Physics 169
Kitt Peak National Observatory
Tuesday, March 29, 16
Luis anchordoqui
1
Question
Steady electric current can give steady magnetic field
Because of symmetry between electricity and magnetism ☛ we can ask:
Steady magnetic field can give steady electric current
OR!Changing magnetic field can give steady electric current
S ECTI O N 31.1 • Faraday’s Law of Induction
969
When a magnet is moved toward a loop of wire ☛ sensitive ammeter deflects
indicating that current is induced in the loop
I
N
Ammeter
S
When magnet is held stationary ☛ there is no induced current in the loop
(a)
even when the magnet is inside the loop
N
Ammeter
S
When magnet is moved away from loop ☛ ammeter deflects in opposite direction
(b)
indicating that induced current is in opposite direction
I
N
Ammeter
Tuesday, March 29, 16
S
(c)
Active Figure 31.1 (a) When a magnet is moved toward a loop of wire connected to a
2
Answer
Steady electric current can give steady magnetic field
Because of symmetry between electricity and magnetism ☛ we can ask:
Steady magnetic field can give steady electric current
✘
OR!Changing magnetic field can give steady electric current
S ECTI O N 31.1 • Faraday’s Law of Induction
969
✔
When a magnet is moved toward a loop of wire ☛ sensitive ammeter deflects
indicating that current is induced in the loop
I
N
Ammeter
S
When magnet is held stationary ☛ there is no induced current in the loop
(a)
even when the magnet is inside the loop
N
Ammeter
S
When magnet is moved away from loop ☛ ammeter deflects in opposite direction
(b)
indicating that induced current is in opposite direction
I
N
Ammeter
Tuesday, March 29, 16
S
(c)
Active Figure 31.1 (a) When a magnet is moved toward a loop of wire connected to a
3
Figure 30.20 The magnetic flux
surface,
shownthat
in Figure
30.20. If the magnetic field at this element is
netic flux through the element is B ! dshaped
A, where
d A is as
a vector
is perpendicular
through an area element dA is
flux
the element
B ! dA, where dA is a vector that is perpend
heBsurface
and
has#,awhere
magnitude
to the area
d A.through
Therefore,
the totalismagnetic
! dA
$ BMagnetic
dA cos
dA
is a equalmagnetic
9.1
Flux
to the surface and has a magnitude equal to the area dA. Therefore, the total ma
the surface
is
"vector
perpendicular
to the
B through
ˆ
surface.
flux "B through the surface is
① Magnetic flux through surface S
!
=
Definition of magnetic flux
"m
B $
B · dA
!
B!dA
S
=
Z
(30.18)
"B $
B!d A
~ · dA
~
B
Consider the special case of a plane
mof area A in a uniform field B that makes an
Consider
thethis
special
e # with d A. The magnetic
plane
case iscase of a plane of area A in a uniform field B that ma
S in
Unit flux
of through
: the
Weber
(Wb)
m angle
# with dA. The magnetic flux through the plane in this case is
"
$
BA
cos
#
(30.19)
B
1Wb
= 1Tm2
Unit of m ☛
"B $ BA cos #
e magnetic field is parallel to the plane, as in Figure 2
30.21a, then # $ 90° and the
If the
magnetic fieldtoisthe
parallel
to as
theinplane,
through the plane is zero. If the field
is perpendicular
plane,
Figureas in Figure 30.21a, then # $ 90° a
fluxplane
through
plane
is zero.value).
If the field is perpendicular to the plane, as in
$ 0 and the flux through the
is BAthe
(the
maximum
1b, then
② #Graphical
2
# $ 0 and
flux(Wb);
through
the$plane is BA (the maximum value).
which then
is defined
as athe
weber
1 Wb
The unit of magnetic flux is T ! m ,30.21b,
2
2
The unit of magnetic flux is T ! m , which is defined as a weber (Wb); 1
m.
(2) Graphical:
m2.
☛ number 1ofT ! magnetic
field lines passing through surface
Weber (Wb)
1 Wb = 1 Tm
m
Flux through plane is zero when magnetic field is parallel
to plane surface
dA
m
= Number of magnetic field lines passing through surface
dA
dA
Faraday’s
law of induction:
B
dA
B
(a)
B
Induced emf(b)|E| = N
d
m
dt
B
(a)
(b)
ve Figure
Magnetic
flux link
through a plane lying in a magnetic field.
At 30.21
the Active
Figures
he
through theFlux
planethrough
is you
zero whenplane
the
magnetic
field
is parallel
to the
plane
isFigure
maximum
when
magnetic
islying
perpendicular
to plane
atflux
http://www.pse6.com,
Active
30.21
Magnetic
flux
throughfield
a plane
in a magnetic field.
ce.
(b)rotate
The flux
the plane is a maximum
when
the magnetic
field
can
thethrough
plane and
(a) The flux
through
the plane
is is
zero when the magnetic field is parallel to the plane
endicular
to
the
plane.
Tuesday,
March
29,
16
4
change the value of the field to
surface. (b) The flux through the plane is a maximum when the magnetic field is
where
N = Number of coils in the circuit.
Ammeter
9.2 Faraday’s Law
Switch
–
+
Primary
coil
Iron
Secondary
coil
At the Active
at http://www.ps
can open and clo
Active Figure 31.2 Faraday’s experiment. When the switch in the primary circuit is
When
switch
in
primary
circuit
is
closed
Faraday’s
experiment
and observe the
closed, the ammeter in the secondary circuit deflects momentarily. The emf induced in
ammeter.
the secondary circuit is caused by the changing
magnetic
through the secondary
ammeter
in field
secondary
circuitcoil.
deflects momentarily
Battery
emf induced in secondary circuit
is caused by changing magnetic field through secondary coil
Faraday’s law of induction
Induced emf ☛
|E| = N
d
m
dt
number of coils in circuit
Tuesday, March 29, 16
5
8.2. LENZ’ LAW
99
!
B = Constant
A = Constant
E =0
B = Constant
 = Constant
dA/dt ⇤= 0
|E| > 0
B̂ = Constant
dB/dt ⇤= 0
A = Constant
|E| > 0
B = Constant
A = Constant
dÂ/dt ⇤= 0
|E| > 0
Note : The induced emf drives a current throughout the circuit, similar to the
Note
function
of a battery.
However,
the di⇥erence
heresimilar
is that the induced emf
Induced
emf drives
a current
throughout
circuit
is distributed throughout the circuit. The consequence
is thatofwea cannot
to function
battery
define a potential difference between any two points in the circuit.
Difference here is that induced emf is distributed throughout circuit
Suppose there is an induced current in the loop, can we
define consequence
VAB ?
we cannot define a potential
Recall: difference between any two points in circuit
Tuesday, March 29, 16
V
=V
V = iR > 0
6
Suppose there is an induced current in the loop, can we
VAB ?
define a potential difference between any two points in the circuit.
Suppose define
there is an
induced
current in loop ☛ can we define
VAB
?
Suppose there is an induced current in the loop, can we
define VAB ?
Recall:Recall
Recall:
VAB = VA
VB = iR > 0
VAB
VAB =
= VV
=BiR=
> 0iR > 0
A A VB V
)⇥VAVA >
> VBVB
⇥ VA > VB
Going
anti-clockwise
as i )
Going
anti-clockwise
(same as(same
i),
ockwise
as A
i),
> VB
If we (same
start
to to
wegetget
B!B, then
If wefrom
start
fromgoing
A, going
then we
VA > VV
B .A
If we start from B, going to A, then we get VB > VA .
> .VA
If we start from B going to A! then we get
B V
f we
start from A, going to B, then we get V V>
A
B
We cannot
definethen
VABwe
!! get V > V .
f we start from B, going
to
A,
A
) We cannot define VAB !!B
This situation is like when we study the interior of a battery.
This situation is like when we study interior of a battery
o We cannot define
n
⇥
AB⌥
A battery ⌥
⌃ provides the energy needed to drive the
⇧ chemical reactions.
A battery
chemical
provides
energy
needed
to
drive
⌥
⌅ charge carriers around the circuit by ⌥
⇤
V
!!
reactions
changing magnetic flux.
n isThe
likeloopwhen we
study
the
interior
of
a
charge carriers around circuitbattery.
by
A loop
sources of emf
changing magnetic flux
non-electric means
⌥
chemical
reactions.
⇧
non-electric
means
needed to drive the
8.2 Lenz’ Law
charge
carriers around the circuit by ⌥
⇤
Tuesday, March 29, 16
sourcesthe
of emf
provides
energy
changing magnetic flux.7
9.3 Lenz’s Law
① Flux of magnetic field due to induced current
opposes change in flux that causes induced current
② Induced current is in such a direction
as to oppose changes that produces it
③ Incorporating Lenz’s law into Faraday’s Law ☛
E =
If
d
m
dt
> 0,
m
" )
N
E
d
m
dt
appears
) Induced current
appears
so that
)
~ -field due to
B
induced current
Tuesday, March 29, 16
) change in
m
)
m
#
8
④ Lenz’s Law is consequence from principle of conservation of energy
Suppose bar is given slight push to right
This motion sets up a counterclockwise current in the loop
R
BUT
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
I
What happens if we assume that current is clockwise
Bin
×
FB
v
×
such that direction of magnetic force exerted on bar is to the right?(a)
This force would accelerate the rod and increase its velocity
×
×
×
This (in turn) would cause area enclosed by loop to increase more rapidly
this would result in increase in induced current
×
×
×
which would cause increase in force
which would produce increase in current ... and so on... R ×
×
×
System would acquire energy with no input of energy I
×
×
×
×
×
×
v
F
×
×
× B
× conservation
This is clearly inconsistent with all experience and violates law× of energy
We are forced to conclude that current must be counterclockwise
×
Tuesday, March 29, 16
×
×
×
×
9
×
×
×
×
×
(a)
Likewise ☛ if bar is push to the left
R
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
I
v
FB
clockwise whe
direction.) If
netic flux thr
field is direc
clockwise if it
the induced c
by the current
Let us exa
given a slight
sets up a coun
(b)
Tuesday, March 29, 16
10
Example
flux due to an external field directed into the page is increas×
× this change, must produce a field
rent,× if it× is ×to oppose
age. Hence, the induced current must be directed counterWhen
magnet
is moved
toward stationary conducting loop
×
×to the
×
×
×
ar moves
right.
(Use
the right-hand rule to verify this
current
is induced
in the direction
shown
is moving to(b)the left, as in Figure
31.13b,
the external
magarea
enclosed
by the loop decreases with time. Because the
ure
31.13
(a) As the
ducting
bar slides
the two of the induced current must
the page,
the ondirection
be
v
dduce
conducting
a fieldrails,
thatthe
also is directed into the page. In either case,
netic flux due to the external
ends
to maintain the original flux through the area
S enclosed
N
I
N
Example
netic field into the page
ugh the area enclosed by the
s increases
situationinusing
considerations. Suppose that the bar is
time. energy
By Lenz’s
the
right. Incurrent
the preceding
the induced
must be analysis, we found that this motion
nterclockwise
soinasfield
to produce
wise current
the
loop.
if we
that (a)
the
Magnetic
lines What
shownhappens
are those
dueassume
to bar magnet
(b)
unteracting magnetic field
ctedThis
out of
the page.
(b) When
induced
current
produces its own magnetic field directed to the left
bar moves to the left, the
that counteracts the increasing external flux
uced current must be clockwise.
?
S
N
N
I
v
I
S
S
N
I
Magnetic field lines shown are those due to induced current in ring
(a)
Tuesday,
March 29, 16
(b)
11
Example
r moves to the left, the
ed current must be clockwise.
When magnet is moved away from stationary conducting loop
current is induced in direction shown
I
S
N
v
I
S
S
N
I
(b)
Magnetic field lines shown are those due to(c)bar magnet
This induced current produces magnetic
field directed
themagnet
right is moved toward the stat
Figure 31.14
(a) Whentothe
loop, a currentand
is induced
in the direction
shown.
The magne
so counteracts
decreasing
external
flux
are those due to the bar magnet. (b) This induced current pr
magnetic field directed to the left that counteracts the increas
magnetic field lines shown are those due to the induced curre
I (c) When the magnet is moved away from the stationary cond
is induced in the direction shown. The magnetic field lines sh
N
S
the bar magnet. (d) This induced current produces a magnet
I
right and so counteracts the decreasing external flux. The ma
shown are those due to the induced current in the ring.
Field lines shown are those due to induced current in ring
Tuesday, March 29, 16
12
⌅ B-field due to
induced current
⌅
change in
so that
m
Question
=⌅
m
⇤
4) Lenz’ Law is a consequence from the principle of conservation of energy.
3
Motional EMF
’s try to look at a special case when the changing magnetic flux is carried by
tion in the circuit wires.
Tuesday, March 29, 16
13
⌅ B-field due to
induced current
⌅
change in
Answer
so that
m
=⌅
m
⇤
4) Lenz’ Law is a consequence from the principle of conservation of energy.
3
Motional EMF
’s try to look at a special case when the changing magnetic flux is carried by
tion in the circuit wires.
Tuesday, March 29, 16
14
9.4 Motional EMF
Straight conductor of length L
is moving
uniform
-field
nductor
ofthrough
length
L B~moving
directed into the page
v in a magnetic field B.
Assume conductor is moving with constant
under influence of some external agent
~
~v ? B
~
F~B = q~v ⇥ B
v and B~
directed along the length L perpendicular to both ~
Electrons in conductor experience force
Under influence of this force
electrons move to lower end of conductor and accumulate there
leaving net positive charge at upper end
~ is produced inside conductor
Because of this charge separation electric field E
Charges accumulate at both ends
until downward magnetic force
Tuesday, March 29, 16
qvB on
charges remaining in conductor
is balanced by the upward electric force
qE
15
At this point ☛ electrons move only with random thermal motion
F~E + F~B = 0
Equilibrium requires that
~ + q~v ⇥ B
~ = 0
) qE
~ =
) E
~
~v ⇥ B
Voltage across ends of conductor ☛
V =
) Voltage ☛
V =
Z
L
0
~ · d~s
E
EL
V = vBL
Potential difference is maintained between ends of conductor
as long as the conductor continues to move
through the uniform magnetic field
Tuesday, March 29, 16
16
0
E · ds
Suppose moving wire slides without friction
EL
age :
on stationary U -shape conductor
V = vBL
Motional emf can drive electric current i in U -shape conductor
without
uctor.
c cur-
) Power is dissipated in circuit
.
Pout = V i ☛ Joule’s heating
)
What is source of this power?
Look
ucting
rod:at the forces acting on conducting rod:
• Magnetic force
L⇥B
LB (pointing left)
~ ⇥B
~
F~m = iL
Fm = iLB
☛
(pointing left)
• For wire to continue to move at constant velocity
we need to apply an external force
e at constant velocity v, we need to apply
= iLB
(pointing right)
Tuesday, March 29, 16
oving:
~ext =
F
~m = iLB
F
☛
(pointing right)
17
) Power required to keep rod moving
Pin = F~ext · ~v
= iBLv
dx
= iBL
dt
d(xL)
= iB
dt
d(BA)
= i
dt
( xL = A area enclosed by circuit)
(BA
=
m magnetic flux)
Since energy is not being stored in system
)
Pin + Pout = 0
iV + i
We recover Faraday’s Law
Tuesday, March 29, 16
d
m
dt
)
= 0
V =
d
m
dt
18
(2) Generators and Motors:
and
Motors
Assume that the circuit loop isGenerators
rotating at a constant
angular
velocity
, (Sourcecircuit
of rotation,
e.g. is
steam
produced by
water falling
Assume
loop
rotating
atburner,
constant
angular
from a dam)
velocity !
(Source of rotation ☛ steam produced by burner or water falling from dam)
changes with time!
✓ = !t
number of coils
Magnetic flux through loop ☛
B
= N
Z
loop
Induced emf ☛ E =
d
B
dt
=
~ · dA
~ = N BA cos ✓
B
d
N BA (cos !t) = N B A ! sin !t
dt
E
N BA !
Induced current ☛ i =
=
sin !t
R
R
Tuesday, March 29, 16
19
Alternating current (AC) voltage generator
ε
εmax
t
External
circuit
Power has to be provided by source of rotation
At the
to overcome torque acting on a current loop in a magnetic
field
at http://w
µ
~
(b)
z }| {
am of an AC generator. An emf is induced
in
~
~
~⌧ =inNthe
iAloop
⇥B
The alternating emf induced
) ⌧ = N iAB sin ✓
can adjust
rotation an
field to see
emf gener
Net effect of torque is to oppose rotation of coil
Tuesday, March 29, 16
20
The net effect of the torque is to oppose the rotation of the coil.
with
Electric motor is a generator operating in reverse
An electric motor is simply a generator
)
Replace loadoperating
resistance
with a battery of emf E
inR
reverse.
⌅ Replace the load resistance R with
t inWith
the battery
coil, and
itthere
experiences
a☛
battery
ofisemf
E.a
a current
in coil
and it experiences torque in B-field
to an induced emf, Eind , in
) Rotation of coil leads to an induced emf ☛ Eind
of the battery.
Law)there is a current in the coil, and it experienc
With (Lenz’
the battery,
in direction opposite that of battery
torque in the B-field.
law
⌅
Rotation of the coil leads to anLenz’s
induced
emf, Eind
the direction opposite of that of the battery.
(Lenz’ L
)
i=
i ⇤
Pelectric =
torque delivered = N iAB sin
Electric power input
Tuesday, March 29, 16
E
E)
ind
R
i2 R
⇤
i =
E
Eind
R
As motor speeds up Eind ",
+
) i #
Pmechanical
Mechanical power delivered
21
8.4
Induced Electric Field
9.5 Induced Electric Field
We have seen that a changing magnetic flux
induces
an emf
and a that
current
in a conducting
o far
we have
discussed
a change
in mag- loop
etic flux will lead in an induced emf distributed
n the loop, resulting from an induced E-field.
In the same way
we can relate induced current in conducting loop to an electric field
4.owever,
INDUCED
even inELECTRIC
the absence FIELD
of the loop (so that there is no induced current)
by claiming that electric field is created in conductor
he induced E-field will still accompany
change
as a aresult
of in
themagnetic
changingflux.
magnetic flux
Consider a circular path in a region
th changing magnetic field.
Tuesday, March 29, 16
22
me. Hence, the net emf induced in
mf
magnetic flux through the loop changes in time.
Induced electric field is nonconservative
unlike electrostatic field produced by stationary charges
We can illustrate this point by considering conducting loop of radius r
and
Electric
Fields
situated
in uniform
magnetic field that is perpendicular to plane of loop
nging magnetic flux induces an emf and a current in a
udy If
of electricity,
related
a current
to an
electric
magneticwefield
changes
with
time
☛ field
on charged particles. In the same way, we can relate an
according to Faraday’s law
ucting loop to an electric field by claiming that an electric
induced
in loop
ductor as aemf
resultEof =
the changing
magnetic
flux.
d B /dt
tudy of electricity that the existence of an electric field is
ce ofInduction
any test charges.
This suggests
that even in the absence
of current
in loop
nging magnetic field would still generate an electric field~in
implies presence of induced electric field E
which
must beunlike
tangent
to the loop
eld is
nonconservative,
the electrostatic
field produced
×
×
×
E
E
×
×
×
×
×
×
×
×
×
×
r
×
×
×
×
×
×
×
×
E
E
B
×
×
×
in
an illustrate
point
considering
a conducting
loop in
of the wire move
becausethis
this
is by
direction
in which
charges
Figure 31.19 A conducting loop
rm magnetic field that is perpendicular to the plane of the in response
electric
force
of radius r in to
a uniform
magnetic
the magnetic field changes with time, then, according to
field perpendicular to the plane of
~
q
Work
done
by
-field
in
moving
test
charge
once
around loop = qE
E
n emf ! " # d $B/dt is induced in the loop. The induction
the loop. If B changes in time, an
pliesBecause
the presence
of an induced
electric field
E, which must ~ electric field is induced in a
electric
force acting
on charge
= q E direction tangent to the
ause this is the direction in which the charges in the wire
work done by electric field in moving charge once
around loop
qE2⇡r
circumference
of the=
loop.
ctric force. The work done by the electric field in moving a
he loop
is equal
q !. Because the
These
two to
expressions
for electric
work force
doneacting
mustonbe equal
done by the electric field in moving the charge once around
Tuesday, March 29, 16
23
2% r is the circumference of the loop. These two expressions
) we see that ☛ qE = qE2⇡r
E
E=
E = E2⇡r
2⇡r
1 d B
=
2⇡r dt
r dB
=
2 dt
In general ☛ emf for any closed path can be expressed as line integral
E=
I
I
~ · d~s
E
~ · d~s =
E
d
dt
Induced electric field is a nonconservative field
that is generated by a changing magnetic field
Tuesday, March 29, 16
24
L.H.S. = Integral
R.H.S. = Integral
I
Z
d
~ · dA
~
B
⇥Cdetermined by dt
S
Direction of dA
direction
of line
) Faraday’s Law becomes ☛
Direction of
~ ind · d~s =
E
L.H.S
= Integral around a closed loop C
R.H.S
= Integral over a surface bounded by C
~ determined by direction of line integration C
dA
(Right-Hand Rule)
Tuesday, March 29, 16
25
created by charges
SUMMARY
~ -field
atedRegular
by charges
E
created by chan
~ -field B-field
created
by changing
Induced
E
by charges
eldcreated
lines start
from +ve and end created by changing B-field
E-field
lines
form
clo
start
from
+ve
and
end
~
~
-field lines start from
form
closedloops
loops
E -field
veE charge
E-field
lineslines
form
closed
rge
+q and end on
q charge
can define electric potential
Electric potential cannot be defined
define
electric
potential
so
that
electric
so that
so thatpotential
we can discuss
potential
Electric
potential
ca
(or, potential
has no
meaning)
Electric
potential
cannot
be
defi
candifference
discuss
potential
difference
between
two points
cuss
potential
difference
(or,has
potential
has no
(or,
potential
no
meaning)
ween
two points
⇓
⇓
o points
Conservative force field
Non-conservative force field
⇥
⇥
⇥
Classification
of⇥electric and magnetic effects
of reference
of observer
!!!
ervative
force fieldforce depend
Non-conservative
force field
Conservative
field on frame
Non-conservativ
e.g. For motional emf ☛ observer in reference frame of moving loop
~ -field but just a regular E
~ -field
will NOT see an induced E
ation
ofcontinued
electricofand
magnetic
depend
on the
frameon
of the
refere
classification
electric
and effects
magnetic
effects
depend
fra
To be
next
semester
in Special
Relativity
same
same
ver.
e.g. Fore.g.
motional
emf, observer
inobserver
the reference
of
e observer.
For motional
emf,bat-time,
inbat-channel
theframe
referen
Tuesday, March 29, 16
26