l e n g t ihs c a l l e dt h e T H R E S H o LWDA V E L E N G T
oH
r C U T - O FW
F AVELENGTH
and is denotedby trg.
C
Ao: r
lo
T h em i n i m u me n e r g yt h a t a p h o t o n m u s th a v et o c a u s ep h o t o e l e c t r oenm i s s i o n
i s d e n o t e db y
$ a n d i s c a l l e dt h e w o r k f u n c t i o ne n e r g yo f t h e m e t a l
Q : hf o
L E A RT
NH I S :
The WoRK FUNCTIoNenergyof a metal is the minimum energythat an incidentphoton
must havein
order to causethe emissionof a photoelectron.
W h e nt h e e n e r g yo f t h e p h o t o ni s e q u a l t ot h e w o r k f u n c t i o ne n e r g yo f t h e m e t a l ,t h e e m i t t e d
p h o t o e l e c t r odno e sn o t h a v ea n y i n c r e a s ien i t s K i n e t i ce n e r g y .
H o w e v e ri,f t h e e n e r g yo f t h e p h o t o n( E : h f ) i s G R E A T Et hRa nt h e w o r k f u n c t i o ne n e r g y
of
t h e m e t a l ,t h e nt h e d i f f e r e n c ien t h e t w o e n e r g i e (si . e .h f g
o
e
s
i
n
t
o
i
n
c
r
e
a
s
i
n
g
t
h
e
K
I
N
ETIC
il
E N E R GoYf t h e p h o t o e l e c t r o n .
T h ep h o t o e l e c t r o nasr e l i b e r a t e d
w i t h a r a n g eo f K i n e t i ce n e r g i e u
s p t o a m a x i m u mv a l u ew h i c h
is denotedby KEr.".
KE."*= hf - O. Thisis EtNSTE|N,S
PHOTELECTRTC
EeUAT|ON.
L E A R NT H I : :
Einstein's
Photoelectric
equation
is:KE.",= hf - e ORKE.."=!^uko*:
189
hf - e
POTENTIAL
STOPPING
W h e n l i g h ts t r i k e st h e m e t a l
akes
p l a t eC , p h o t o e m i s s i ot n
C o n s i d et rh e f o l l o w i n gc i r c u i t :
p l a c e( a s s u m i nt gh a t t h e
V a c u u mc h a m b e r
f r e q u e n c yo f l i g h ti s
s u f f i c i e n t lhyi g h ) .
P h o t o e l e c t r o nasr e l i b e r a t e d
from C and moveto strikethe
p l a t eA .
T h e m i g r a t i o no f e l e c t r o n s
f r o m C t o A c o m P l e t etsh e
c i r c u i ta n d a c u r r e n ti s
d e t e c t e da n d d i s P l a Y eodn t h e
ammeter.
a
e c r o s sA Ca s s h o w n :
C o n s i d enr o w t h e c a s ew h e r et h e r ei s a p o t e n t i adl i f f e r e n c a
negative
adjusteduntilthe plateA is sufficiently
The reversepotentialis continuously
t o i u s t r e p e la n y p h o t o e l e c t r o nf rso m r e a c h i n igt , t h u sc a u s i n tgh e p h o t o c u r r e n t
r e c o r d e db y t h e a m m e t e rt o f a l lt o z e r o
A t t h i s p o i n t ,t h e r e v e r s ep o t e n t i aal c r o s sA Ci s c a l l e dt h e s t o p p i n gp o t e n t i aal n d i s
to crossthe gapacross
denotedby V, , and the energyneededby the photoelectrons
- O
ACwould havebeenequafto ] mvlro, - llf
the energyireededby an electronto overcomea barrierpotentialis equal
Recallr!-rat
the stopping
s u s th a v et o o v e r c o m e
i o e V ,i l e n c et h e K i n e t i ce n e r g yt h a t t h e e l e c t r o n m
p o t e n t i ails e q u a lt o e V '
190
H e n c e€ V , - K E ^ o * - ! * u , ^ q x :
hf - O.
i cq u a t i o nc a nt h e r e f o r eb e r e = w r i t t e na s :
E i n s t e i n 'psh o t o e l e c t r e
'J-
't
r
€rV
,: ; m v k o * - h f - 0 .
L
B u t@: h f o ;
€Vr: hf - hfo: h(f - f).
T h i se q u a t i ocna nb e r e -a r r a n g eadsf o l l o w s :
v' : h f - h f o
ee
l f a g r a p ho f S t o p p i n gP o t e n t i avl e r s u sf r e q u e n c iys p l o t t e d i, t w o u l dh a v et h e f o l l o w i n g
shape:
Stopping
Potential
fo
frequency
The pointwherethe graphcutsthe x-axisrepresents
a stoppingpotentialof zeroand this is
p o i n tw h e r et h e f r e q u e n c o
y f t h e i n c i d e nrt a d i a t i o ni s e q u a lt o t h e T H R E S H o LFDR E Q U E N c y
F O RT H EM E T A L .
T h i sc a nb e d e d u c e df r o m t h e e q u a t i o n :
v' : h f - h f o
e
V ,w i l l b e e r o w h e n
hf
l"l_-
hfo
ee
W h i c hm e a n st h a t
f =foat V, =0
191
Fromthe previousgraph,the gradientwould be equalto
I
can
The stoppingpotentialfor a metal irradiatedwith photonsof sufficientlyhigh energy
that must be appliedacross
DIFFERENCE
POTENTIAL
thereforebe definedas the REVERSE
preventthe
the metal surfaceand a collectorplate in an evacuatedchamberto iust
photoelectronsfrom reachingthe collector'
T h e e n e r g yo f a p h o t o nc a nb e e x p r e s s eidn J o u l e so r a l t e r n a t i v e liyn E L E C T R o N V o L T S ( e V )
L e V= e l e c t r o n i c h a r g ex l V o l t = 1 ' 6 x L 0
1eJ'
LRAPHS:
O T H E RU S E F UG
rrent
Photocu
4rlntensitYr (C)
Intensityz(B)
\
I n t e n s i t v 3( A )
A<B <C
-vr.
t atures:
T h eg r a p hs h o w s3 s i g n i f i c a nf e
( 1 ) A s t h e i n t e n s i t yi s i n c r e a s e dt h, e m a x i m u mp h o t o c u r r e ni tn c r e a s e s '
s u r f a c ea, n d a s a
A g r e a t e ri n t e n s i t ym e a n st h a t t h e r ea r e m o r e p h o t o n sp e r s e c o n ds t r i k i n gt h e m e t a l
> T h i sl e a d st o a
r e s u l t h e r ew o u l d b e m o r ee l e c t r o n sp e r s e c o n db e i n ge j e c t e df r o m t h e m e t a ls u r f a c e
g r e a t e rp h o t o c u r r e n t .
( 2 ) T h e p h o t oc u r r e n ti n c r e a s efsr o m z e r o( a t - V , ) u p t o a m a x i m u ma t
the
V = 0 .A s t h e p . d i s m a d el e s sa n d l e s sn e g a t i v et ,h e r e i s l e s sr e s i s t i v feo r c ea c t i n go n
p h o t o e l e c t r o nhse n c em o r eo f t h e m w o u l d b e a b l et o t r a v e lf r o m t h e m e t a ls u r f a c et o t h e
f r e et o
a n o d e .A t V = 0 , t h e r ea r e n o r e s i s t i v feo r c e sa c t i n go n t h e p h o t o e l e c t r o nasn d a l l a r e
e f V b e y o n dz e r od o e sn o t
t .h e i n c r e a s o
m o v et o t h e a n o d et o p r o d u c et h e p h o t o c u r r e n T
remain
i n c r e a s teh e p h o t o c u r r e nat st h e n u m b e ro f p h o t o e l e c t r o nl se a v i n gt h e m e t a ls u r f a c e
s n l y i f i n t e n s i t yc h a n g e s )
u n c h a n g q (dt h i sc h a n g e o
( 3 ) R e g a r d l e sosf h o w t h e i n t e n s i t yi s c h a n g e dt,h e s t o p p i n gp o t e n t i a l( - V , )r e m a i n su n c h a n g e d '
which is
T h e s t o p p i n gp, o t e n t i a li s o n l y d e t e r m i n e db y t h e e n e r g y o f t h e p h o t o e l e c t r o n s
d e t e r m i n e db y t h e f r e q u e n c yo f t h e i n c i d e n rt a d i a t i o n
192
TUTORIAL
SHEET
(DATA: speed"of li"ghtls 3 x 1O8ms-1,Planck constanth _ 6.63 x 10-34Js
m a s s o f e l e c t r o n : 9 , 1 x 1 0 - 3 t k g , c h a r g eo f e l e c t r o n : r . 6
L0-1eC)
i
(1) A metalsurfacehasa cut off wavelengthof O.7zgm.calculate
( a )T h ec u t o f f f r e q u e n c y
( b )T h ew o r k f u n c t i o ni n b o t hJ o u l e sa n d e l e c t r o n v o l t s
( c )T h e m a x i m u mK Eo f t h e p h o t o e l e c t r o nwsh e nt h e i n c i d e nlti g h th a sa w a v e l e n g t h
of 0.42;rm.
( d )T h em a x i m u ms p e e do f t h e e m i t t e dp h o t o e l e c t r o nwsh e nt h e s u r f a c ei s
i r r a d i a t ew
d i t h l i g h to f w a v e l e n g tO
h .42gm
(Ans4
; . r 7 x l , o r a H z ; 2 . 7x61 0 - 1 s1J .; 7 3 e V; 1 . 9 gx 1 O - i e; 6
J .6 x 10sms-1)
( 2 ) l f a m e t a ls u r f a c eh a sa w o r k f u n c t i o no f 3 . 2 e Vf,i n d :
(i)
T h et h r e s h o l dw a v e l e n g t h
(ii)
T h et h r e s h o l df r e q u e n c y
( i i i ) T h e m a x i m u mK Eo f t h e e m i t t e dp h o t o e l e c t r o ni fst h e s u r f a c ei s i r r a d i a t e d
with ligfrtof wavelength
O.2Zp,m.
( A n s :3 . 9x 1 , 0 m
- 7; 7 . 7 xl O r a H z ; 3 . gx21 0 - 1 e J )
( 3 ) A m e t a ls u r f a c eh a sa w o r k f u n c t i o no f 4 e V .W h a t m u s tb e t h e f r e q u e n c ya n d
w a v e l e n g to
h f i n c i d e nlti g h to n i t t o c a u s ep h o t o e l e c t r o nt os b e e m i t t e dw i t h a
'
maximum speedof 4.4x1,0sms-t?
( A n s t: . L x L O 1 s H
; 2z. 7 3x 1 0 - 7 m )
( a ) T h e m a x i m u mK Eo f p h o t o e l e c t r o nesm i t t e df r o m a m e t a ls u r f a c ei s 1 . 8e V w h e nt h e
frequencyof the incidentlightis 7.9 xi,OraHz.
Calculate
(i)
T h ew o r k f u n c t i o no f t h e m e t a l
(ii)
T h e m a x i m u ms p e e do f t h e p h o t o e l e c t r o n s
( i i i ) T h et h r e s h o l df r e q u e n c y
(iv)
The cut off wavelength
(v)
T h e m a x i m u mK Ea n d s p e e do f t h e p h o t o e l e c t r o nwsh e nt h e i n c i d e n lti g h t
h a sa f r e q u e n c o
y f 8 . 8x 1 0 ' o H z .
(Ans:2.35x 10 1eJ;
7.96x 10sms-1;3.54
x l-o1aHz
;8.47 x r-0-7m
;3.4gx 10-1sJ
;
8.7 xl-Osms-1)
193
is
when lightof frequency3x 101sHz
(5) A metalsurfacehasa work functionof 4.4eV,and
flowsin the circuitwith the metaland a
incidenton it, a photocurrentof O.221tA
c o l e c t o rp l a t e .D. e t e r mni e :
(i)
The KEof the photoelectrons
(ii)
T h e n u m b e ro f p h o t o n ss t r i k i n gt h e m e t a ls u r f a c e a c hs e c o n d .
(Ans:1,.2gx 10-18J;
1012electronsper second)
1..375x
a
(6) When lightof frequency5.8x L0isHzstrikesa rnetalsurface,photoelectrons
l i b e r a t e dw i t h a m a x i m u mK Eo f 2 . 8x 1 0 - 1 8 J
yf light?
( a ) W h am
t u s tb e t h e s t o p p i n gp o t e n t i aflo r t h i sf r e q u e n c o
( b )W h a t i s t h e w o r k f u n c t i o no f t h e m e t a l ?
(c)What is the cut off wavelengthof the metal?
( A n s: 1 7. 5 Y ; 6 . 5 3 e V1;. 9x 1 0 - 7 m )
electronsare
(7) When a metalsurfaceis irradiatedwith lightof frequency5.8 x LOlsHz,
l i b e r a t e dw i t h a m a x i m u ms p e e do f 1 . 8x 1 0 6 m s - t . D e t e r m :i n e
(i)
T h ew o r kf u n c t i o no f t h e m e t a l
(ii)
T h es t o p p i n gp o t e n t i arl e q u i r e df o r t h i sf r e q u e n c y
( i i i ) T h et h r e s h o l df r e q u e n c fyo r t h e m e t a l
( A n s :2 . 3 7x 1 o - 1 8 J. 2; 9Y ; 3 . 6x 1 o 1 s H z )
194
S O L U T I OG
NU i D E L I N E S
3x108
t; ^ - c
fo
i:
(l)a. lo:7.2xL0-7m;c:3x10ems-'
,.r r r*
- 4'1'7x Lol4Hz
(b) 0 - hfo : 6.63 x 10-34x 4.1'7x 1014- 2.76 x !0-7eJ
L e V - t . 6 x r 0 - r r J , l t e n c ez . T 6 x 1 0 - 1 e
6 . 6 3x 1 0 - 3 4x 3 x 1 0 8
Itc
KE:hf-0-T-Q-
4.2x L0-7
1'
rrn
KE -;mv-;
2
v -
2'76x 1'0--1e
- L.7Z6eV
t.6 x L0-1e
- 2 . 7 6 x 1 o - l e - 1 . 9 8x 1 0 - 1 e /
-2:K E
zxr9Bx1o-1e
m
9.1x1o-31
-
6 . 6x l , O s m s - 1
( 2 ) 0 : 3 . 2 e v- 3 . 2 x L . 6x ' J , 0 - 7 e-J 5 . L Zx 1 0 - 1 e J
hc
UseQ:10,
tro:
hc
6 . 6 3x l - 0 - 3 4x 3 x 1 0 8
e:
5 . 1 , 2x 1 0 - 1 e
c
3x108
:7'7x1'o14Hz
fo:i-
( 3 ) E n e r g yo f p h o t o e l e c t r o -n
).*u'
-)x
: 3.9 x 1-0-7m
o . t X L 0 - 3 tx ( + . +x L 0 s ' ) : B . Bx L 0 - 2 o J
Work functton - 4eV - 4 x 1,.6x L o - r s J : 6 . 4 x 1 ' o - r s J
hf - Q-KE;
ft-
hf :KE+
7 . 2 8x 1 0 - 1 e6 . 6 3x 1 0 - 3 4
O - B . B x 1 ' 0 - 2 0 + 6 . 4 x 1 0 - 1 e- 7 . 2 8 x L } - r e J
t . 1 x 1 0 7 s H z ; I J s1e- r t u
l
195
fhdwavelength
K E - l . B e V: 1 . 8x I . 6 X 1 0 - 1 e- 2.BBx L'0-1sJ
(4)
K E * o , - h f - Q ; O - h f - K E - ( 6 . 6 3 x 1 0 - 3 4x 7 . 9 x 1 g t o )_ Z . B Bx 1 0 - 1 e
: 2 . 3 5x 1 0 - 1 e J
( t t ) U s eK E -
),^r'
2KE
m
: Z . B B x 1 0 - t e; u
2x2.BB x 10-1e
9.1 x 10-31
7.g6x l-0sms-1
(i.ii")Thecutof f frequency ts the frequency correspondtng to the work functfnn
6
hfo - Q t fo :;-
(tv)l,:
C
n:
2 . 3 5x L 0 - 1 e
: 3 . 5 4x L 0 1 4 H z
ffi
3 x l-08
: 8.47 x 10-7m
3 . 5 4x 1 0 1 4
-3a
x B.Bx L0t4 : 5.8 x 1,0-1e
(v)lnci"d"ent
energy E - hf : 6.63 x 10
J
KE^o* - Itf'- Q - 5.8 x 10-1e- 2.35 x 10-1e : 3.48 x 1.0-1eJ
2KE
-:
m
2 x 3 . 4 8 x 1 0 - 1 e9.t x L0-31
8.7x l0sms-1
( s ) I ( E - t r f - Q , ( w h e r e Q- 4 . 4 x 1 . 6 x 1 0 - I e : 7 . 0 4 x t 0 - 1 e 1 )
K E : ( 6 . 6 3x 1 0 - 3 ax 3 x 1 0 1 s )- 7 . 0 4 x 1 0 - 1 e- L . 2 9x L 0 - 1 8 /
The number of photons striktng each second ts equal to the number of electrons leavtng
each second.
Stncethe photocurrent ts 2.2 x l0-7 A,ilre current - rate of f low of charge.
I -
n :o n
tt
where Q - electronf"ccharge and - - nltmber of electrons per second
h':nce
n
I
2 . 2x ! 0 - 7
t
a
L . 6x 1 0 - 1 e
- 1.375 x 1.012
per second
196
(6)eV, : KE^ox iVs :
(it) K Emax - h f - Q ;
KE*o,
2.8 x 1018
e
1 , . 6x 1 0 - 1 e
- 17.5V
Q - I t f - K E ^ o , - ( 6 . 6 3x L 0 - 3 4x 5 . 8 x 1 0 1 s )_ Z . B x 1 0 - 1 8
0 - 10.4x 1,0-7e
J
1 0 . 4x L 0 - 1 e
Q_
(tll)Q-
hc
1,
) 7o:
1 , . 6x 1 0 - 1 e
hc
- 6 . 53 e V
6 . 6 3x 1 0 - 3 ax 3 x 1 0 8
- 1 . 9x I 0 - 7 m
1 , 0 . 4x 1 0 - 1 e
O:
( 7 ) I n c i " d e nE
t n e r g y - l ^ L -f 6 . 6 3 x 1 0 - 3 4x 5 . 8 x 1 0 1 s: 3 . 8 4 x L 0 - 1 8 /
u m o r : 1 . 8x t O 6 m s - . ; K E * o r : ; m v f r " o * - ) r 9 . I x l - 0 - 3 1
x ( 1 . 8x t 0 6 ) z
_ 1,.47x 10-18/
KE^o' -hf
-Q;
Q -hf
-KE^o*:3.84x
(i.i) eV, - KE^oy; W (tli)Q:hfoifo:
1 0 - 1 8- 1 . 4 7 x 1 0 - 1 8- 2 . 3 7x L 0 - 1 8 /
KE*o,
_t :r." :9-:rJ_ e.Zv
e
L6 x 10-1e
a
2 . 3 7x 1 0 - 1 8
-
It
6 . 6 3x 1 0 - 3 4
1.97
3.57 x 1,01s
Hz
X.RAYS
Production:
T h e P r i n c i p loef X - R a Y
such
strikea metal.target
Whenvery highenergyelectrons(in the orderof kiloelectron-volts)
s e c e l e r a trea p i d l ya, n dw h i l em o s to f t h e i rk i n e t i ce n e r g yi s
a st u n g s t e nt,h e e l e c t r o n d
in the form of heat,a smallpart (aboutt%|, is convertedto X radiation.Additionally,
dissipated
s i l l s o m e t i m ecsa u s ee n e r g yl e v e tl r a n s i t i o nosf e l e c t r o n isn t h e
t h e h i g he n e r g ye l e c t r o nw
targetatomsand causethe emissionof X rays
two processes
At the pointof collisionbetweenthe electronand the tungstenatom,thereare
that leadto X-raYProduction:
(1) Bremsstrahlun
Pgr o c e s s
s
h
e
l
l
(2) K
emission
leadto two differenttypesof X-rayspectra.
Thesetwo processes
w h, i l et h e K - s h e lel m i s s i o n
pg
r o c e s lse a d st o a C O N T I N U O USSP E C T R U M
T h eB r e m s s t r a h l u n
a l s oc a l l e da c h a r a c t e r i s tsi cp e c t r u m )
l e a d st o w h a t i s c a l l e da L I N GS P E C T R U(M
LUNG PROCESS:
BREMSSTRAH
m e a n s" r a d i a t i o n ' j ) , .
i sgo f G e r m a no r i g i n (B r e m sm e a n s" b r a k i n g a" n d s t r a h l u n g
Bremsstrqhlun
X radiation
a n d i s d u et o t h e c o n v e r s i oonf t h e k i n e t i ce n e r g yo f t h e h i g he n e r g ye l e c t r o nt o
' b r a k e sa' r e i m p o s e do n t h e e l e c t r o na s i t c o l l i d e w
s ithanobstacle.
w h e nt h e
atoms,the
When an incominghighspeedelectronof kineticenergyEostrikesone of the target
of anX-ray
e l e c t r o nw i l l l o s ea n a m o u n to f e n e r g yA E, a n d p a r to f t h i se n e r g yw i l l b e t h e e n e r g y
photon.(Therestof AEwill be convertedto heat)
Theelectronwill be deflectedby the collisionand it is now referredto asa SCATTERED
electron.
electron'whoseenergyis now lessthan Eswill now go on to collidewith
This.scattered
, h i c hw i l l h a v ea n e n e r g y t h a its d i f f e r e n t
a n o t h e ra t o m a n d p r o d u c ea n o t h e rX - r a yp h o t o n w
from that of the first Photon.
processwill continueand a seriesof photonsof differentenergiesare
Theelectronscattering
p r o d u c e dc,a u s i n tgh e f o r m a t i o no f a c o N T I N U o U xS r a ys p e c t r u mT. h ee l e c t r o nw i l l s t o p
producingphotonsrvhenit hascometo rest'
198
THEK-SHELL
EMISSION
PROCESS:
W h e na n i n c o m i n gh i g hs p e e de l e c t r o ns t r i k e sa targetatom,that electronmay knockout one
o f t h e e l e c t r o n isn t h e K - s h e l(ll o we n e r g ys h e l w
l h i c hi s c l o s et o t h e n u c l e u s, )l e a v i n g
a h o l ei n
t h a ts h e l l .
A n e l e c t r o ni n o n e o f t h e h i g h e rs h e l l sw i l l n o w j u m p i n t ot h a t l o w e rK s h e l tl o f i l l t h e v a c a n c y ,
a n d i n d o i n gs o ,i t w i l l g i v eu p s o m eo f i t s e n e r g yb y e m i t t i n ga n X - r a yp h o t o n .
T h e K s h e l li s t h a t s h e l fl o r n = 1 , t h e L s h e l li s t h e o n ef o r w h i c hn = 2 a n dt h e M s h e t fl o r n = 3 .
W h e na v a c a n c yi s c r e a t e di n t h e K s h e l ld u e t o t h e e j e c t i o no f a n e l e c t r o na, n d a n e l e c t r o n
f r o m t h e L s h e l fl i l l st h a t v a c a n c yt ,h e e m i t t e dr a d i a t i o ni s c a l l e dt h e K ol i n ei n t h e s p e c t r u m .
l f t h e v a c a n c yi n t h e K s h e l li s f i l l e db y a n e l e c t r o nf r o m t h e M s h e l l t, h e nt h e e m i t t e dr a d i a t i o n
i s c a l l e dt h e K sl i n ei n t h e s p e c t r u m .
T h ed i s t r i b u t i o on f w a v e l e n g t hi sn t h e e n t i r es p e c t r u ms h o w i n gb o t ht h e C O N T IU
NO U S
S P E C T R UaMn dt h e L I N ES P E C T R Ui M
s g i v e nb e l o w .
T h e K pl i n eh a s
greaterenergy(due
Relative
t o i t ss m a l l e r
intensity
w a v e l e n g t hb) u t
h a sa s m a l l e r
INTENSITY.
T h e s h a r pp e a k sa r e
Continuous
d u et o K s h e l l
e m i s s i o nw, h i l et h e
continuous
s p e c t r u mi s d u e t o
B16msstrahlune
,
trrin
Wavelength(pm)
E n e r g yl e v e l t r a n s i t i o n
Diagram:
M ( n = 3)
L(n=2)
K(n=1)
199
PRODUCTION
OFX.RAYS
Target
anode
Coolingoil+
Orwater
:--------------*
I
E v a c u a t etdu b e
C a t h o d e( t o e m i t e l e c t r o n s )
Lead
T h ec a t h o d ei s a c o i lo f t u n g s t e na c r o s sw h i c hi s c o n n e c t e tdo t h e m a i n ss u p p l y( 2 2 0 Va . c . ). A s
. h e s ee l e c t r o n as r et h e n a c c e l e r a t e d
a c u r r e n ft l o w st h r o u g hi t , e l e c t r o nas r e e m i t t e d T
t h r o u g ha p o t e n t i adl i f f e r e n c o
e f a b o u t1 0 0 k Vw h i c hi s a p p l i e da c r o s st h e a n o d et o t h e
cathode.
T h ea n o d ei s m a d eo f c o p p e ra n d h a sa s m a l lb l o c ko f t u n g s t e nf i x e do n i t . T h et u n g s t e ni s t h e
t a r g e tf o r t h e h i g hs p e e de l e c t r o n sA. st h e e l e c t r o ncso l l i d ew i t h t h e t u n g s t e n9, 9 %o f t h e i r
I % i s c o n v e r t e di n t o X r a y p h o t o n s .
k i n e t i ce n e r g yi s c o n v e r t e di n t o h e a ta n dt h e r e m a i n i n g
i s a s u i t a b l em a t e r i a l f o r t h et a r g e ts i n c ei t c a nw i t h s t a n dv e r yh i g ht e m p e r a t u r e s
Tungsten
w i t h o u tm e l t i n g .
T h el e a ds h i e l di s a n e c e s s a rsya f e t yd e s i g nt o p r e v e n t h e e s c a p eo f x r a y sf r o m a l l p o i n t s
o t h e rt h a nt h e w i n d o w .
T h eh i g ht e m p e r a t u r edse v e l o p e d
b y t h e c o l l i s i o nbse t w e e nt h e e l e c t r o nas n dt h e t a r g e t ,
m u s tb e c o n t r o l l e db y m e a n so f a c o o l i n gf l u i d ,w h i c hi s t y p i c a l l o
y i l .O i l d o e sn o t e v a p o r a t e
as
e u i l du p .W a t e rc a nb e u s e d
w a t e rd o e s ,a n d h e n c et h e r ei s n o d a n g e ro f a v a p o u rp r e s s u r b
',,vhen
there is a veryefficientflow design.
I
t_
200
ABSORPTION
OFX RAYS
X r a y sa r ea b s o r b e db y m e t a l s ,w h i c hm e a n st h a t t h e i ri n t e n s i t yd e c r e a s easst h e y p a s st h r o u g h
m e t a l sa n dw e s a yt h a t t h e y u n d e r g oA T T E N U A T I O N .
Attenuation
i s t h e g r a d u adl e c r e a s ien i n t e n s i t y .
l f a b e a mo f X - r a y sh a sa n i n c i d e nitn t e n s i t y1 oo n a m e t a ls u r f a c et ,h e n a s i t p a s s etsh r o u g ha
d e p t hx o f t h e m e t a l ,t h e i n t e n s i t y w o u ldde c r e a s e t o
a n e w v a l u el w h e r e I : I o € - v r .
T h eq u a n t i t yp i s c a l l e dt h e L T N E AARB S O R P I OcNo E F F t c l E N T .
TUTORIAT
SHEET
(1) A beamof X raysof intensity
leis irradiated
on a pieceof material
thathasan absorotron
coefficient
of 0.18cm-1.
what mustbethethickness
of the material
to causethe
intensitv
of the beamto fallto:
(a) 4Oo/o
(bl 76%
(c) 0.03%
(Ans:5.1cm;10cm;45.Ocm)
(2) What is the valueof the absorptioncoefficient
of a materialif for a thickness
of 0.4rnm
'the
intensityof a beamof X raysis reducedto 4o/o
of its initialvalue.
(Ans8.05mm-1)
( 3 ) A b e a mo f X r a y so f i n t e n s i t y5 5 W m m - i2s i n c i d e not n a m a t e r i aw
l h o s el i n e a ra b s o r p t i o n
coefficientis 0.44mm-1.
What is the intensityof the beamat a depthof :
(a)0.75mm
(b)2.6cm
(Ans3
: 9 . 5 W mm - 2 ; 5 . x9 1 0 - 4 W m m - 2 )
( a ) W h e na b e a mo f X r a y s i s i n c i d e not n a s p e c i m e nt h
, e intensitd
y e c r e a s eBsY3 5 %o f i t s
i n i t i a vl a l u ef o r a d e p t ho f 0 . 8 5 c m .
calculate
t h e l i n e a ra b s o r p t i o cno e f f i c i e notf t h e m a t e r i a l .
( A n s0 . 5 L c . n ' 1 )
20r
SOLUTIONS
( 1 ) 1 : I o € - F x ; I = 0 . 4 I o . , l t e n c e0 . 4 l l 0 = -I s e - t t x ; 0 . 4 :
s-Lrx
where F : 0.1Bcm-7
T a k t n g I o g s : l n ( 0 . + )- - P x ,
-0.92 - -0.18x ; x - 5.1,cm
-0.L8x ;
( i i ) l n ( 0 . 1 , 6 )-
-1.8 : -0.18x
; x - 10cm
-0.18x ; -8.11 :0.LBx ;x = 45cm
( l i l ) l n ( 0 . 0 0 0 3 )-
(2)l-lse-ttx
; I=0.04 Is,x-0.4mm
h e n c e0 . 0 4 / s = - I o e - j ' + u i 0 ' 0 4 - e - o ' 4 t t
ln(0.04) -
(3)1
;
-0.4t ,
-3.21t -
-0.4tt ; lt:
B'\Smm-r
I s e - t r x ; I : $ J s - @ ' + + x 0 ' 7:s ) 3 9 . 5 W m m - 2
I z : 5 $ s - ( o ' + + x 2 6 -) 5 . 9 x L 0 - 4 W m m - 2 ( N o t e t h a t 2 . 6 c m - 2 6 m m )
(4)A d"ecreaseby 350/omeans that the new Intenstt! : 650/oIo
I : I o e - F * ; 0 . 6 5 1 s: I o e - F x
l n 0 . 6 5-
-0.85p ; p:
202
ln 0.65
- S.Lcm-'
_0S5
APPLICATION
OFX RAYS
IN MEDICINE
C A N C EFRA C T N
S a t i o n aCl a n c e !rn s t i t u t eN a t i o n aIln s t i t u t eos f
H e a l t hR a d i o t h e r o payl,s oc a l l e dr a d i a t i o n
t h e r a p y i,s t h e
treatmeno
t f cancea
r n d o t h e rd i s e a s ews i t h i o n i z i n gr a d i a t i o n .
l o n i z i n gr a d i a t i o nd e p o s i t e
s n e r g yt h a t i n j u r e so r d e s t r o y cs e l l s
i n t h e a r e ab e i n gt r e a t e d( t h e" t a r g e t i s s u e "b) y d a m a g i n g
t h e i rg e n e t i cm a t e r i a lm
, a k i n gi t i m p o s s i b lfeo r t h e s ec e l l st o
c o n t i n u et o g r o w .A l t h o u g hr a d i a t i o nd a m a g e b
s o t hc a n c e r
c e l l sa n d n o r m a lc e l l st,h e l a t t e ra r ea b l et o r e p a i rt h e m s e l v e s
A s * s n n s d ' l * v e rs l i c e
How doesa CTscannerwork?
l n 1 9 6 7a n E n g l i s h m abny t h e n a m eo f G o d f r e yH o u s n f i e lm
d u l l e do v e rt h e p o s s i b i l i toyf
producing
a c r o s s - s e c t i o ni m
a la g eo f a n o b j e c t .C o m p u t e da x i a tl o m o g r a p h (yC A Tt)h u sb e c a m e
t h e f i r s tn o n - i n v a s i vt e c h n i q u e
f o r d i s p l a y i nagn i m a g eo f t h e h u m a nb o d yf r e eo f
s u p e r i m p o s i t i of rno m o v e r l y i n g
organs.
T h ef i r s tC Ts c a n n eur s e di n 1 . 9 7 2
w a se x t r e m e l yr u d i m e n t a r yb y t o d a y ' ss t a n d a r d s T
. o
p r o d u c eo n e r e c o g n i z a bilm
e a g et o o k n i n ed a y s i, m a g er e c o n s t r u c t i ot w
n o a n d a h a l fh o u r sa n d
p u n c ht a p ef e e d - i nt i m e ' ag r e a td e a ll o n g e r E
. a r l ys c a n n e rcso u l do n l yd i s p l a yi m a g e si n o n e
p l a n e( a x i apl l a n e )h, e n c et h e t e r m C A Ts c a n- ' A ' m e a n i n ga x i a lS
. c a n n e rnso w c a nd i s p l a y
i m a g e si n a n yp l a n e i,n c l u d i n3gD s o t h e ' A ' w a sd r o p p e df r o mt h e t i t l e .
T o d a yC
, Ts c a n n e rcsa na c q u i r ea n i m a g e ( si )n l e s st h a no n e s e c o n dw
, i t h t h e i m a g ed i s p l a y e d
immediatelM
y .o d e r n ' h i g h
e n d ' s c a n n e rcsa na c q u i r eu p t o 6 4 i m a g e a
s t onceS
. u c hs c a n n e r s
a r ec a l l e dM u l t iS l i c eu n i t s .
A C Ts c a n n eer s s e n t i a l cl yo m p r i s eosf a t a b l e ,a c o m p u t e ra, n d a g a n t r y( t h e' d o n u t ' s h a p e rdi n g
c a s i n g )l.n s i d et h e g a n t r yi s a n x - r a yt u b e a n d a r i n go f r a d i a t i o nd e t e c t o r w
s h i c hs u r r o u n dt h e
t a b l e . D u r i n ga C Ts c a n t, h e x - r a yt u b e s p i n sa r o u n dt h e p a t i e n te m i t t i n gx - r a y sw
, h i c hp a s s
t h r o u g ht h e p a t i e n t ' sb o d y . T h ea m o u n to f r a d i a t i o na b s o r b e d
b y t h e p a t i e n ti s m e a s u r e d
by
t h e d e t e c t o r sT. h i si n f o r m a t i o n
i s t h e n s e n tt o a c o m p u t e rw
, h i c hu s e st h e s em e a s u r e m e n t s ,
t h e p o s i t i o no f t h e p a t i e n ta n dt h e a n g l et h e r a d i a t i o n
w a sd e t e c t e da t , t o d e c i d ew h e t h e re a c h
p a r to f t h e p a t i e n ts c a n n e d
c o m p r i s eosf a i r ,f l u i d ,b o n e ,f a t , o r m u s c l e .T h i si n f o r m a t i o n
is
t h e n d i s p l a y e ad sa p i c t u r e .I n t h i sw a y ,a C Ts c a n n eer n a b l e sa d o c t o rt o l o o ka t t h e b o d ys l i c e
byslice.
a n df u n c t i o np r o p e r l y .
l o c a l i z esdo l i dt u m o r s ,s u c ha s c a n c e r o
s f t h e s k i n ,t o n g u e ,l a r y n xb, r a i n ,b r e a s to, r u t e r i n e
c e r v i xl.t c a na l s ob e u s e dt o t r e a t l e u k e m i a n d l y m p h o m a( c a n c e rosf t h e b l o o d - f o r m i ncge l l s
a n d l y m p h a t i sc y : ie m , r e s p e c t i v e l y ) .
203
LINESPECTRA
. v i d e n coef t h i s i s
t o e x i s ti n p a r t i c u l aer n e r g yl e v e l sE
T h ee l e c t r o n isn a t o m sa r e b e l i e v e d
o f l i n es p e c t r a .
o b t a i n e db y t h e p r o d u c t i o n
E l e c t r o nws i l l t e n d t o p o p u l a t et h e l o w e s te n e r g yl e v e l sf i r s t ,a n d a st h e s eb e c o m e f i l l e dt h, e
e l e c t r o n s t a r to c c u p y i ntgh e h i g h e rl e v e l s .
d ith
W h e na n e l e c t r o na b s o r b se n e r g y f o re x a m p l ei f t h e m a t e r i ails h e a t e do r i f i t i s i r r a d i a t e w
l
o
w
e
n
e
rgy
e l e c t r o m a g n e twi ca v e s a, n e l e c t r o nm a ya b s o r be n e r g ya n d m o v ef r o m i t s i n i t i a l l y
s t a t e( l e v e lt)o a h i g h e rl e v e l
o b e u n s t a b l e* .T h e
I n t h i sh i g h e rl e v e lt,h e e l e c t r o nh a sm o r ee n e r g ya n d i s c o n s i d e r et d
b y t h e a m o u n to f e n e r g yi t p o s s e s s etsh;e g r e a t e r t h e
s t a b i l i t yo f t h e e l e c t r o ni s d e t e r m i n e d
a m o u n to f e n e r g yt,h e m o r eu n s t a b l ei t i s .
T h ee l e c t r o na t t h e h i g h e rl e v e tl h e n t r i e st o r e g a i ni t s s t a b i l i t ya, n d i t d o e st h i s b y e m i t t i n gt h e
e n e r g yt h a t i t p r e v i o u s layb s o r b e dT. h i se n e r g yi s e m i t t e di n t h e f o r m o f l i g h to r s o m eo t h e r
f o r e x a m p l eX - r a y sg, a m m ar a y se t c .
t y p e o f e l e c t r o m a g n e triac d i a t i o n
F o rd i f f e r e n et n e r g yl e v e tl r a n s i t i o nisn t h e a t o m ,d i f f e r e n at m o u n t so f e n e r g i eas r ee m i t t e d ,
t a v e l e n g tahn dh e n c ea d i s t i n cct o l o u r
a n de a c hp a r t i c u l aarm o u n t b fe n e r g yh a sa d i s t i n cw
a s s o c i a t ewdi t h i t .
is
H e n c ef o r a s e r i e so f e n e r g yl e v e tl r a n s i t i o n sa,S P E C T R UOMFB R I G H T LCYO L O U R ELDI N E S
produced.
T h ef o l l o w j n gd i a g r a ms h o w ss o m ep o s s i b l e n e r g yl e v e tl r a n s i t i o nisn a n a t o m .
n=5
^-A
tt -
F o ra n e n e r g yl e v e lt r a n s i t i o n
f r o m E 4t o E s, t h e e n e r g y
r a d i a t e db y t h e a t o m i s g i v e n
a
n=3
by:
l t
-
AE:hf:
L
Ea-En
t n e r g yt r a n s i t i o ni s f r o m E 4t o E sa n d t h i s t r a n s i t i o nw i l l h a v e t h e
I n t h e a h r o v ed i a g r a m , r e g r e a t e se
hct
t a v e l e n g t h( A E - h f : = l
g r e r r. ' , ' r tf r e q u e n c y a n d t h e s h o r t e sw
204
A
L i n es
dencefor
E a c hl i n ei n t h e s p e c t r u m
i n d i c a t eas p a r t i c u l aern e r g yl e v e tl r a n s i t i o n
a n d a n o b s e r v a t i oonf t h e
s p e c t r u mo f I S O L A T EADT O M Sr e v e a l tsh a t t h e l i n e sa r es e p a r a t e
a n dd i s t i n cftr o m e a c ho t h e r .
T h ef a c tt h a t t h e l i n e sd o n o t t o u c he a c ho t h e rt o f o r m a c o n t i n u o ubs a n d ,i n d l c a t etsh a t t h e
e n e r g yl e v e tl r a n s i t i o nasr eq u i t ed i s c r e t ea, n dt h e r ei s n o t a c o n t i n u u m( a c o n t i n u o u rse g i o n i)n
t h e t r a n s i t i o np r o c e s s .
The EMISSION
SPRECTRUM
for any atom is seenas a seriesof brightlycolouredlinesagainst
a dark background.
ABSORPTION
SPECTRA
E a c he l e m e n th a si t s o w n d i s t i n cst e t o f s p e c t r al li n e sa s s o c i a t ewdi t h t h e e n e r g yl e v e l
t r a n s i t i o nt sh a t a r eu n i q u ef o r e a c he l e m e n t .
R e c a ltlh a t e a c hc o l o u ri n t h e s p e c t r u mc o r r e s p o n dt o
s a p a r t i c u l aar m o u n to f e n e r g yt h a t i s
r a di a t e d ,
T h ee l e m e n st o d i u mh a s p n l yo r a n g es p e c t r al il n e s .
C o n s i d enro w a n e l e m e n itn i t sg a s e o ufso r m (f o r e x a m p l e
s o d i u mv a p o u r b) e i n gi r r a d i a t e d
w i t h w h i t e l i g h t .T h ew h i t e l i g h ti s m a d eu p o f a s e r i e so f c o l o u r s( V I B G y O R
w)h i c hc a nb e s p l i t
u p b y m e a n so f a p r i s ma n dt h e l i n e so f t h e d i f f e r e nct o l o u r sc a nb e v i e w e d t h r o u g a
h device
c a l l e da L I G H S
T PECTROMETER.
W h e nt h e w h i t el i g h tp a s s etsh r o u g ht h e s o d i u mv a p o r ,t h e a t o m so f s o d i u ma b s o r bt h e e n e r g y
c o r r e s p o n d i nt og t h e o r a n g el i g h t , c a u s i nt hge e l e c t r o n isn t h e a t o mt o m o v et o a h i g h e rl e v e l .
A t t h i s h i g h e rl e v e tl h e e l e c t r o ni s u n s t a b l e
a n df a l l sb a c kd o w nt o a l o w e rl e v e l a
, n dr a d i a t e s
t h e e n e r g ya s s o c i a t ewdi t h t h e o r a n g ew a v e l e n g t h .
H o w e v etrh i s r a d i a t i o no f o r a n g el i g h to c c u r si n A L Ld i r e c t i o nasn d h e n c et h e s p a c et h a t w a s
p r e v i o u s loyc c u p i e db y t h e o r a n g el i g h ti n t h e V I B G Y OsRp e c t ; u ma, p p e a r sa s a d a r kr e g i o n .
T h i sr e s u l t a nst p e c t r u mw h i c hi s m a d eu p o f a b r i g h t l yc o l o u r e db r i g h tb a c k g r o u n d
w,i t h a d a r k
b a n di n i t , i s c a l l e da n A B S O R P T I O
N
SpECTRUM.
An AbsorptionSpectrumfor sodium
T h i si s t h e r e g i o nw h e r eO R A N GsEh o u l d
h a v eb e e n
205
OF MAfiER
NATURE
THEWAVE-PARTICLE
e f f e c t i, t i s p o s s i b l teh a t
s i n c ew a v e se x h i b i tp a r t i c l ep r o p e r t i eas ss e e ni n t h e P h o t o e l e c t r i c
t a v ep r o p e r t i e s '
p a r t i c l essh o u l de x h i b i w
e x p e r i m e n tosn E L E C T R O N
E v i d e n coef t h i sW A V EN A T U R Eo f m a t t e rw a s p r o v i d e db y
DIFFRACTION.
f i l m o f g r a p h i t ei,t w a sf o u n dt h a t a
w h e n a b e a mo f h i g hs p e e de l e c t r o n iss d i r e c t e da t a t h i n
g r a p h i t ef i l m '
r i n g sw a sp r o d u c e do n a s c r e e np o s i t i o n e bd e h i n dt h e
s e r i e so f d i f f r a c t i o n
Electron
beam
ring
Diffraction
pattern
film
Graphite
and by
meansthat they haveWAVEproperties,
Thefact that electronscanundergodiffraction,
t a v ep r o p e r t i e s '
e x t e n s i o na,l l m a t t e rs h o u l de x h i b i w
OnNd t h e s et w o
a cn E
dD I F F R A C T I a
E l e c t r o m a g n e rt iacd i a t i o n( e . g .l i g h t )e x h i b i tI N T E R F E R E N
w h i l et h e
i n d i c a t et h a t e l e c t r o m a g n e triacd i a t i o nh a v ea W A V EN A T U R E ,
phenomena
radiation'
of electromagnetic
NATURE
photoelectric
effectindicatesa PARTIcuLATE
Lu
Ea l i t yo f m a t t e r .
r t c a nb e d e d u c e dt h e n ,t h a t t h e r ei s a w A V E - p A R T l c d
p, the wavelengthassociated
DEBrogliesuggestedthat for eachmasshavinga momentum
with its momentumis givenbY:
h
1- p
V , t l - , , ,h- e= P l a n c k 'cso n s i a n t .
206
TUTORIAL
SHEET
(1) An electronis accelerated
througha potentialdifferenceof 2000Vin a vacuum.
Determinethe speedof the electronand its corresponding
wavelength.(themassof
t h e e l e c t r o ni s 9 . 1 x 1 0 - 3 1 kagn d i t s c h a r g ei s 1 . 6x 1 0 - 1 e c )
( A n s: 2 . 6 5x 1 0 7 m s - 12;J 5 x L 0 - 1 1 m )
(2) A protonis accelerated
througha potentialdifferenceV and acquiresa speedthat
allowsit to havea wavelengthof 8.7 x 10-13
m.
The massof the protonis 1..67x Lo-27kg
and its chargeis 1.6x l_0-1ec.
Determine
t h e p o t e n t i adl i f f e r e n c teh a t c a u s e dt h e p r o t o n ' sa c c e l e r a t i o n :
(Ans:1087V)
(3) A particlehavinga chargeq is accelerated
througha potentialdifferenceof 2500Vso
that its speedallowsit to havea wavelength
of 4.9! x !O-12m.
lf the massof the
particleis 4.55.xL0-30kg,
determineits chargeq.
(Ans8 x 10-1eC)
207
SOLUTIONS
( I ) q = ! . 6 x 1 . 0 - r s C; p . d . : 2 0 0 0 V ; m a s s
- g . 1x 1 0 - 3 r k g
done onthe charge is gtue:nby
When a cltarge q f"saccelerated"through ap.d.V,the work
of the charge.
W - q V .T h i sw o r k d o n ei s e q u a lt o t h e i n c r e a s ien K i n e t i ce n e r g y
h e n c e W- q' zV : 1 * u '
1 . 6x L 0 - 1 ex 2 0 0 0- 1 * 9 . 1 . x 1 0 - 3 x1 v 2
2
2 x 1 . 6x 1 0 - 1 ex 2 0 0 0:
9 . 1 ,x 1 0 - 3 1
2.65 x 1'07ms-r
(ti")Thewavelength ts f ound f rom the DeBrogltef ormulatr
lL
p
TI
P:mv
; h e n c e1 - mu
6 . 6 3x 1 0 - 3 4
9 . 1 , x 1 0 - 3 1x 2 . 6 5 x 1 0 7
2.75 x I0-11m
wher t , p - m o m e n t u m ;
)-
tr
(2) ustngthe DeBroglteequatton,
Hencev :
6 . 6 3x L 0 - 3 4
lt
p
ILh
mv'
- :"
mA
- 4 . 5 6x L 0 s m s - 1
tSZx!0-27x8.7x10-13
s n i n c r e a s ien i t s
s u e s t i o nt ,h e w o r k d o n ei n a c c e l e r a t i nt hge p r o t o nc a u s e a
A si n t h e p r e v i o u q
K i n e t i cE n e r g Y .
t
1?
1,
zmu- 2x
V_
(3)1 --
t^u'
- qv
1 . 6 7 x 1 . 0 - 2 7x ( 4 . 5 6 x 1 0 s ) 2
1 . . 6x L 0 - 1 e
6 . 6 3x 1 0 - 3 4
1,087V
: 2.97 x L07ms-I
+ s s r i o - 3 0x 4 . 9 !x 1 ' o - 1 2
1 0 - 3 0x ( z . o l x 1 , 0 7 ) 2
208
: B x 1,0-1eC
ATOMICSTRUCTURE
E a r l yT h e o r i e o
s n A t o m . iS
c tructure:
( 1 ) T h o m s o nM o d e l :C a l l e d
t h e p l u m - p u d d i nmgo d e l
T h i sm o d e lp r o p o s e d
t h a t t h e p o s i t i v ec h a r g eo f t h e a t o mw a ss p r e a do v e r t h ee n t i r e
v o l u m eo f t h e s p h e r eo f t h e a t o m a n dt h e e l e c t r o nw
s e r ed i s p e r s erda n d o m l yl i k ep l u m s
w i t h i nt h i ss p h e r e .T h ee l e c t r o nw
s e r et h o u g h t o b e i n c o n s t a nvt i b r a t i o na b o u tt h e i r
f i x e dp o s i t i o n s .
electron
O
O'O
OO
S p h e r eo f p o s i t i v e
charge
(2) The Rutherford-Bohr model:
T h ee l e c t r o n os r b i tt h e p o s i t i v e lcyh a r g e dn u c l e u sl i k ep l a n e t sa r o u n dt h e s u n .
209
The Geiger-MarsdenExPeriment:
NxGp e r i m e n t .
T h i se x p e r i m e nits k n o w na st h e A L P H AS C A T T E R I e
was directedto strikea thin sheetof
ln an evacuatedmetalbox,a streamof alphaparticles
s e r ed e t e c t e db y z i n cs u l p h i d es c r e e nw h i c hw a s m o u n t e da t t h e
g o l df o i l . T h ea l p h ap a r t i c l ew
s c r e e na t i n y f l a s ho f l i g h t
e n d o f a m i c r o s c o p eW. h e na n a l p h ap a r t i c l es t r i k e st h e z i n cs u l p h i d e
r f t h e k i n e t i ce n e r g yo f t h e a l p h a
. h i sf l a s hi s d u et o t h e t r a n s f e o
c a l l e da s c i n t i l l a t i oi ns s e e n T
a n dt h e
s n e n e r g yl e v e tl r a n s i t i o n
p a r t i c l et o a n e l e c t r o ni n t h e z i n cs u l p h i d ew, h i c hc a u s e a
emissionof a photonof light.
foil
Gotd
P l a nv i e w
screen
Z i n cs u l o h i d e
Metal box
with alpha
source
P i p et o v a c u u mp u m p
revealedthat whilemostof the alphaparticlespassedstraightthroughwith
Theobservations
90'andeven
littleor no deflection, a few were deflectedin the rangefrom 5" to approximately
fewer were deflectedat anglesgreaterthan 90".
increased.
T h en u m b e rt h a t w a sd e f l e c t e dd, e c r e a s eadst h e a n g l eo f d e f l e c t i o n
2r0
T h e r e s u l t so f t h e e x p e r i m e ni tn d i c a t et h a t :
( 1 ) T h ea t o m c o n t a i n e d
a p o s i t i v e lcyh a r g e dn u c l e u w
s h i c hw a sr e s p o n s i b fl e
or deflection
of
t h e p o s i t i v e lcyh a r g e da l p h ap a r t i c l e .
( 2 ) T h i sp o s i t i v ec h a r g ew a s c o n t a i n e di n a v e r y t i n y v o l u m eo f t h e a t o m , w h i c hw a s t h e
s x p e r i e n c eadn y d e f l e c t i o n . ( t h ew
r e a s o nw h y o n l yf e w a l p h ap a r t i c l e e
r ea s o n l y a s m a l l
p r o b a b i l i toy f t h e a l p h ap a r t i c l ec o m i n gc l o s et o t h i s p o s i t i v e lcyh a r g e dn u c l e u s . )
( 3 ) M o s t o f t h e m a s so f t h e a t o m w a sc o n t a i n e di n t h e n u c l e u s i n c ei f t h e m a s sw a ss p r e a d
o u t o v e r t h e e n t i r ea t o m a s p r o p o s e db y T h o m s o nm
s ould
, o s t o f t h e a l p h ap a r t i c l e w
h a v es t r u c kt h i s m a s sa n dw o u l dn o t h a v eb e e na b l et o p a s st h r o u g ht h e a t o m .
USEFUL
TERMSIN ATOMICPHYSICS:
(1) NucleonT h ew o r d n u c l e o ni s a c o l l e c t i vtee r m f o r b o t h p r o t o n sa n d n e u t r o n sA. n u c l e o ni s a
particle
inthe NUCLEUS.
(2) AtomicnumberT h i si s t h e n u m b e ro f p r o t o n si n t h e n u c l e u sa, n d i s r e p r e s e n t ebdy t h e s y m b o l Z .
( 3 ) N e u t r o nn u m b e rT h i si s t h e n u m b e ro f n e u t r o n si n t h e n u c l e u a
s n d i s r e p r e s e n t ebdy t h e s y m b o N
l .
( 4 ) N u c l e o nn u m b e r T h i si st h e t o t a ln u m b e ro f N U C L E O Nl NST H EN U C L E UlSt i.s a l s oc a l l e dt h e M A S S
n u m b e ra n d i s r e p r e s e n t ebdy t h e s y m b o A
l .
T h ef o l l o w i n gr e l a t i o ni s u s e f u l t or e m e m b e rA: - Z + N
i.e.Nucleonnllmber - Number of Protons * Number of Neutrons(5) NuclideT h i si s a n a t o m c h a r a c t e r i z ebdy t h e n u m b e ro f p r o t o n sa n d n e u t r o n si n t h e n u c l e u s .
T h i st e r m i s u s e di n p r e f e r e n cteo t h e w o r d " a t o m "s i n c et h e t e r m a t o m i n c l u d etsh e
o r b i t a le l e c t r o n as n df o r n u c l e a P
r h y s i c st h, e o r b i t a le l e c t r o nds o n o t c o m ei n t o p l a y .
(6) lsotopeA n u c l i d eh a sd i f f e r e n its o t o p e isf i t c a nh a v ed i f f e r e n m
t a s sn u m b e r sw h i l eh a v i n g t h e
s a r n ea t o m i cn u m b e rT
. h ed i f f e r e n c ien t h e m a s sn u m b e ri s d u et o t h e n u c l i d eh a v i n g
d i f f e r e nn
t r i u t r o nn u m b e r sE. . g .
t t c ; ' Z C ; t Z C a r ei s o t o p eos f c a r b o na n d
l H ; ? n ; I U a r ei s o t o p eosf h y d r o g e n .
211,
OFCHARGE
THEQUANTIZATION
t i l l i k a ni n 1 9 1 0
w)a sd o n eb y R o b e r M
T h e m e a s u r e m e notf t h e e l e m e n t a rcyh a r g e ( 1 .x61 0 - 1 s C
a n dt h e e x p e r i m e ni st n o w c a l l e dt h e M i l l i k a no i l d r o pe x p e r i m e n t .
T h ea r r a n g e m e notf t h e a p p a r a t ufso r t h e e x p e r i m e ni ts g i v e nb e l o w :
O i ld r o p l e t s
ee
€€
€
e
A t o m i s e re . g .p e r f u m e
s p r a yb o t t l e
P o s i t i v e lcyh a r g e dm e t a l
plate
X raysfor
i o n i z i n tgh e a i r
mtcroscooe
N e g a t i v e l yc h a r g e dm e t a l
plate
L i g h ts o u r c ef o r
illuminating
droplets
. h ef i r s tp a r ti n v o l v etsh e d e t e r m i n a t i oonf t h e t e r m i n a l
T h ee x p e r i m e ni ts d o n ei n t w o p a r t s T
a
l
l
o
w st h e R A D I UoSf t h e d r o pt o b e d e t e r m i n e d .
s p e e do f a n o i l d r o p ,w h i c ht h e n
T h ec i r c u i ti s l e f t O P E Ni n i t i a l l ys o t h a t t h e m e t a l p l a t e sa r e u n c h a r g eadn dw h e nt h e a t o m i z e r
f ol r c ea c t i n go n
s p r a y sd r o p l e t si n t ot h e e m p t ys p a c e , t h ed r o p l e t sf a l ld u e t o t h e g r a v i t a t i o n a
them
S o m eo f t h e d r o p l e t sp a s st h r o u g ht h e o p e n i n gi n t h e u p p e rp l a t ea n dt h e i rm o t i o ni s t i m e d
"d".
w h e nt h e yf a l lt h r o u g ha d i s t a n c e
T h et e r m i n a sl p e e dc a nt h e n b e f o u n df r o m 1 ) .:t #
T h ev i s c o u fso r c ea c t i n go n t h e d r o p i s g i v e nb y F - 6 r c n r uw h e r eq i s a c o n s t a nat n d i s t h e
v i s c o s i toy f a i r ,r i s t h e r a d i u so f t h e d r o pa n dv = t e r r n i n asl p e e dw h i c hw a sp r e v i o u s l y
determined.
T h e r a d i u so f t h e d r o p h a st o b e f o u n da sf o l l o w s .
W h e nt h e d r o pf a l l si n a i r i t s w e i g h ti s e q u a lt o t h e v i s c o u fso r c ea c t i n gu p w a r d sh, e n c e
nng - 6n4re 1. or mg = kvr where k is a constant.
212
T h ed r o pi s s p h e r i c aaln d i t sv o l u m ei s V H e n c ef r o m t h e e q u a t i c j *n
, a n d m a s s =V o l u m ex d e n s i t y =
p.
tnrt
t"f
t o r = p g = 6 n 4 r v t ,h e n c er c a nb e d e t e r m i n e d .
l n t h e s e c o n dp a r to f t h e e x p e r i m e n t , t hpeo w e rs u p p l yi s s w i t c h e do n
a n dt h e p l a t e sb e c o m e
potentialdifferencebetweenthe platesis about
charged.The
O n ep a r t i c u l adr r o p l e ti s s e l e c t e d
a n dt h e p . d .b e t w e e nt h e p l a t e si s a d j u s t e ds o t h a t t h e
t e r m i n a vl e l o c i t yo f t h e d r o p l e td e c r e a s etso a v a l u €v z . T h ev i s c o u fso r c e
n o w a c t i n go n t h e
d r o pi s k v 2
T h et o t a l u p w a r df o r c ei s E q+ k v zw h i l et h e d o w n w a r df o r c ei s m g .
HenceEq - mg - kur; butmg : kur; HenceEq _ k(v, _ v)
By varyihBE,a seriesof corresponding
valuesof (v1- vz)are obtainedand a graphof E versus
(vt - v2)givesa straightline with a gradientof I fror which q
can be determined.Thisgraph
q
is only to be usedfor a particulardroplet,sinceeachdifferentmass
will haveits own
particularset of valuesof (v1- vz).
A seriesof graphscan be drawn and the valuesof q can be compared.
In all casesq will be
found to be an integralmultipleof 1.Gx 10-1s
C.
H o wd i d t h e d r o p l e t a
s c q u i r ea c h a r g e ?
A n s w e rS
: o m eo f t h e d r o p l e t sw e r ea b l et o g e t a c h a r g ew h e nt h e y m o v e d
t h r o u g ht h e a i r ,
k n o c k e de l e c t r o n o
s u t o f s o m eo f t h e a i r m o l e c u l easn dt h e s ef r e ee l e c t r o nw
s e r et h e n , U i " i o '
a d h e r et o s o m eo f t h e o i l d r o p l e t s .
H o w e v e irn o r d e rt o i n c r e a steh e n u m b e ro f c h a r g e os n t h e d r o p l e t sa n d
t o f a c i l i t a t em o r e
d r o p l e t sb e i n gc h a r g e dt ,h e a i r i n t h e v e s s esl h o u i db e l o N l S E Db y X - r a v s
o r b y u s i n ga
radioactivesourceto ionizethe air.
lmportant: The air will causethe drop to experience
an upthrust,howeverthe magnitudeof
t h i s i s s m a l la n d c a nb e A S S U M E tDo b e n e g l i g i b l e .
(vr- vr)
213
SHEET
TUTORIAL
( 1 ) ( i )C a l c u l a tteh e r a d i u so f a d r o po f o i l o f d e n s i t y8 7 0 k g m -itf i t f a l l sw i t h a t e r m i n a l
Nsm-2'
is 1'.7gx l-O-s
in air wherethe viscosity
velocityof 2.88mms-t
( i i ) l ft h e d r o p h a s2 0 e l e c t r o n os n i t a n dt h e p l a t es e p a r a t i oins 4 m m ,c a l c u l a tteh e
p . d .t h a t m u s tb e a p p l i e db e t w e e nt h e t w o p l a t e si f t h e d r o pi s t o b e h e l ds t a t i o n a r y .
( A n s :5 . 2 x 1 0 - 6 m6; 2 6 8 V )
y h e n i t a c q u i r efso u r e l e c t r o n s
t , d r o p i s h e l ds t a t i o n a rw
( 2 ) l nt h e M i l l i k a n ' e
s x p e r i m e na
a n dt h e v o l t a g ea c r o s st h e p l a t e si s 2 2 5 0 V .
(i)
(ii)
W h a t i s t h e p l a t es e p a r a t i oinf t h e r a d i u so f t h e d r o p i s 1 . 7 5x 1 0 - 6 ma n dt h e
o f o i l i s9 2 0 k g m - ' .
densitY
w h a t m u s tb e t h e r a d i u so f a n o t h e rd r o pi f i t i s t o b e h e l ds t a t i o n a ruy n d e r
t h e s a m e c o n d i t i o nas sa b o v ei f i t h a s8 e l e c t r o nas t t a c h e dt o i t ?
( A n s7 . L m m ; 2 . 2 P m )
. a l c u l a tteh e
( 3 ) E l e c t r o nasr ea c c e l e r a t ei dn a n X r a yt u b et h r o u g ha p . d .o f 8 0 0 0 V C
of electrorl= 9.1 x L0-31kg,
the motionof the electrons.(mass
wavelengthassociafed
charge=l-.6x10-1sC)
11m)
( A n s: L . 3 7x 1 0
, a s sa n d
0 . C a l c u l a tteh e m o m e n t u mm
h f 3 . 6x L 0 - 1m
(4) A n X r a y p h o t o nh a sa w a v e l e n g t o
energyof the X raYPhoton.
1;6.14
5 . 5x 1o-16J;
x 1 0 - 3 3 k; g
A n s . (1 , . 8 4 x1 0 - 2 a k g m s
Ca
. l c u l a tteh e m i n i m u mp o t e n t i a l
( 5 ) T h e K o Xr a y sh a v ea w a v e l e n g t oh f 6 . 8 8x L O - 1 0 m
differenceacrossan X raytube that will producetheseX rays'
W h a t i s t h e e n e r g yo f t h e X r a y p h o t o n si n e V ?
(Ans:1,807V;
1807eV)
s e a ta t
( 6 ) A n X - r a y t u b e i s o p e r a t e do n a d . cp o t e n t i adl i f f e r e n c e ' o3f8 0 0 0 Va, n d p r o d u c e h
a rate of 660w. lf only 0.4%of the energyof the incidentelectronsis convertedto X
r a y s c, a l c u l a t e :
(i)
T h e r a t ea t w h i c he l e c t r o n s t r i k et h e t a r g e t '
(ii)
T h es p e e do f t h e i n c i d e net l e c t r o n s
s e r s e c o n d; 1 . 1 8x l - 0 8 m s - 1 )
7 l e c t r o np
( A n s : 1 . 0x91 0 1 e
heat is generatedin the
(7) An x ray tube operatesat a potentialdifferenceof 55000V,and
of the energyof the electronsis
what percentage
target at a rrte of 85OW.Determine
t o X r a y si f t h e c u r r e n ti n t h e X r a y t u b ei s 0 . 0 1 8 A(. A n st 4 % )
converted
214
SOLUTIONS
(1) At Termtnalveloci,ty,theviscousForce - wetght of drop
VtscousForceF - 6rnrv,the drop ts a spherei"tsmass- vol x densi.ty.
V o l u m eo f s p h e r e- ! o r = ,
Hence 6m7rv
' 3 -!nr=
,
IBrTv
l - a l
pg
1 8 x 1 . 7 9x 1 0 - sx 0 . 0 0 2 8 8
- 5.2 x 1,0-6m
4 x 8 7 0x 9 . 8 1
+pg
(ti")lf the drop i"sheld stati,on&ry,then the electrf.cf orce acttng upward"s,
would be equal to the wei"ght
Hence mg = Eq where E f"sthe Electric f teld strength
Vq
mqd
.
E_V
ms-7; v -;
7;
V_
1"" psd
! r c 6 . 2 x 1 0 - 6 ) 3x 8 7 0x 9 . 8 1x 0 . 0 0 4:
20x1,.6x10-1e
6268V
(2) When the drop f"sstattonary,wei.gltt(acti.ngdown) - Electri"c orce(acting up)
f
h e n c e m g : E qo r m g - r y t
u
d-Y-
2250x6'4x\o-Ie
-
m
" LAa
+
//^
2ftQ.75
-7.Lx!o-3m
x 1 0 - 6 ) 3x 9 2 0 x 9 . 8 1
( i " i ) C h a r g e- B x 1. 6 x ! } - l e , p l a t e s e p a r a t i o n : 7 . t x L 0 - 3 m i p . d .
mg
Vq 4
g
d ' 1 o ' =P
Vq
d'
2 2 5 0x B x 1 , . 6 xL 0 - 1 ex 3
: 2.2 x L0-6 m
4 x 7 . 1 x 1 0 - 3x n x 9 2 0 x 9 . 8 1
215
2250V
(3)Work done on electron - Increase tn i,tsKE
h e n c eq V :
1,
r*u,
mornentumP: mv and 1-
(4) 1-
2 x I . 6 x 1 0 - 1 ex 8 0 0 0
;v -
h
mv
6 . 6 3x 1 0 - 3 4
9 . 1 x, 1 0 - 3 1x 5 . 3 x 1 0 7
mv
: L.37 x't0-11m
6.63 x 1,0-34
- 1 . 8x ! 0 - 2 a k g m s - 1
3 . 6x 1 0 - 1 0
mv *
Momentum - mass x speed,
(ir)mass -
: 5,3 x 107ms-I
9 . 1x l - 0 - 3 1
hence ma"ss-
1 . 8x ! 0 - 2 4 -
momentum
speed(of ltght)
6 . 1 4x 1 0 - 3 3 k g
3x 108
hc 6 . 6 3 x 1 0 - 3 a x 3 x 1 0 B
: 5.5 x 1,0-16J
(iii)Energy of a photon 1
3 . 6x 1 0 - 1 0
(S)TheEnergy of the X - rays would have come fromthe work d"onei"naccelerattng
the electrons that produce them.
Itc
The energy of an X ray photon U :
T
'
The work done tn accelerattna the electron - q V
henceqV:;i
Itc
1
' V:
hc
6 . 6 3x 1 0 - 3 4x 3 x L O B
":
qA
1 . 6 x 1 - 0 - 1 xe 6 . 8 8x L 0 - 1 0
: 1807V
T h ec h a r g eo f a n e l e c t r o ni s e a n dt h e p . d .a c r o s si t i s 1 8 0 7 Vh, e n c et h e e n e r g ys u p p l i e d
to the
electronis 1807eV.
2r6
(6) 4o/oconverted"to X ray meqns that 99.6%Power conv'erted.to heat
.Hence 660W - 99.60/o
of totat power suppti.ed"
:
Totalpowersuppli"ed
ffi
- 66ZrSW
Totalpower - (number of electrons per second) x (Energy of 1,electron)
6625W - number of electrons per second x qV
number of electrons per second :
( i i , )q v - ) ^ u '
;v -
662.65
I . 6 x 1 0 - 1 ex 3 8 0 0 0
2qv
2 x 1 . 6 x 1 0 - 1 ex 3 8 0 0 0
-
m
9 . 1x 1 - 0 - 3 1
: 1 . 0 9x 1 0 1 7
1 . 1 5x L 0 8 m s - 1
( 7 ) P o w e r s u p p l t e dP - V I - 5 5 0 0 0 x 0 . 0 1 8- 9 9 0 W
Power lost to heattng : 850W,
Hence Power avai"lablef or Xray production - 990W - BS\W _ 'J.40W
Percentage of power used"f or Xray prod"ucti.on- Y
x I00o/o990
217
L4o/o
VIASSDEFECT
AND BINDINGENERGY
ble,i.e.Masscanbe
showsthat massand energyare interchangea
theorvof Relativity
Einstein,s
convertedinto energy,and energycan be convertedinto mass'
by a releaseof
is accompanied
a decreasein massAm,this decrease
When particlesexperlence
an amountof energyAE.
The equationrelatingmassand Energyis
AE = Amcz
WhereE is the energyin Joules,M is the massin kg and C = 3x 108ms-1
with 1
aswell asthe Energyassociated
Thiseouationallowsthe massof a Jouleto be calculated
kg to be calculated.
E.g(1) Findthe massof l loule:
L e t A E= l J o u l e ; L = A m c 2 .
H e n c e/ m : ; =
,'
.,;|ry
= 1 . 1 1x L o - l 7 k s
with the changein massof 1kg.
2) Findthe energyassociated
'
A E= 1 x c 2= 1 x ( 3 x 1 0 8 ) ' z9=x 1 0 1 6 J .
THEATOMICMASSUNIT:
r h y s i ciss t h e U n i f i e dA t o m i cM a s sU n i t ,
A v e r yu s e f u u
l n i t o f m a s sa n d e n e r g yi n N u c l e aP
'u'.
which is alsocalledthe atomicmassunit and abbreviated
T h ea t o m i cm a s su n i t i s d e f i n e da s o n et w e l f t ho f t h e m a s so f t h e c a r b o na t o m
'ZC.
I n a m o l eo f c a r b o nt,h e r ea r e 6 . 0 2x 1 0 2 3a t o m sa n dt h e m a s so f 1 m o l ei s L 2 9 .
Hencethemassof1carbonatom''ffie=ffikg_12u;Hence
12
Lu:
- L.66 xl0-27kg.
ffikg
S i n c e1 l < g= 9 x l - 0 1 6 ia, n d 1 M e V = l - ' 6 x L 0 - 1 3 J
1 ' 6 6 x L O - 2 7x 9 x 1 0 1 6
t-
l'L
-
r-6x10-13
21.8
-93LMev
R e c a l:l
T h ep r o t o n sa n d n e u t r o n si n t h e n u c l e u sa r ec o l l e c t i v e l y
r e f e r r e dt o a s n u c l e o n s
T h ee n e r g y t h a m
t u s tb e s u p p l i e dt o t h e n u c l e u tso c o m p l e t e l y
s e p a r a t teh e nvuvc'lsevuJsi n t o i t s
"'l'|
i n d i v i d u anlu c l e o n si ,s c a l l e dt h e B I N D I N G
E N E R Goyf t h e n u c l e u s .
A c c o r d i ntgo E i n s t e i n 'm
s a s s -e n e r g yr e l a t i o n s h i p
i f ,e n e r g yi s s u p p l i e d
t o t h e n u c l e u st ,h e n
w h e nt h e i n d i v i d u anl u c l e o n as r es e p a r a t e dt h, e t o t a l
m a s sw o u l di n c r e a s ei ..e i f e n e r g yi s
added,then this energyis convertedinto extramass.
T h i sm e a n st h a t t h e s u mo f t h e m a s s e o
s f t h e i n d i vei vdvu' a'In'lvu c l e o niss g r e a t e tr h a nt h e
m a s so f
the nucleus.
T h eB I N D I N G
E N E R GcYa nt h e r e f o r e
a l s obe definedasthe ToTALenergythat is reteased
when
t h e i n d i v i d u anl u c t e o ncso r n et o g e t h e rt o f o r m t h e
nucleus.
T h ed i f f e r e n c ien m a s sb e t w e e nt h e s u mo f t h e m a s s e s
o f t h e i n d i v i d u anl u c l e o n as n d t h e m a s s
o f t h e n u c l e u si s c a i l e dt h e M A S SD E F E C T .
= (sum of massesof Nucleons (Mass
MAss DEFECT
of Nucleus)
)T h eb i n d i n ge n e r g yi s t h e e n e r g ya s s o c i a t ewith
d the
Binding Energ!
massdefect,hence
: mqss def ect x (3 x 108)2
Example:
C a l c u l a tteh e b i n d i n ge n e r g ya n dt h e m a s sd e f e c tf o r
t h e h e l i u mn u c l e u s .
s y m b o fl o r h e l i u mn u c l e su i s
l H e ; t h e n u c l e o n u m b e ri s 4 a n dt h e p r o t o nn u m b e ri s 2 . H e n c e
i t h a s2 p r o t o n sa n d 2 n e u t r o n s .
T h em a s so f a p r o t o ni s 1 ' 0 0 7 3 ua n dt h e m a s so f a n e u t r o n
is1.00g7u.
T h et o t a l m a s so f t h e i n d i v i d u anl u c l e o n iss e q u a l t o(
m a s so f 2 p r o t o n s+) ( m a s so f 2 n e u t r o n s )
T o t a lm a s s= ( 2 x 1 . 0 0 7 3 u+) ( Z x 1 . 0 O g 7 u=)4 . 0 3 2 0 u
T h eh e l i u mn u c l e u si s f o u n dt o h a v ea m a s so f 4 . 0 0 1 5 u .
T h eM A S SD E F E Ci sTt h e r e f o r e4 . 0 3 2 0u - 4 . 0 0 1 5 u= 0 . 0 3 0 5 u .
R e c a ltih a t L u h a sd F €
: Q u i v a l e netn e r g yo f 9 3 1 M e V ,h e n c eo . 0 3 0 5 uw i l l
h a v ea n e n e r g y
€eu,''.,alence
of
0.0305x 931MeV= 28.4MeV.
2t9
OFTHENUCLEUS
STABILITY
Nh. eg r e a t etrh e
GN E R GpYe r N U C L E O T
T h eS t a b i l i toyf a n u c l e u iss d e t e r m i n e bd y i t s B I N D I N E
d sf o l l o w s :
r i l l b e t h e s t a b i l i t yT. h i sc a nb e e x p l a i n e a
b i n d i n ge n e r g yp e r n u c l e o nt,h e g r e a t e w
s i l l h a v et h e g r e a t e sst t a b i l i t ya, n d b o d i e st e n d
, o d i e st h a t h a v et h e l o w e s te n e r g i ew
In natureb
t o l o s ee n e r g yi n o r d e rt o i n c r e a steh e i rs t a b i l i t y .
formed
l n t h e a t o m ,t h e n u c l e o n cs o n g r e g a tteo f o r m t h e n u c l e u sa n d i n d o i n gs o ,t h e n u c l e u s
g i v e su p s o m ee n e r g yt o a t t a i ns o m es t a b i l i t yT. h en u c l e ti h a t g i v eu p t h e m o s te n e r g y ,a r et h e
m o s ts t a b l eo n e s .
H e n c et h e e l e m e n t st h a t h a v et h e g r e a t e sbt i n d i n ge n e r g yp e r n u c l e o nw i l l h a v et h e m o s t
s t a b l en u c l e i .
t a s sn u m b e r t, h e f o l l o w i n g
W h e na g r a p ho f B i n d i n gE n e i g yp e r n u c l e o ni s p l o t t e da g a i n sm
s h a p ei s o b t a i n e d :
B i n d i n ge n e r g y
p e rn u c l e o n
238
56
M a s sn u m b e r
t eh a st h e h i g h e sbt i n d i n ge n e r g yp e r n u c l e o na n di s
T h eg r a p hs h o w st h a t t h e e l e m e n F
t h e r e f o r et h e m o s ts t a b l e .
that
L i g h t eer l e m e n t s u c ha s H y d r o g e nh, e l i u m , A,l L i , e t c .a l l h a v em a s sn u m b e r sl o w e r t h a n
c
an
T
h
e
y
p
e
r
o f F e ,a n d a c c o r d i ntgo t h e g r a p h t, h e yw i l l h a v el o w e rb i n d i n ge n e r g i e s n u c l e o n .
i n c r e a steh e i rb i n d i n ge n e r g i epse r n u c l e o nb y i n c r e a s i nt hg e i rm a s sn u m b e rc l o s etro 5 6 ,a n d
f o r t h i st o h a p p e nt,h e ym u s tu n d e r g oF U S I O N '
220
Elements
with massnumbersgreaterthan 56 for exampleUranium,havebindingenergiesper
nucleonlessthan that of Fe,and in orderfor elementssuchastheseto becomgmorestable(by
increasing
their bindingenergyper nucleon),
they mustdecrease
their massnumberto a varue
closerto 56.Thiscanhappenif the elementundergoes
FISSION.
In FUsloN,two lightnucleiarefusedto form a heavierone with the releaseof bindingenergy.
In FlssloN,a heavynucleusis splitinto two fragmentsof approximately
equalmass,with the
releaseof bindingenergy.
In the both FISSION
and FUSIONas well as all other nuclearprocessesthere is conservationof
AtL of the following quantities:
NUCLEONNUMBER,PROTONNUMBER, ENERGY-MASS
and CHARGE.
Example:
Findthe energyreleasedwhen a Poloniumnuclideemitsan alphaparticle.
tt|po -+ 23tpb+
lHe + E
T h em a s s eos f t h e v a r i o u sn u c l i d easr eg i v e na sf o l l o w s :
L e a d' | u r p a- 2 0 5 . 9 6 9 u ; q p a r t i c l e | H e - 4 . 0 0 4 u , p o l o n t u m z [ o n p-o z 0 g . 9 8 2 u
T h eT o t a lm a s so n t h e l e f t s i d eo f t h e e q u a t i o ni s 2 0 9 . 9 g 2 u
Thetotal masson the rightsideis (205.969+
4.004)u= 209.973u
Thereis a massdefecton the rightsideof (209.982- 209.973)u
= 0.009u
A n E N E R GTYE R Mm u s tb e i n c l u d e od n t h e r i g h ts i d es o t h a t i t s m a s se q u i v a l e n cweh e na d d e d
t o t h e o t h e rm a s s e o
s n t h e r i g h ts i d eo f t h e e q u a t i o nw, i l l b e e q u a lt o t h e m a s so n t h e l e f ts i d e
of theequation
Q u e s t i o n:
W h e na n a l p h ap a r t i c l es t r i k e sa n i t r o g e nn u c l e u st ,h e f o l l o w i n gN U C L E ArR
e a c t i o no c c u r s :
l u e + ' \ w - , 1 $ o+ \ n
B yc o n s i d e r i ntgh e r e s tm a s s e o
s f t h e p a r t i c l egs i v e nb e l o w ,d e t e r m i n et h e K Ea n ds p e e dt h a t
t h e a l p h ap a r t i c l em u s th a v ei n o r d e rt o e f f e c t h e r e a c t i o n .
.
M a s s o f H e = 4 . 0 0 2 : t 0mua; s s o f N = 1 4 . 0 0 3 0 7 u ; m a s s o f H = 1 . 0 0 7 8 3 u ; m a s s o f O = 1 6 . 9 9 9 1 3 u
\''l-,:ts
of Alphaparticle= 4 x 1.66xtO 27kg;
( A n s: 1 , . 9 2
x L O - 1 ;37J. 6 x l - 0 6 m s - 1 )
221,
RADIOACTIVITY
process.
R a d i o a c t i v iitsya S P O N T A N E OaUnSd R A N D O M
s i l ld i s i n t e g r a t e
I t i s S p o n t a n e o ubse c a u s teh e r ea r e n o i n d i c a t o rasst o w h e n a n u c l e u w
t h i c hp a r t i c u l aar t o m sw i l l u n d e r g or a d i o a c t i v e
I t i s R a n d o mb e c a u s iet i s i m p o s s i b lteo p r e d i c w
decay.
Therearetwo factorsthat affectnuclearstability:
(i)
(ii)
p e r n u c l e o nt,h e
T h e b i n d i n ge n e r g yP e r n u c l e o n -T h e g r e a t e rt h e b i n d i n ge n e r g y
g r e a t e ri s t h e s t a b i l i t Y .
and t[O tfre ratio is ]-.
as 12uC
i
T h en e u t r o n -p r o t o nr a t i o .F o rl i g h tS T A B LnEu c l e such
F o rh e a v yn u c l e it,h e r a t i oi n c r e a s et so w a r d sl - ' 6 '
i i t h a h i g h e rs t a b i l i t yT' h e
i i l l s p o n t a n e o u sdl ye c o m p o steo f o r m n u c l e w
a t o m i cn u c l ew
Unstable
s h i c ha r e r e l e a s e d
. ee n e r g ya n d p a r t i c l ew
d e c o m p o s i t i opnr o c e s iss c a l l e dr a d i o a c t i v i tTy h
. h e nu n s t a b l en u c l e di e c o m p o s ien
d u r i n gt h e d e c o m p o s i t i opn[ o c e s sa r ec a l l e dr a d i a t i o nW
y .h e nt h e u n s t a b l en u c l e ai r e
n a t u r et,h e p r o c e s iss r e f e r r e dt o a s n a t u r a rl a d i o a c t i v i tW
p r e p a r e di n t h e l a b o r a t o r yt h, e d e c o m p o s i t i oi snc a l l e di n d u c e dr a d i o a c t i v i t y .
'.
T h e r ea r et h r e em a j o rt y p e so f n a t u r a rl a d i a t i o n
1. nlpha Radiation
A l p h ar a d i a t i o nc o n s i s tos f a s t r e a mo f p o s i t i v e lcyh a r g e dp a r t i c l e sc,a l l e da l p h a
p a r t i c l e sw, h i c hh a v ea n a t o m i cm a s so f 4 a n da c h a r g eo f + 2 ( a h e l i u mn u c l e u s ) .
i s e j e c t e df r o m a n u c l e u st h, e m a s sn u m b e r o ft h e n u c l e u s
W h e na n a l p h ap a r t i c l e
d e c r e a s ebsy f o u r u n i t sa n dt h e a t o m i cn u m b e rd e c r e a s ebsy t w o u n i t s .F o r
example:
z l f t fn ; T h e h e l i u mn u c l e u si s t h e a l p h ap a r t i c l e .
, 3 3 U|lle +
, h i c hw o u l dc a u s e
A l p h ap a r t i c l ee m i s s i o no c c u r sw h e nt h e n u c l e u sh a st o o m u c hp r o t o n sw
causes
p a r t i c l ee m i s s i o n
t h e n e u t r o n -p r o t o nr a t i ot o b e t o o s m a l fl o r h e a v yn u c l e iT. h ea l p h a
b
v
2
'
d
e
c
r
e
a
s
e
t h e p r o t o nn u m b e r t od e c r e a sbey 2 a n da l s o t h en e u t r o nn u m b e r t o
2 3 r l Ut s ' f f
- 1'587
F r o mt h e a b o v ee x a m p l et,h e n e u t r o n -p r o t o nr a t i o no f
- r ' 6 , h e n c et h e a l p h ad e c a yc a u s e dt h e
2l[rn tt'#
T h en e u t r o n -p r o t o nr a t i oo r
n e u t r o n- p r o t o nr a t i ot o i n c r e a steo w a r d sL . 6w h i c hm e a n sa n i n c r e a s ien s t a b i l i t y .
222
(21
B e t aR a d i a t i o n, w h i c hh a st w o t y p e si . e .p a n d g * e m i s s i o n s .
(i)
B e t a . ( m i n u rsa) d i a t i o ni s a s t r e a mo f e l e c t r o n sc,a l l e dp p a r t i c l e s .
W h e na
beta particleis ejected,a neutronin the nucleusis converted
to a proton,so
t h e m a s sn u m b e ro f t h e n u c l e u si s u n c h a n g e db,u t t h e a t o m i c
number
, n c r e a s ebsy o n e u n i t .F o re x a m p l e :
,
2llra;The
e r e c t r o ni s t h e b e t a ( m i n u sp)a r t i c r e .
33rn - -?e +
T h i so c c u r sw h e nt h e n u c l e u sh a st o o m a n yn e u t r o n sf o r s t a b i l i t y
a n dt h e n e u t r o n protonratio is decreased
by F-emission
( i i ) 9 * e m i s s i o nw h e r e t h ep * p a r t i c l ei s c a l l e da p o s l T R o N
w h i c hi s e f f e c t i v e l y a
p o s i t i v ee l e c t r o nI.n t h i s p r o c e s sa, p r o t o nc h a n g e isn t o
a neutron.
s c c u r sw h e nt h e n u c l e u si s d e f i c i e nitn n e u t r o n s .
T h i sp r o c e s o
tu -[n+le
I t i s s o m e t i m e cs o n v e n i e nt to a s s u m et h a t a n e u t r o ni s m a d e
u p o f a p r o t o na n d a n
electroncombinedtogether,and when p-emissionoccurs,
it is the electronfrom
t h e n e u t r o nt h a t i s e j e c t e dl,e a v i n gb e h i n da p r o t o n .H o w e v e r t h i s
a s s u m p t i oins
incorrect.
l n i t i a l l yi t w a sp r o p o s e d
t h a t t h e n e u t r o ni s r e a l l ym a d eu p o f s i xe l e c t r o nas r r a n g e d
s o t h a t t h e i re l e c t r i cf i e l d sc a n c eel a c ho t h e r ,w h i l ea p r o t o n
a l s oh a sa s i m i l a r
a r r a n g e m e notf t h e s i xe l e c t r o n b
s u t w i t h a p o s i t r o na t t h e c e n t r eo f t h e
a r r a n g e m e nH
t . o w e v e trh e d i s c o v e royf t h e s u b - a t o m ipc a r t i c t e
c a l l e dt h e q u a r k
h a sl e dt o t h e d o w n f a lol f t h i se a r l i e rp r o p o s a T
l . h en e u t r o ni s n o w t h o u g h t o b e
composed
o f 3 q u a r k s . E x p e r i m e ndtaatlah a ss h o w nt h a t t h e n e u t r o nh a s
a negative
chargeboth in its innercoreand its outer edgewith a positive
.
chargesandwiched
i n t o m a k et h e p a r t i c r e r e c t r i c a inl ye u t r a r .
(3) GammaRadiation
Gammaraysare high-energy
photonswith a veryshortwavelength(0.0005to 0.1
nm)' Theemissionof gammaradiationresultsfrom dn energv
changeof electrons
w i t h i n t h e a t o m i cn u c l e u sG
- a m m ae m i s s i o cn h a n g e ns e i t h r et h
r e a t o m i cn u m b e r
n o r t h e a t o m i cm a s sA
. l p h aa n d b e t ae m i s s i o n
a r eo f t e na c c o m p a n i e d ' b
g ya m m a
e m i s s i o na,sa n e x c i t e de l e c t r o ni n t h e n u c l e u sd,r o p s t oa l o w e r a n d
m o r es t a b l e
energystate.
A l p h a ,b e t a ,a n dg a m m ar a d i a t i o na l s oa c c o m p a niyn d u c e dr a d i o a c t i v i t y .
R a d i o a c t i vi seo t o p e s
a r e p r e p a r e di n t h e l a b u s i n gb o m b a r d m e nrte a c t i o ntso c o n v e r t
a s t a b l en u c l e u si n t oo n e
whichis radioactive.
Positron(particlewith the samemassas an electron,but a charge
of +1
i n s t e a do f - 1 ) e m i s s i o n
i s n ' to b s e r v e di n n a t u r a rl a d i o a c t i v i tby u
, t i t i s a c o m m o nm o d eo f d e c a y
in inducedradioactivity.
Bombardment
reactions
can be usedto produceveryheavyelements,
i n c l u d i nm
g a n yw h i c hd o n ' to c c u ri n n a t u r e .
223
ACTIVITY
s i l l d e c a y b, u t i f a s a m p l ec o n t a i n sN
t h e n a p a r t i c u l anr u c l e u w
I t i s i m p o s s i b lteo p r e d i c w
g i v e nb y
n u c l e it,h e n t h e r a t eo f d e c a yc a l l e dt h e A C T-IdVt I TiYs p r o p o r t i o n at ol N w h i c hi s
A=
dt
x N.
t h a t a st i m e i n c r e a s etsh, e v a l u eo f
s i g nt o i n d i c a t e
T h e r a t eo f d e c a yi s g i v e na n e g a t i v e
decreases.
T h ep r o p o r t i o n a l i st yi g nc a nb e r e m o v e dt o g i v et h e g e n e r ael q u a t i o n :
-dN
= iN= A
N
I
dt
. a su n i t so f ( t i m e ) - '
I OONN S T A NlTt h
Y O N S T A NoTr D I S I N T E G R A TC
W h e r e . tirs c a l l e dt h e D E C A C
will u
lar nucl
d as the probabil
is
Y CONSTANT
The
radioactivedecavat anv giventime.
i n a s a m p l ea f t e r
T h ee q u a t i o nc a nb e r e a r r a r l g et do d e t e r m i n et h e n u m b e ro f n u c l e ri e m a i n i n g
s o m et i m e t , a n dt h e g e n e r ael x p r e s s i ofno r t h i s i s
N - No e - A t
sinceActivitYAo(N
Similarly
A - Aos-At
l c t i v i t yo f t h e
i r e s e n (t a tt = 0 ) , a n dA o i st h e i n i t i a A
w h e r eN oi s t h e i n i t i a ln u m b e ro f n u c l e p
s am p l e .
RATEand is measuredin the numberof
ORDECAY
TheActivityis alsocalledthe coUNT RATE,
disintegration
Pse rs e c o n d .
o n e d i s i n t e g r a t i opne r s e c o n di s c a l l e da B e c q u e r e( lB q ) .
224
H A L FL I F E .
T h eh a l fl i f eo f a s a m p l ei s t h e t i m e t a k e nf o r b o t h t h e A c t i v i t yo f t h e s a m p l ea n dt h e n u m b e ro f
r a d i o a c t i vneu c l eti o f a l lt o h a l fo f t h e i n i t i a vl a l u e .
l e a t h a l fl i f eN s ) f
a n dA s ) + .
l f t h e s er e l a t i o n a
s r e s u b s t i t u t e idn t h e p r e v i o u e
s q u a t i o ntsh e n ;
A t t = T r / r , * = T , h e n c e b y s u b s t i t u t i n g N = f i n t h e e q u a t i o n w i t h A- / N s e - t t , w e g e t :
:-u -
N69-AT112
z
B ye l i m i n a t i n N
g ow e g e t ;
t 2
,-ATr/z
T a k i n gl o g sg i v e s- l n Z -
-7Ttr OR
/2
LEARN
A very usefulequation
to learnis :
TL
r1t
A _ or\z)
where n is the number of half ltves
si.mi.tarlyN"r(;)"
No
The graphto the left represents
the
v a r i a t i o no f t h e n u m b e ro f a c t i v e
n u c l ew
i i t ht i m e .
I t i s a l s oa r e p r e s e n t a t i oonf t h e
v a r i a t i o no f t h e A c t i v i t yw i t h t i m e , i n
w h i c hc a s et h e N 6o f l t h e y - a x i s ,c a n
No
b e r e p l a c e dw i t h 4 6 .
1
No
4
A/o
a
I
2Ttr
T t^ ,/ 2
/2
22s
3Ttr
/2
SOMEPROPERTIES
OF RADIOACTIVE
EMISSIONS
ALPHAPARTICLES:
( 1 ) A l p h ap a r t i c l easr eh e l i u mn u c l e i .
(2) Theyhavea chargeof +2e (2 x L.6x10-1eC1.
( 3 ) F r o ma n yp a r t i c u l asra m p l et,h e a l p h ap a r t i c l eas r ee j e c t e dw i t h a p p r o x i m a t etl hy e s a m e
velocity.
( a ) T h ee n e r g i e os f a l p h ap a r t i c l e rsa n g ef r o m 4 M e Vt o 8 M e V ,a n dt h e r a n g ei n a i r i s a b o u t
8cm.
(5) Theyare deflectedby both magneticand electricfields.
( 6 ) A l p h ap a r t i c l eas r et o t a l l ya b s o r b e db y a t h i c ks h e e to f p a p e ra sw e l l a s b y h u m a ns k i n .
( 7 ) C a u s es i g n i f i c a ni ot n i z a t i o dn u et o i t s r e l a t i v e llya r g em a s sa n ds i z e -t h e y h a v ea h i g h
p r o b a b i l i toy f c o l l i d i n g
w i t h g a sm o l e c u l easn d k n o c k i n o
g f f e l e c t r o n sa,st h e i r
m o m e n t u mc a u s et h e m t o k e e po n t r a v e l l i n a
g f t e ri t s f i r s tf e w c o l l i s i o n s .
Beta Particles:
(1) A beta (-) particlei, .n electron
(2) Theyare ejectedfrom a samplewith a rangeof speedsfrom about0.2 timesthe speed
o f l i g h tt o a b o u t0 . 8t i m e st h e s p e e do f l i g h t .
( 3 ) T h e ya r e m u c hm o r ep e n e t r a t i ntgh a n a l p h ap a r t i c l eas n d c a nb e s t o p p e db y a b o u t
3 m mo f a l u m i n i u m
.
( a ) T h er a n g ei n a i r i s f r o m a b o u t1 0 c m t o a b o u t1 0 0 c m .
( 5 ) T h e yc a u s es i g n i f i c a n tlley s si o n i z a t i otnh a n a l p h ap a r t i c l eds u et o t h e i rs m a l lm a s s t-h e y
h a v e a m u c hs m a l l e p
r r o b a b i l i toy f c o l l i d i n w
g i t h g a sm o l e c u l e s .
(6) Theyare deflectedby both electricand magneticfieldsin directionsoppositeto thoseof
a l p h ap a r t i c l e sT.h e i rm a s s e as r e a b o u t7 0 0 0t i m e sl e s st h a n t h a t o f a l p h ap a r t i c l e sa,n d
hencethey experience
muchgreaterdeflections.
GammaRays:
(1) Theyare not particlesbut insteadthey are electromagnetic
wavesof very highenergy.
(2) Theytravelat the speedof light.
(3) Theyare muchmore penetrating
than beta particlesand canpenetratea few metresof
concrete.Theycanbe stoppedby a wall of leadthat is at least10cmthick.
( a ) T h e r a n g ei n a i r i s u p t o s e v e r ahl u n d r e dk i l o m e t e r s .
( 5 ) T h e yc a u s ev e r yl i t t l ei o n i z a t i o on f g a sm o l e c u l e s .
(6) Theyare deflectedby neitherelectricnor magneticfields.
226
EXPERIMENT
TO DISTINGUISH
BETWEEN
THETHREE
TYPES
OF RADIATION:
sincethe 3 typesof rad.iation
havedifferentpenetratingpowers,then this property
can be
incorporated
in an experimentto distinguish
betweeneachtype.
A n a r r a n g e m e notf t h e a p p a r a t uiss g i v e nb e l o w :
C{
88
A l u m in u m
A G e i g eM
r u l l e rt u b e i s p t a c e db e h i n de a c hb a r r i e (r G M
t , G M z , G M :G
, M+).
F o ra l l r a d i o a c t i vm
e e a s u r e m e n ttsh,e B A C K G R o u NcDo u N T
R A T Ei s f i r s tm e a s u r e da,n dt h i s i s
t h e n s u b t r a c t efdr o m a l l s u b s e q u e nr te a d i n g s .
l f t h e G M r g i v e so n l yt h e v a l u eo f t h e b a c k g r o u ncdo u n t
r a t e ,t h e n t h e s a m p l ew o u l dh a v eh a d
o n l ya l p h ap a r t i c l e s .
l f h o w e v e rt,h e r e a d i n go n G M 1i s g r e a t e tr h a n , t h eb a c k g r o u n d
r a t et h e n t h e r ew o u l db e
p a r t i c l eps a s s i n tgh r o u g ht h e t i s s u ew h i c hw o u l d
b e e i t h e rb e t ap a r t i c l eos r g a m m ar a y so r
b o t h 'T h er e a d i n gos n G M za n d G M : w i l l a l l o wt h e o b s e r v e r
t o d e t e r m i n et h e n a t u r eo f t h e
e m i s s i o ntsh a t a r e n o t a l p h a .
227
from
T h ed e f l e c t i o nisn e l e c t r i ca n d m a g n e t i fci e l d sa l s oa l l o wt h e e m i s s i o ntso b e d i s t i n g u i s h e d
o n ea n o t h e r :
Alpha, Beta,and Gamma
Historically,,the productsof radioactivitlzwere called alpha,beta,and gammawhen it was found
thatthey couldbL analyzedinto threedistinctspeciesby eithera magneticfield or an electric
field.
XXXX
tx
X
X
x
Y
X
XXXI
fie1r1
1"1;:r1rrtti':
XXX
ij1,,;i3U f t-r:rTft
X
',,"igr,',r;:1-
XX
+
+
+
+
+
+
I
I
I
I
-
FX
:;riflt trlE
FiArlir:r;:r.:ti.',"E
.
i n I *id trlirr::l::
228
EXPERIMENT
TO DETERMINE
THEHALFLIFEOFA RADIOACTIVE
ISOTOPE
WITHA SHORTHALFLIFE:
T h eh a l fl i f ec a nb e m e a s u r e d
by monitoring
t h e i o n i z a t i o cnu r r e n tp r o d u c e db y t h e s a m p l eo f
r a d i o a c t i vm
e aterial.
A t y p i c a sl a m p l et h a t i s u s e di n t h e l a b o r a t o riys R a d o ng a s .
Centra I e l e c t r o d e
valves
1-2Vbattery
m i c r o am m e t e r
d . c .a m p l i f i e r
Polythene
bottle
R= 1011e
w i t h r a d o ng a s
T h ev a l v e sa r es e tt o t h e o p e np o s i t i o na n dt h e p o l y t h e n eb o t t l e
i s s q u e e z etdo f o r c es o m e
r a d o ng a si n t ot h e i o n i z a t i o cnh a m b e rm
( e t a lc a n ), a n dt h e v a l v e sa r et h e n c l o s e d .
T h ec u r r e n to n t h e a m m e t e ri s r e c o r d e da n d a s t o p w a t c hi s
started.
R e a d i n gosf c u r r e n ta r e r e c o r d e da t e v e r y1 0 s e c o n dfso r a p e r i o d
o f a b o u t5 m i n u t e s .
229
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SAFETYPRECAUTIONS
FOR HANDLINGAND DISPOSALOF RADIOACTIVE
MATERIAL
S i n c ea l p h aa n d b e t ap a r t i c l e d
s o n o t p e n e t r a t et h e s k i nt h e y a r e u n l i k e l yt o b e h a r m f u fl r o m E X T E R N A L
S O U R C Ei .Se .f r o m s o u r c e so u t s i d eo f t h e h u m a nb o d y .
T h e yw i l l h o w e v e rp o s ea s e r i o u st h r e a ti f t h e y a r e i n g e s t e do r i n h a l e d .
Personh
s a n d l i n gr a d i o a c t i vm
e a t e r i a l sm u s tn o t e a t s m o k eo r d r i n ki n t h e v i c i n i t yo f t h e r a d i o a c t i v e
s a m p l e sM
' a s k sa s w e l l a s d i s p o s a b lgel o v e sa n d p r o t e c t i v ec l o t h i n gs h o u l db e w o r n .
I n t h e c a s eo f X r a y sa n d g a m m ar a y s ,p e r s o n sm u s tb e s h i e l d e dt,h e r em u s tb e a l a r g ed i s t a n c eb e t w e e n
t h e p e r s o na n d t h e s o u r c ea, n d e x p o s u r et i m e sm u s tb e a s s h o r ta s p o s s i b l e .
P e r s o nw
s o r k i n gi n r a d i o a c t i vaer e a ss h o u l dw e a r a F I L MB A D G Ew h i c hd a r k e n sa s i t i s e x p o s e dt o
r a d i a t i o nT. h e d a r k n e s o
s f t h e f i l m g i v e sa n i n d i c a t i o n
o f t h e e x p o s u r et i m e .
l n t h e c a s eo f r a d i o t h e r a payn d d i a g n o s i st h, e i s o t o p et h a t i s c h o s e nm u s t h a v ea s h o r th a l f l i f ea sw e l l a s
i t m u s tp a s st h r o u g ht h e b o d ya n d b e e x c r e t e do u t i n a v e r ys h o r tt i m e .
W o r k e r sm a y a l s oh a n d l er a d i o a c t i vm
e a t e r i a l sb y u s i n gr e m o t ec o n t r o l l e dt o o l s .
RADIOACTIVE
WASTE
RadioactiveWaste Description
R a d i o a c t i vwea s t ec a nv a r y! r e a t l y i n i t s p h y s i c aal n d c h e m i c aflo r m . l t c a nb e a s o l i d ,l i q u i d ,g a s ,o r e v e n
s o m e t h i n gi n b e t w e e n s, u c ha s s l u d g eA. n y g i v e nr a d i o a c t i vw
e a s t ec a nb e p r i m a r i l yw a t e r ,s o i l ,p a p e r ,
p l a s t i cm
, e t a l ,a s h ,g l a s sc, e r a m i co, r a m i x t u r eo f m a n yd i f f e r e n tp h y s i c af lo r m s .T h ec h e m i c a l
form of
r a d i o a c t i vw
e a s t ec a nv a r ya s w e l l .R a d i o a c t i vwea s t ec a nc o n t a i nr a d i o n u c l i d eosf . v e r yl i g h te l e m e n t s ,
s u c ha s r a d i o a c t i v he y d r o g e n( t r i t i u m ) o, r o f v e r y h e a v ye l e m e n t ss, u c h ' a su r a n i u m R
. a d i o a c t i vwea s t ei s
c l a s s i f i eads h i g h ,i n t e r m e d i a t eo,r l o w l e v e l .D e p e n d i n og n t h e r a d i o n u c l i d ecso n t a i n e di n i t , a w a s t ec a n
r e m a i nr a d i o a c t i vfer o m s e c o n d st o m i n u t e so, r e v e nf o r m i l l i o n so f y e a r s .
Radioactive
Waste Management
R a d i o a c t i vwea s t em a n a g e m e ni tn c l u d e st h e p o s s e s s i o tnr,a n s p o r t a t i o nh,a n d l i n gs, t o r a g ea, n d u l t i m a t e
d i s p o s aol f w a s t e .T h e s a f em a n a g e m e not f r a d i o a c t i vw
e a s t ei s n e c e s s a rtyo p r o t e c tp u b l i ch e a l t h .l f
h a n d l e di m p r o p e r l yp, o t e n t i ael x p o s u r eosf h u m a n st o h i g h - l e v er la d i o a c t i vw
e a s t ec a nb e d a n g e r o u s ,
e v e nd e a d l y S
. o m er a d i o a c t i vw
e a s t e ss u c ha s c e r t a i nt y p e so f t r a n s u r a n i w
c a s t ec a nc a u s eb i o l o g i c a l
e f f e c t si n h u m a n so n l y i f t h e r a d i o n u c l i d ecso n t a i n e di n t h e w a s t ea r e d i r e c t l yi n h a l e do r i n g e s t e dM
. ost
l o w - l e v erla d i o a c t i vw
e a s t e sc a n b e h a n d l e db y h u m a n sw i t h o u ta n y m e a s u r a b lbei o l o g i c ael f f e c t s .
'Neverthelesg
s ,o o dh a n d l i n gp r a c t i c eos f a l l r a d i o a c t i vm
e ateriala
s n d w a s t es h o u l db e t h e g o a lt o
p r o v i d eo p t i m u mp r o t e c t i o nt o h u m a n sa n d t h e e n v i r o n m e n tT. h e r eh a v eb e e nh i s t o r i cp r a c t i c e s
a s s o c i a t ewd i t h t h e u s eo f r a d i o a c t i vm
e a t e r i aw
l h e r ew o r k e r sw e r e u n a w a r eo f p o t e n t i a rl i s k sT. h e
r a d i u mw a t c hd i a lp a i n t e r so f t h e 1 9 2 0 si l l u s t r a t e
t h e h e a l t he f f e c t st h a t c a nb e a s s o c i a t ew
di t h
i m p r o p e rh a n d l i n gp r a c t i c e sT.h e p a i n t e r se x p e r i e n c ehdi g ho c c u r r e n c eosf c a n c e ro f t h e l a r y n xa n d
t o n g u ed u e t o i n g e s t i o n
of radium.
T h et r a n s p o r t a t i ocn. r a d i o a c t i vw
e a s t ec a no c c u rv i a r o a d w d va, i r c r a f ts, h i p / b a r g ea,n d r a i l .T h e
c l a s ' ; i f i c a t i oan d p h y s i c asl i z eo f r a d i o a c t i vw
e a s t ed i c t a t et h e m e t h o do f t r a n s p o r tt,h e p a c k a g i n g
i - € ( ; r : i r eadn, d t h e l a b e l i n gn e c e s s a rtyo a l l o wf o r t h e s h i p m e n o
t f a s p e c i f i cw a s t e .T h e r ea r e
i n t e r n a t i o n at rl a n s p o r t a t i orne q u i r e m e n tfso r r a d i o a c t i vw
e a s t e ,a sw e l l a s m o r es p e c i f i cr e g u l a t i o nisn
i n d i v i d u aclo u n t r i e s .
23r
RS
RADIOACTIVITYDETECTO
( 1 ) T h eS p a r kC o u n t e r :
T h i sw o r k so n t h e p r i n c i p l teh a t r a d i o a c t i veem i s s i o ncsa u s ei o n i z a t i o on f t h e a i r a n d c a nc a u s e
t h e a i r t o c o n d u citn a r e g i o nw h e r ea h i g hp o t e n t i adl i f f e r e n cies a p p l i e d .
T h ea p p a r a t ucso n s i s tos f a f i n ew i r e g a u z ew h i c ha c t sa sa c a t h o d eb e i n gm o u n t e dcloseto a
. h i g hv o l t a g ev a r i a b l ed . c .s o u r c e( 0 ) 5 0 0 0 V ) i s connected
m e t a lr o d w h i c ha c t sa s a n a n o d e A
a c r o s st h e a n o d ea n d c a t h o d e .
anode +
cathode€
tI
I n s u l a t i nbga s e
Radium
-+
source
H i g hv o l t a g e
supply
I n i t i a l l yw, i t h n o r a d i o a c t i vseo u r c ep r e s e n tt,h e v o l t a g ei s i n c r e a s eadc r o s st h e a n o d ea n d
c a t h o d eu n t i ls p a r k sj u m p a c r o s st h e g a p .T h ev o l t a g ei s t h e n r e d u c e ds l o w l yu n t i lt h e s p a r k s
j u s tf a i lt o b e p r o d u c e d .
T h er a d i u ms o u r c ei s t h e n i n t r o d u c e cd l o s et o t h e w i r e g a u z ea, n dt h e s p a r k sr e a p p e a r .
W h e nt h e r a d i o a c t i vseo u r c ei s p r e s e n tt,h e a i r b e c o m e si o n i z e da n dt h e i o n sa n df r e ee l e c t r o n s
nf sparks.
t n dt h i sc o n d u c t i o cna u s e tsh e p r o d u c t i o o
c a i r s et h e a i rt o c o n d u c a
232
( 2 ) T h e D I F F U S I OCNL O U D
CHAMBER:
T h i si n s t r u m e nits u s e dt o m a k et h e t r a c k so f r a d i o a c t i veem i s s i o n tso b e c o m ev i s i b l e .
I t w o r k so n t h e f o l l o w i n gp r i n c i p l e :
W h e na n i o n i z i n gp a r t i c l ep a s s etsh r o u g ha g a s ,i t l e a v e sa t r a i lo f p o s i t i v e
g a si o n sa l o n gi t s
path due to electronsbeingknockedoff from the gasparticles
when the radioactive
particle
c o l l i d ew
s ith it.
l f t h e g a sc o n t a i n sa s a t u r a t e d
v a p o u r f o re x a m p l ea l c o h ovl a p o u r a, l c o h odl r o p l e t sw i l l
condensa
e r o u n dt h e t r a i lo f p o s i t i v ei o n st o s h o wa v i s i b l et r a c k .
perspex
Saturated
vaDour
\
F e l ts o a k e dw i t h a l c o h o l
Metal base
R a d i o a c t i vseo u r c e
Dry ice
sponge
T h ea p p a r a t ucso n s i s tos f a P e r s p ecxh a m b e w
r i t h a s t r i po f f e l t a t t h e t o p t h a t h a sb e e n
soakedin a mixtureof alcoholand water.The.darkmetalbaseat the bottom is kept at a
temperaturo
e f a b o u t- 5 0 ' Cb y d r y i c e .T h e r ei s a c o n t i n u o uds i f f u s i o no f a l c o h ovl a p o u r f r o m
' t h e t o p o f t h e c h a m b e tro t h e
bottom.
T h e r ew i l l b e a s u p e r s a t u r a t ev da p o u r a b o v e t h e
c o l db a s e a
, n dt h e r a d i o a c t i vseo u r c ew i l l
p r o d u c ev i s i b l et r a c k st o a n yo b s e r v elro o k i n gf r o m t h e t o p o f t h e c h a m b e r .
g l p h ao n l y ,t h e r ea r es o m e t i m eds i f f e r e n lte n g t h so f t r a c k sd u e t o t h e
F o ra s o u r c ep r o d u c i n a
d i f f e r e n et n e r g i e os n e m i s s i o n .
233
( 3 ) T h e G E I G E R - V I u L lrrU
nBE:
l i r e A w h i c ha c t s
s t h i n c e n t r aw
T h i si n s t r u m e nits a l s or e f e r r e dt o a st h e G M t u b e .l t c o n t a i n a
r w h i c ha c t sa st h e c a t h o d e .
g E T A Lc y l i n d eC
a sa n a n o d ea n d i s i n s u l a t e fdr o m t h e s u r r o u n d i nM
+400V
relativeto the cathodewhichis earthed.
The anodeA is kept at a potentialof about
A r g o na n d
A n o d eA
Halogen
Protective
To amplifier
gauze
Mica
window
cathode C
W h e na n a l p h ao r b e t ap a r t i c l ee n t e r st h e t u b e v i at h e m i c aw i n d o w ,e l e c t r o n as n d i o n sa r e
p r o d u c e dw h e nt h e i o n i z i n gp a r t i c l ek n o c k so f f e l e c t r o nfsr o m t h e g a sm o l e c u l e s .
s n d i o n sc a nb e i n c r e a s esdi g n i f i c a n tilfyt h e p o t e n t i a+l V o n t h e
T h i sn u m b e ro f e l e c t r o n a
v o l t a g ei s t h e v o l t a g e
v o l t a g eo f t h e g a s .( t h eb r e a k d o w n
a n o d e ,i s l a r g e rt h a nt h e b r e a k d o w n
a t w h i c ht h e g a sb e g i n st o c o n d u c t .
T h ee l e c t r o n as r ea t t r a c t e dt o w a r d s t h eA N O D Ea n dt h e p o s i t i v ei o n sa r e a t t r a c t e dt o w a r d s t h e
ODE.
CATH
T h e r ew o u l db e a p u l s ec u r r e n tp a s s i n tgh r o u g hR a n d h e n c et h e r ei s a p o t e n t i adl i f f e r e n c e
acrossR.
t o a n e l e c t r o n icco u n t e r .
T h ep u l s ev o l t a g ei s a m p l i f i e da n d p a s s e d
234
lmportantfeaturesof the GM tube:
( 1 ) W h y i s a r g o na n d h a l o g e nu s e d ?
T h ed i s c h a r goec c u r so n l y f o r a s h o r t t i m e ,a n d s e c o n d a reyl e c t r o nas r ee m i t t e db y t h e c a t h o d e
w h e nt h e p o s i t i v ei o n sa r r i v et h e r e .
particles.
T h i sw o u l dt h e n u p s e t h e r e c o r d i n o
gf t h e i o n i z i n g
T h eg a si n t h e t u b e s h o u l dt h e r e f o r eb e a m i x t u r eo f a r g o na n d h a l o g e nw h o s em o l e c u l e s
a b s o r be n e r g yo f t h e p o s i t i v ei o n so n c o l l i s i o a
n n d h e n c et h e d i s c h a r gies q u i c k l yq u e n c h e d .
( 2 ) T h e a n o d ew i r e m u s t b e t h i n s o t h a t t h e c h a r g eo n i t c a n p r o d u c ea n e l e c t r i cf i e l d o f
v e r y h i g h I N T E N S I TaYs a l l o f t h e c h a r g ew o u l d n o w b e c o n c e n t r a t e idn a v e r y s m a l l
v o l u m eo f w i r e . l f t h e c h a r g e sw e r e d i s p e r s e d
o v e r a l a r g e rv o l u m e ,t h e e l e c t r i cf i e l d
i n t e n s i t ya r o u n dt h e w i r e w o u l dd e c r e a s e .
T h eh i g he l e c t r i cf i e l dw o u l dt h e n c a u s et h e e l e c t r o ni-o n p a i rt o b e a c c e l e r a t et d
o very
h i g he n e r g i e n
s e a rt o t h e w i r e a n dw o u l dt h e r e f o r ec a u s et h e p r o d u c t i o n
o f m u c hm o r e
i o n p a i r sb y c o l l i s i o nws i t h g a sm o l e c u l e s
BACKGROUND
RADIATION
ORIGINS
OF BACKGROUND
RADIATION:
( 1 ) T h e r ea r e i n t e r n asl o u r c e o
s f r a d i o a c t i vneu c l i d e ss,u c ha s p o t a s s i u m
f l f w n i c hi s
p r e s e nitn t h e h u m a nb o d y .
( 2 ) G a m m ar a y sa n d r a d o ng a si n t h e a i r c o m ef r o m n a t u r a l l o
yccurring
radioactive
m a t e r i a lssu c ha s r a d i u mt,h o r i u ma n du r a n i u mi n t h e e a r t h ' sc r u s t .
( 3 ) G y p s u mw a l l se m i t r a d i a t i o n .
( 4 ) C o s m i cr a d i a t i o ni s m o s t l ya b s o r b e db y t h e a t m o s p h e r eb,u t a s m a l lp e r c e n t a gree a c h e s
t h eg r o u n d .
( 5 ) I n d u s t r i aslo u r c e s u c ha s n u c l e a p
r o w e rp l a n t st e n d t o l e a ks o m er a d i a t i o ni n t ot h e
atmosphere.
( 6 ) N u c l e afra l l - o u tf r o m b o m bt e s t st e n d st o r e m a i ni n t h e a t m o s p h e rfeo r m a n yy e a r s .
( 7 ) L u m i n o ups a i n te m i t sr a d i a t i o n
( 8 ) T e l e v i s i osnc r e e n a
s n d o t h e rc a t h o d er a yt u b e se m i t g a m m aa n dX r a d i a t i o n .
EFFECTS
OF RADIATION
ON LIVINGTISSUE:
d h i c ht h e ne i t h e rk i l l t h ec e l l so r c h a n g e
W h e nc e l l sa b s o r br a d i a t i o ni o, n sa r ep r o d u c e w
them
i n a n u n d e s i r a bw
l ea y .
W h e na l a r g ea m o u n to f c e l l sa r e d e s t r o y e dt h
, e o r g a n i s mm a yd i e .
T h e r ea r ea l s om a n yd e l a y e de f f e c t ss u c ha s c a n c e rl,e u k a e m i aa,n d h e r e d i t a rdy e f e c t sa n d
deformities.
235
U S E F UALP P L I C A T I OO
NF
SR A D I O I S O T O P E S :
(1) Tracers:
A t r a c e ri s t h e t e r m u s e df o r a p a r t i c u l acro m p o u n dt h a t c a nb e d e t e c t e da s i t m o v e s
t h r o u g ha l i v i n gb o d y
Radloisotopa
e rse u s e f u al st r a c e r sb e c a u s a
e p a r tf r o m b e i n ge a s i l yd e t e c t e di n t h e
b o d y ,t h e y h a v et h e s a m ec h e m i c apl r o p e r t i eas st h e i r n o n - r a d i o a c t i vi seo t o p e s .
F o re x a m p l es,o d i u mh a sa r a d i o i s o t o poef l l N a a n dt h i si s a l s oa b l et o f o r m N a C .l
H e n c ei f a p a t i e n ti n g e s t s o m er e g u l a sr a l tt h a t h a s . as m a l la m o u n to f s o d i um - 2 4 ,a
Geiger-Mulle
t ur b e c a nb e u s e do v e rt h e p a t i e n t ' sb o d yt o t r a c et h e p a t ho f t r a v e lo f t h e
s o d i u mc h l o r i d e .
( 2 ) D i a g n o s t ilcm a g i n g :
l f a p a t i e n ti n g e s t a
s c o m p o u n dt h a t c o n t a i n a
s s m a l la m o u n to f g a m m a -e m i t t i n g
r a d i o i s o t o p teh, e n w h b nt h e c o m p o u n dr e a c h e as p a r t i c u l aor r g a nf o r e x a m p l et h e l i v e r ,
t h e g a m m ar a y se m i t t e dc a nb e u s e dt o c o n s t r u cat n i m a g eo f t h e o r g a n w
, i t h t h e a i do f
a g a m m ac a m e r a .
T h eg a m m ae m i t t e ri s c h o s e nb e c a u s e
i t e s c a p efsr o m t h e b o d yw i t h o u tc a u s i n g
any
i o n i z a t i od
n a m a g eA. l s oa r a d i o i s o t o pwei t h a s h o r th a l fl i f ei s u s e ds o t h a t t h e i n t e n s i t y
r e a c h ea
s v e r yi n s i g n i f i c a vnat l u ea f t e r2 4 h o u r s .
(3) CancerTreatment:
G a m m ar a d i a t i o ins u s e di n a h i g hi n t e n s i t by e a mt o k i l lc a n c ecr e l l s .
T h e ' g a m m raa y sh a v eh i g hp e n e t r a t i npgo w e r sa n d c a nb e u s e dt o t a r g e ta l o c a t i o n
i n s i d et h e b o d y ,u n l i k et h e c a s eo f a l p h aa n db e t ap a r t i c l ews h i c ha r es t o p p e db y t h e
s k i no f a p e r s o n .
P r e v i o u s lXy- r a y sw e r e u s e dt o k i l lc a n c e cr e l l sb u t X r a y m a c h i n eas l a r g ea n d
c u m b e r s o mteo u s e , a n dt h e y a l s or e q u i r ev e r yh i g hv o l t a g e tso o p e r a t e .
A g a m m as o u r c eo n t h e o t h e rh a n dc a nb e v e r ys m a l w
l i t h d i m e n s i oonf a f e Wc m ,a n d
d o e sn o t r e q u i r ea h i g hv o l t a g es o u r c e .
236
TUTORIAL
SHEET
( S o l u t i o nsst a r tf r o m p g 1 8 9 )
( 1 ) R a d i u m ( Rdae) c a y tso r a d o n( R n )b y t h e r e a c t i o n :
'33no--' ,|ZRn+
/He+ y
T h ea t o m i cm a s s e a
s r e ;R a d i u
m 3 . 7 5 2 4x L O - 2 s kr ga ;d o n3 . 6 5 6x 1 0 - 2 s k hg e; l i u m0 . 0 6 7x
10 2skg.
P l a n c kc o n s t a n=t 6 . 6 3x 1 0 3 a i sa; n d s p e e do f l i g h ti n a v a c u u m= 3 x i - 0 8 m s - 1 .
(i)
E s t i m a t teh e e n e r g yi n J o u l e sw h e na n a t o mo f 2 2 " $ ndae c a y s .
(ii)
E s t i m a t teh e w a v e l e n g t oh f t h e g a m m ap h o t o ne m i t t e dd u r i n gt h i sd e c a yg, i v e n
t h a t 4 % o f t h e e n e r g yr e l e a s etdu r n st o g a m m ar a d i a t i o n .
( i i i ) w h a t h a p p e ntso t h e r e m a i n i n9g6 %o f t h e e n e r g y ?
;
( A n s: 2 . 6 4 6x l - 01 0 J; 1 . 9x 1 0 1 a mC
o n v e r t etdo K i n e t i ce n e r g yo f t h e p a r t i c l e s
a n da l s ot o h e a te n e r g y . )
( 2 ) A r a d i o a c t i vsea m p l eo f a n e l e m e n X
t d e c a y st o a n o t h e re l e m e n Y
t witha decay
c o n s t a not f
.
0 . 0 2 2h o u r s1 .
(i)
D e t e r m i nteh e h a l fl i f eo f X .
(ii)
W h a tf r a c t i o no f t h e o r i g i n asl a m p l ew i l l r e m a i na f t e r4 . 5 h a l fl i v e s ?
( i i i ) W h a tf r a c t i o nw i l l r e m a i na f t e r1 _ d
4a y s ?
(Ans3
; 1 . 5h o u r s 0
; . 0 4 4 ; 6x 1 - 0 - 4 )
( 3 ) O n eo f t h e d a n g e r sa s s o c i a t ewdi t h n u c l e a pr o w e rp l a n t si s t h e l e a k a g o
ef plutonium
w h i c hi s a b y - p r o d u c ti n t h e r e a c t o rl.t h a sa h a l fl i f eo f a p p r o x i m a t e 2l y. 4 x 1 - g a y e a rAs .
l e t h a dl o s eo f p l u t o n i u mi s 1 . 6 5 m g .
(i)
D e t e r m i nteh e a m o u n to f p l u t o n i u m
n u c l eiin a l e t h a dl o s ei f i t s r e l a t i v e
atomic
m a s si s 2 3 9 .
(ii)
C a l c u l a tteh e d e c a yr a t eo f t h i sd o s e
( A n s: 4 . 1 6x 1 0 1 81;, . 2x l - 0 l 4 d i s i n t e g r a t i o
pn
e sry e a r )
.
( 4 ) A r a d i o a c t i vsea m p l et h a t i s u s e df o r m e d i c i n apl u r p o s e hs a st o b e p r e p a r e di n a
l a b o r a t o r yb e f o r ei t s i n t e n d e du s e .T h es a m p l eh a sa h a l fl i f eo f 9 6 . 7h o u r s .
(i)
W f r a ts h o u l di t s i n i t i a a
l c t i v i t yb e i f t h e s a m p l em u s th a v ea n a c t i v i t yo f 6 . 6x
L 0 ' B q ,w h e n i t i s u s e d 3 0 h o u r sa f t e ri t h a sb e e np r e p a r e d ?
(ii)
W h a tw i l l t h e a c t i v i t yo f t h e s a m p l ei n p a r t ( i ) b e 2 4 h r sa f t e ri t h a sb e e n
administered?
( A n s 8: . 1 8x 1 0 s B q5; . 5 6x t O e a q . )
237
( 5 ) A r a d i o a c t i vsea m p l ei n a l a b o r a t o r y h aas m a s so f 2 0 ga n da n a c t i v i t y o f2 . 3 5x
1 0 6 B q . T hsea m p l eh a sa r a d i o a c t i vies o t o p ew h o s en a t u r aal b u n d a n cies 2 t % a n dw h o s e
m a s sn u m b e ri s 1 4 8 . C a l c u l a t e :
(i)
T h ed e c a Yc o n s t a n t
(ii)
T h eh a l fl i f ei n Y e a r s
( i i i ) T h et i m e r e q u i r e df o r t h e a c t i v i t yt o f a l lt o 1 . 2x 1 0 3 B q .
16s1 I.6 x 1Osyea
rs;t'75x l0eyears)
(Ans:1'.37x 10
;
e a t e r i a lcso n t a i n s2 0 p g o fa
t , m i x t u r eo f r a d i o a c t i vm
( 6 ) A t t h e s t a r to f a n e x p e r i m e n a
r a d i o i s o t o pAe w h i c hh a sa h a l fl i f eo f 9 0 s e c o n d as n d 3 2 0I E o f r a d i o i s o t o pBew h i c hh a s
a h a l fl i f eo f 3 0 s e c o n d s .
sf each
(i)
A f t e rw h a t p e r i o do f t i m e w i l l t h e m i x t u r ec o n t a i ne q u a lm a s s e o
W h a t i s t h e m a s so f e a c hi s o t o p ea t t h i st i m e ?
isotope?
g h e nt h e
(ii)
C a l c u l a tteh e r a t ea t w h i c ht h e a t o m so f i s o t o p eA a r ed e c a y i n w
=
Avogadro's
and
massesare the same,giventhat the molarmassof A 2209
c o n s t a n=t 6 . 0 2x 1 0 " )
4,
( A n s: 1 8 0s d c o n d ;s5 t t g; 1 . 0 5 x1 0 ' " B q )
materialscontains160pgof a
a mixtureof radioactive
( 7 ) At the startof an experiment,
r a d i o i s o t o pAe w h i c hh a sa h a l fl i f eo f 8 0 s e c o n d as n d 8 0 0 p go f r a d i o i s o t o pBew h i c h
h a sa h a l fl i f eo f 4 0 s e c o n d s .
sf each
( i ),
A f t e rw h a t p e r i o do f t i m e w i l l t h e m i x t u r ec o n t a i ne q u a lm a s s e o
w h a t i s t h e m a s so f e a c hi s o t o p ea t t h i st i m e ?
isotope?
g h e nt h e
(ii)
C a l c u l a tteh e r a t ea t w h i c ht h e a t o m so f i s o t o p eA a r ed e c a y i n w
, i v e nt h a t t h e m o l a rm a s so f A = L 4 0 ga n dA v o g a d r o ' s
m a s s ea
s r et h e s a m e g
constant=6.02x1-0")
(186seconds
; 32Vg;1'.rcx 101sBq)
( g ) ( a )E x p l a i n
w h a t i s m e a n tb y t h e h a l fl i f ea n dt h e d e c a yc o n s t a not f a r a d i o i s o t o p e .
t h a sa h a l fl i f eo f 5 8 d a y s .l t i s u s e dm e d i c a l l iyn t h e d i a g n o s iosf
A r a d i o a c t i veel e m e n X
t sr ec o m p l i c a t ebdy t h e f a c tt h a t X i s e x c r e t e d, i . e .
b l o o dd i s o r d e r sM. e a s u r e m e n a
r e m o v e df r o m t h e b o d ya t a r a t es u c ht h a t 8 7 d a y sa f t e ra d m i n i s t e r i nagd o s e ,h a l fo f
t h e X a t o m si n t h e d o s eh a v eb e e ne x c r e t e dT. h a ti s e q u i v a l e n t t os a y i n g t h a t Xh a sa
b i o l o g i c ahla l fl i f eo f 8 7 d a Y s .
(i)
l f t h e c o u n tr a t ef r o m a b l o o ds a m p l ei s 1 4 0 0c o u n t sp e r m i n u t e ,w h a t w i l l i t b e
f r o m a s i m i l a rb l o o ds a m p l et a k e n1 7 4d a y sl a t e r ?
(ii)
H o w ' o n ga f t e rt h e f i r s ts a m p l ew a st a k e nw o u l da f u r t h e rs i m i l a sr a m p l eg i v ea
couni rateof 1'2countsPerminute?
(Ans:44 countsPerminute;65 daYs.)
238
(9)
T h er a d i o a c t i vneu c l i d e' 3 2 U i s p r e s e n itn s o m er o c k sa n d h a sa h a l fl i f eo f 4 . 5 x
lOeyears
as it
decaysinto zf;lPb(lead)according
to the nuclearequation,32u- 2fl$eb
+
decay products
A s a m p l eo f r o c ki s f o u n dt o c o n t a i n4 g m go f 2llu ana32mgot zflfnn.
(i)
(ii)
(iii)
(iv)
C a l c u l a tteh e n u m b e ro f U r a n i u ma t o m si n t h e s a m p l e
c a l c u l a tteh e n u m b e ro f l e a da t o m si n t h e s a m p l e
Determine
t h e i n i t i an
l u m b e ro f u r a n i u ma t o m si n t h e s a m p l ea t f o r m a t i o n
a s s u m i ntgh a t t h e r ew e r e n o i n i t i a l e a da t o m sp r e s e n t .
Determine
t h e a g eo f t h e r o c k .
(Ans:1,.21,
xL020;9.35
x 101e;
2.1,5x L020;3.73
x 10syr)
(1 0 )
I n r a d i o a c t i vdea t i n gt,h e r a d i o i s o t oep, 2 c i s u s e da s i t h a sa longhalf life of
5 5 7 0y e a r s l. n a l l l i v i n go r g a n i s mtsh e c o n c e n t r a tni oo f L [ c t o l f C i s 1 . 1 6: l - 0 1 2W. h e n
t h e o r g a n i s md i e st h e I [ c i s n o t r e p l e n i s h eadn d d e c a y st o t Z c .
l n a n a r c h a e o l o g i csaal m p l eo f b o n e ,t h e c o n c e n t r a tni oo f r / C t o r f C w a sf o u n dt o
be
3.4: 1,014
Determine
t h e a g e . o tf h e b o n e .
( A n s :2 8 , 3 5 5y r )
239
SOLUTIONS
WORKED
( 1 ) T h ed e c r e a s ien m a s sf r o m R a d i u mt o i t s o t h e rp r o d u c t si s g i v e nb y :
L m - t 0 - 2 sG . T S Z 4- [ 3 . 6 5 6+ 0 . 0 6 7 ] ): 0 . 0 2 9 4x 1 0 - 2 s k g
L E : L m c z : 0 . 0 2 9 4x 1 0 - 2 5x 9 x 1 0 1 6 2 . 6 4 6 x 1 0 - 1 0 /
EnergyduetoGammaradiationon|y=0.04x2.646x]-0-10=1.06x].0-1iJ
o'osx 19-34
I ? x 1oB
E -\,/ . t -rynsi.ns
1.06rro_rr::r.9x1o-14m
E
The 96%ofthe Energyis convertedinto heataswell as Kineticenergyof the emitted
particles.
(2)
(i) Tr 2
Use the equatton N : No
InZ
1
0.693
: 3l-.Shr
0.022
wltere nrepresents the number of half ltves
"(;)"
t
( 1 ) - : 0 . 0 4 4.
i sg i v e nb y
( i i ) T hfer a c t i o o
n f t h e o r i g i n asla m p l e
#
(iii)
F i r s tf i n dt h e n u m b e ro f h a l fl i v e si n 1 ' 4d a y s .
Numberof hoursin 3,4days= 1'4x24=336hr
336
- L0.67Tt
31,5
z
Number of half ltves tn336 hr - -
H ence
(3)
#:(;)""
:6x
L0-4
239 g contatn 6.02 x !\z3parti"cles
6.02 x 1,023
1g contai,ns
239
6 . 0 2x 1 0 2 3
x 1 . 6 5x 1 0 - 3- 4 . t 6 x L 0 l 8 n u c l e i "
0 . 6 93
InZ
x 4.16 x 1018
(tt)Acttvtty - )"N,where7 ; A:
Z+ x tOT,
1,.65m9contatn
239
= '1..2x l-o1adi,s/yr
240
2
(4) (i)Halflife= 96.7hr.
Aftera 30 hourperiod,the numberof halflivesisequalto
30
: 0.31half li.ves
g6t
(Jsing the equati"onA - o,
(:)" ,
we setAo:
(ii)
t
1l#-11)
- B.1B
x Los
Bq
24 hrsafterit trasbeenadm,r,r,!?/O
meansthata totattimeof 54 hr woutd
h a v ee l a p s eadf t e rp r e p a r a t i o n .
'A-
- 0.558half ti"ves
54tv - Y
g6i
/l\o'sse
/ 1 1 0 ' s s-sB . 1 B x
or\il
= 5.56xtoeBq
lorx(;,)
( 5 ) l f t h e s a m p l eh a sa m a s so f 2 O ga n dt h e r a d i o a c t i vpea r t i s c o n t a i n e d
i n 2 L % o ft h e
then
the
mass
of
the
radioactive
isotope
.sample,
is 20x0.21,=
4.2g
T h em a s sn u m b e ro f t h e r a d i o i s o t o pies L 4 g 9
1,48g contains6.02x L023particles.
4 . 2 e w i l cl o n t a i n
6.02 x 1,023
x4.2 - 1.7Ix1,\zzparti"cles
MB
(i)Usi.ngtheequationA_trN';^_#-m_]-.37x1g-16,-1
(ii)
The half tif e i.sgi"venby Tr - Y
";1
0 . 6 93
=
- 5 x L0lsseconds - L.6x !\Byears
T57 x 1n-r6
241
(iii)
Useeither A -
Ao(J)" or A :
u stns A -
Aoe-At
Ao()" , si"ves
I.Z x 103
*ffir-t
(})"
*:
where n is the no.of ttalf ltves
L
/1\n
: (;)
; n lni : ln(s'1x 10-a); n - 10'e4
HenceLO.g4halflivesare requiredfor the activityto fall to the desiredcount
rate.
; 10.94half lives=10.94x1.6x \0Byears
Sincet half Iife= I.6 x 1.0Byears
1.75x 10syears.
(6) (i)
B
A
20 +
10
90 sec
320+
160
30sec
After 180 secondstheYwill
both have5;rg
10+
5
180sec
60sec
160+80
80 +
420
40
90sec
1-20sec
20
10
10 +$
150sec
L8Osec
(ii)22Oghas6.02x1.023particles;5pgwitln^u"ffx5x].0-6-I.37x].016
particles.
- ' # x 1 , . 3 7 x 1 0 1 6: 1 . 0 5 x 1 0 1 4 B q
A c t i " u tA
t y- ) . N: ' # x N
t
242
(7) Usethe equationN = Noe-rt
si.ncema"ssis proporti.onal to N we canwrite m:
rftse-At
The bothmasses are tlte same after the ttme t so that:
m:ffte€-trrt-Tftge-lzt
Hence 160e-Art - B00e-1zt
160
800
-:
et(^r^z)
;h! - t(Ar- Ar):r(#-'#)
-1.61 : (-8.67 x 10-3)t
'J.86
t =
seconds
(ii)
The ma-ssx - 160 ,(#)186 - 3zttg
1,409has6.02xL023atoms
32Vghas1.376x1,017
atoms
A = L N = X x 1 . 37 6 x 1 0 7 7- 1 . . 1 . 9
x 1,0rsBq
(8) 174 daysrepresents
3 half livesfor process(1) and two half livesfor process(2).
H e n c ei n a l l t h e s a m p l ew o u l dh a v eg o n et h r o u g h5 h a l fl i v e s .
lf initialactivityis 1400per minute,then after5 half lives,the activitywould be
A - 1400(+)"= 44 countsper mi"nute.
\2/
(iii)
T h i sp a r ti sa b i t m o r ec o m p l i c a t eadn dr e q u i r etsh a ty o uf i n dt h e E F F E C T hI V
a lEf
lifefromthe bothprocesses.
For process(1')87days gives 1,.sTrwhi.tef or proces.s
B,BTdays - LTt
HenceinBT days there are z.s EFFECTIVEhaIflives.
1 ef f ectivehatf ti"fe i.sequaltu Y - 34.Bdays
z.J
usins
A - o,(;)" ; 12= 44(;)",^(#)-= n("I) ; n: rB7
Hence1..87half liveswere neededto reachthe desiredactivity.
T h et i m e f o r t h i s i s e q u a lt o 1 , . 9 7
x 3 4 . g= 6 5 d a y s
243
z
( 9 ) ( i ) 2 3 8 g o f U r a n i u mc o n t a i n6 . 0 2x 1 0 2 3p a r t i c l e s
49mg wi"Ucontain
6.02 x L023
x 48 x 10-3 : 1.2! x 1-\2oparticles
Zn
(ii)
206 gof leadcontain6.02x 1023particles
6.02 x !023
32mgwtttcontatnTx32x10_3:9.35x1019parti'cles
(iii)
T o t a li n i t i a ln u m b e ro f U r a n i u mp a r t i c l eas t f o r m a t i o n=
1 , . 2 Lx L 0 2 0+ 9 . 3 5 x 1 0 1 e- 2 . 1 5 x 1 , \ z o p a r t i " c l e s .
(iv)
S0
de:creasedfrom 2 . 1 5x 1 0 2 0t u 1 . 2 1 x L o 2 0
T h en u m b e ro f ur r a n i u mp arrtric lC:es
i v ) S la sf o l l ;
H e n c ef i n dt h e nr u m b e ro f Ih1aar fllive:
r1'. ITn
N_ro[z
)'
t
1,.2
2L1 x 1 0 20 '
^ ^r L \ '
;l
2 . I 5 x 1 0 2 0(,r2)
L
l''L . I2L. x 1 0 2 0
l=: = n I n 7
hn t lr,
\ z . L Sx 1 0 2 )0I
n - 0 . 8 |3
,y,
s = 0 . 8 3 x4 ,. 5Xx 1 O e = 3 . 7x31 0 / o tars
T o t a ln u m b . , o f V . . rS
Initial amount tn samPrc: #
(1 0 )
Final amount in SamPIe:
#
trtn 3.4 L.I6 /L\'
:
u stng N - tr
;
1gr+ to- \Z)
[Z,l
,n
"'"
t'lrr=-nm1
L 0 0x L . L 6
/.
;n-s.I
Total numberof years - 5.1x 5570: 29365yr
244
BLANKPAGE
THE END
245