Physics 169
Kitt Peak National Observatory
Luis anchordoqui
9.19.1 Inductance
Inductance
An inductor stores energy in magnetic field
just
a capacitor
in electric
An inductor stores energy in
theasmagnetic
fieldstores
just asenergy
a capacitor
stores field
energ
n the electric field.
AWe
changing
B-field
willthat
leada to
an induced
a circuit
have shown
earlier
changing
B-fieldemf
willinlead
to an induced emf in
circuit.
Question
Question
: If generates
a circuit generates
a changing
field, does it lead to an
If a circuit
a changing
magneticmagnetic
field
it lead to
an induced
emf in same circuit?
induced emf in the samedoes
circuit?
YES!
Self-Inductance
YES! Self-Inductance
The inductance L of any current element is
Inductance L of any current element is
di
The negative sign
di
EL = VL = L
Negative sign
from from
LenzLenz
LawLaw.
E L = VL = L
dt comes comes
dt
VS Vs Vs
Unit
of
L:
Henry(H)
1H=1·
Unit of L : Henry (H)
1H = 1 1· H = A1
A
A
• •All
(including
have
some
inductance.
Allcircuit
circuitelements
elements
(includingresistors)
resistors)
have
some
inductance
Commonlyused
usedinductors:
inductors:solenoids,
solenoidstoroids
and toroids
• •Commonly
circuitsymbol:
symbol
• •circuit
Vs
1H=1
A
Example
: Solenoid
Example
Solenoid
EL = V B
VA =
L
VB < VA
di
<0
dt
EL = V B
VA =
VB > VA
L
di
>0
dt
Recall Faraday’s Law
EL =
where
N
d
B
dt
B is magnetic flux ☛
=
N
d
(N
dt
B)
B is flux linkage
) Alternative definition of Inductance
d
di
N
(N B ) = L
) L =
dt
dt
i
) Inductance is also flux linkage per unit current
B
(1) Solenoid:
To first order approximation,
Calculating Inductance:
① Solenoid
B = µ0 ni
where n = N/L = Number of
coils per unit length.
To first order approximation
B = µ0 ni
Consider a subsection of length l of the solenoid:
n = N/` ☛ number of coils per unit length
Flux linkage = N
B
= nl · BA
where A is
cross-sectional area
Consider a subsection of length l of solenoid
Flux linkage = N
L=
B
N
B
= µ0 n2 lA
i
L
= µ0 n2 A where
= Inductance per is
unitcross-sectional
length
l
= nl · B A
Notice :
)
Note
❑L
/ n2
N
A
area
2
(i) L ⇤ n2 B
0
(ii) The inductance, like the capacitance, depends only on geometric
factors, not on i.
L =
i
= µ n lA
L
= µ0 n2 A = Inductance per unit length
l
❑Inductance (like capacitance) depends only on geometric factors (not on) i
I
We see that L depends only on the geometrical factors ( n , R and l ) and is independent
of the current I .
② Toroid
Inside toroid
µ0 iN
B =
2⇡r
Example
11.3 Self-Inductance
a Toroidare
Recall
☛ B-fieldoflines
concentric circles
Outside toroid
B = 0
9.1. INDUCTANCE
Flux linkage through Ztoroid
(2) Toroid:
N
B
= N
~ · d~a
B
2
n~
(a)
B k d~a
109
+
Figure 11.2.3 A toroid with N turns
da = h dr
Recall: B-field lines are concentric circles.
Insidebthe toroid:
Z
(b)
µ0 iN
h dr
µ iN
=
B=
2⇥r
2⇡
r
a
(NOT a constant)
⌘
µ0 iN 2 hwhere r ⇣
b
is the distance from center.
=
ln
2⇡ Outside theatoroid:
B =20
⇣b⌘
N B
µ0 N h
=
ln
) Inductance! L =
i
2⇡
a
2
Flux linkage through the toroid
Again L / N
KEY
11-7
0
ˆ
⇤
⇤
⌅
4. In the RL circuit show in Figure 11.14.1, can the self-induced
than the emf supplied by the battery?
9. 2 LR Circuits
(A) Charging an inductor
When switch is adjusted to position a
By loop rule (clockwise)
E0
VR +
VL = 0
a
Figure 11.14.1
E0
)
iR
di
L
= 0
dt
di
R
E0
+
i =
dt
L
L
First Order Differential
Equation
Similar to equation for charging a capacitor!
changing variables
x = (E0 /R)
dx =
i
L dx
x+
=0
R dt
Z
x
xo
0
dx
=
0
x
ln(x/x0 ) =
R
L
Z
t
dt
0
Rt/L
x = x0 e Rt/L
i = 0 @ t = 0 ) x0 = E0 /R
E0
R
E0
i= e
R
Rt/L
di
di R
E0
First Order Differ+ i=
ential Equation
dt
L
E0 L
t/⌧L
Solution
☛
i(t)
=
(1
e
)
Similar to the equation for charging a capacitor!
(Chap5)
R
⇥
E0
Solution: i(t) =
1 e t/ L
⌧L = L/R ☛ R
Inductive time constant
where ⇥L = Inductive time constant = L/R
| VR |
| VL |
=
iR
| VR | =
=
E0 (1
e
t/⌧L
t/
)
iR = E0 (1 e
)
di
di
E 0 E0 1 1
t/ L
L =
=| VLL| = L= L= · L · · · · ·ee t/⌧
=E0 eE0t/e L t/⌧L )
dt
dt
R R⌧L⇥L
L
(B) Discharging
inductor
”Discharging”
an an
inductor
When
switch
is adjusted
position
b afterinductor
the inductor
has charged
been
Whenthe
switch
is adjusted
at at
position
has been
b after
”charged” (i.e. current i = E0 /R is flowing in the circuit.).
i.e. current i = E0 /R is flowing in circuit
Byloop
looprule:
rule
By
VLVL
⇤
L
di
di L dt
VRVR= 0=
⇤
iR
0
= 0
iR
=
dt inductor as source of emf)
(Treat
Treat inductor as source of
di
didt +
)
+
dt
0
emf
R
Discharging a capacitor
Ri = 0
(Chap5)
L i = 0 Discharging
an inductor
L
i(t) t/⌧
= i0 e
i(t) = i0 e
t/
L
L
where i0 = i(t = 0) = Current when the circuit just switch to position b.
where i0
= i(t = 0) = Current when circuit just switch to position b
+ i=0
(Chap5)
where i0 = i(t = 0)
the circuit just switch to position b.
dt = LCurrent when
i(t) = i0 e
t/
L
where i0 = i(t = 0) = Current when the circuit just switch to position b.
Summary
Summary : During charging of inductor,
During charging of inductor
1. At t = 0, inductor acts like open circuit when current flowing is zero.
1. At t = 0 inductor acts like open circuit when current flowing is zero
2. At charging
t ⇥ ⌅, ofinductor
Summary : During
inductor,acts like short circuit when current flowing is
stablized at maximum.
2. At t !1. 1
inductor
actslikelike
circuit
flowing
At t =
0, inductor acts
open short
circuit when
current when
flowing iscurrent
zero.
2. At t ⇥ ⌅, inductor acts like short circuit when currentisflowing
is
stabilized
at maximum
stablized at maximum.
3. Inductors are used everyday in switches for safety concerns.
3. Inductors are used everyday in switches for safety concerns.
3. Inductors are used everyday in switches for safety concerns
Summary
11.6.1 Rising Current
(a)
I ! = I 2 R + LI
dI
.
dt
(b)
(11.6.6)
The11.6.1
left-hand
side represents
therule
ratefor
at inductors
which the(a)
battery
delivers energy
to the
Figure
Modified
Kirchhoff’s
with increasing
current,
and circuit.
(b)
On
the
other
hand,
the
first
term
on
the
right-hand
side
is
the
power
dissipated
in the
with decreasing current. See Section 11.4.2 for cautions about
use of this modified
rule.resistor in the form of heat, and the second term is the rate at which energy is stored in
the inductor. While the energy dissipated through the resistor is irrecoverable, the
stored
in the inductor
can be released
later. The polarity of the selfThemagnetic
modifiedenergy
rule for
inductors
may be obtained
as follows:
induced emf is such as to oppose the change in current, in accord with Lenz’s law. If the
rate of change of current is positive, as shown in Figure 11.6.1(a), the self-induced emf
11.6.2
Decaying
up an
inducedCurrent
current I ind moving in the opposite direction of the current I to
! L sets
oppose such an increase. The inductor could be replaced by an emf
Next we consider the RL circuit shown in Figure 11.6.5. Suppose the switch S has been
| ! L | = L | dI / dt | = + L(dI / dt) with the polarity shown in Figure 11.6.1(a). On the1 other
closed
for circuit
a long timewith
so thatrising
the current
is at its equilibrium
value ! / R . circuit
What happens
RL
current
and equivalent
I
hand,
if
,
as
shown
in
Figure
11.6.1(b),
the
induced
current
set
by modified
the
dI
/
dt
<
0
Figureto11.6.2
(a) RL
Circuit
rising Scurrent.
(b) and
Equivalent
circuitindusingupthe
0 switches
the current
when
at t = with
is opened
S2 closed?
1
Kirchhoff’s
loop
self-induced
emfrule.
! L flows in the same direction as I to oppose such a decrease.
RL circuit
Consider
the whether
shown
in Figure
11.6.2.
At t = 0 the
We find
We see that
the rate
of change
of current
in increasing
( dI /switch
decreasing
dt > 0 )isorclosed.
a .toThis
b along
bothnot
cases,
change in potential
when moving
from
theto the
< 0 ), indoes
that( dIthe/ dtcurrent
risethe
immediately
to its maximum
value
is due
!/R
presence
ofofthe
inductor.
the have
modified Kirchhoff’s rule for
direction
theself-induced
current I is Vemf
Thus, we
! Vina =the
! L(d
I / d t) .Using
b
increasing current, dI / dt > 0 , the RL circuit is described by the following differential
equation:
Kirchhoff's Loop Rule Modified for Inductors (Misleading, see Section 11.4.2):
dI
! "the
IR"direction
| ! L | = ! of
" IR
= 0 the
. “potential change” (11.6.1)
If an inductor is traversed in
the" L
current,
is
dt
! L(dI / dt) . On the other hand, if the inductor is traversed in the direction opposite of the
RLtheFigure
circuit
with
decaying
andand
equivalent
circuit
current,
“potential
change”
iscircuit
.
+ L(dIwith
/ dt)current
(a) RL
decaying
current,
(b) equivalent
Note that there
is11.6.5
an important
distinction
between
an inductor
and a circuit.
resistor. The
9.3 Energy Stored in Inductors
Inductors stored magnetic energy through magnetic field stored in circuit
Recall equation for charging inductors
E0
Multiply both sides by i
E0 i
|{z}
Power input by emf
(Energy supplied
one charge = qE0 )
iR
=
2
i R
|{z}
di
L
= 0
dt
+
Joule’s heating
(Power dissipated
by resistor)
di
Li
| {zdt}
Power stored in inductor
) Power stored in inductor
Integrating both sides and use initial condition
At
t = 0,
i(t = 0) = UB (t = 0) = 0
) Energy stored in inductor ☛ UB = 1 Li2
2
Energy Density Stored in Inductors
Consider an infinitely long solenoid of cross-sectional area
A
For a portion l of solenoid
L = µ0 n2 lA
) Energy stored in inductor:
1 2
1
UB = Li = µ0 n2 i2 |{z}
lA
2
2
Volume of solenoid
) Energy density (= Energy stored per unit volume) inside inductor
UB
1
uB =
= µ 0 n2 i 2
lA
2
Recall magnetic field inside solenoid
B = µ0 ni
)
uB
B2
=
2µ0
This is a general result of energy stored in a magnetic field
Inductance and Magnetic En
9.4 Mutual Inductance
Very often the magnetic flux through the area enclosed by a circuit varies with time
because of time-varying currents in nearby circuits
11.1 Mutual Inductance
Suppose two coils are placed near each other, as shown in Figur
mutual inductance depends on interaction of two circuits
Consider two closely wound coils of wire shown in cross-sectional view
Current I1in coil 1 which has
N1 turns
creates a magnetic field
Some magnetic field lines pass through coil 2 which has N2 turns
Figure
11.1.11 Changing
current
in coil 1coil
produces
The magnetic flux caused by the current
in coil
and passing
through
2 is changing
12
We define the mutual inductance ofThe
coilfirst
2 with
respect
toand
coilcarries
1
coil has
a current I1 which give
N1 turns
!
B1 . The
Nsecond coil has N 2 turns. Because the two coils are cl
2
12
M12the⌘magnetic field lines through coil 1 will also pass through
I1 through one turn of coil 2 due to I . Now, by v
magnetic flux
1
We shall see that the mutual inductance M12 depends only on th
of the two coils such as the number of turns and the radii of the t
If current I1 varies with time
In a ssimilar
manner,
there
a current
we see from Faraday’
law that
emfsuppose
inducedinstead
by coil
1 iniscoil
2 is I 2 in
varying with time (Figure 11.1.2). Then the induced emf in coil 1
E2 =
N2
d
12
dt
=
d
N2
dt
✓
◆! = " N d# = " d B!
M12 I1
dt dI1dt %%
= M12
N2
dt
21
21
1
!
$ dA1 ,
2
coil 1
and a current is induced in coil 1.
If current I2 varies with time ☛ emf induced by coil 2 in coil 1 is
E1 =
dI2
M21
dt
In mutual induction emf induced in one coil
11.1.2
Changing
current
coil 2 coil
produces
changing m
is always proportional toFigure
rate at
which
current
in in
other
is changing
It is easily seen that
flux in coil 1 is proportional to the changing curre
M12 =This
Mchanging
=
M
21
t
=
0
b
is suddenly
thrown
to
at
.
9. 5 LC Circuit (Electromagnetic Oscillator)
ies:
Figure 11.13.6 LC circuit
After the capacitor is charged we move the switch to position
b
9.4
LC Circuit (Electromagnetic Oscillator)
Initial charge on capacitor = Q
Initial charge on capacitor = Q
current== 00
InitialInitial
current
No battery.
No battery
current
i tobe
be in
in the
direction that
that increases
charge charge
on the positive
AssumeAssume
current
direction
decreases
i to
capacitor plate.
on positive capacitor plate
dQi = dQ
dt
) i =
dt of the inductor.
By Lenz Law, we also know the ”poles”
⇤
(9.1)
(10.1)
By Lenz Law we also know poles of inductor
Loop rule:
Loop rule ☛
V C + VL =
VC + VL = 0
Q
di
L
0C dt = 0
(9.2)
Combining equations (9.1) and (9.2), we get
Q
C
di2
L d Q=+ 01 Q = 0
dtdt2 LC
Combining equations (10.1) and (10.2) we get
This is similar to the equation of motion
2
of a simple harmonic
oscillator:
d Q
1
+
Q= 0
2
dt
LC d2x k
2
+
x=0
(10.2)
This is similar to equation of motion of
ilar
to theharmonic
equationoscillator
of motion
a simple
harmonic oscillator:
d2 x
k
+
x= 0
2
dt
m
Another approach (conservation
of energy)
2
dx
k
+ x=0
Total energy stored in circuit
2
dt
m = UE
U
pproach (conservation of energy)
gy stored in circuit:
U =
+
UB
Q2
1 2
+
Li
2C
2
U =is zero
UE no
+ energy
UB is dissipated in circuit
Since resistance in circuit
) Energy contained in circuit ⇥is conserved
⇥
2
Q
1 2
dU
)
U= =
0 2C + 2 Li
dt
⌘ the circu
dQis zero, di
dQin
esistance in the Q
circuit
no energy is⇣ dissipated
)
·
+ Li
= 0
* i =
y contained in the
C circuit
dt is conserved.
dt
dt
)
)
di
Q
L
+
= 0
dt
C
d2 Q
1
+
Q = 0
2
dt
LC
Solution to this differential equation is in form
Q(t) = Q0 cos(!t + )
dQ
)
= !Q0 sin(!t + )
dt
d2 Q
2
)
=
!
Q0 cos(!t + )
2
dt
= !2 Q
d2 Q
2
)
+
!
Q = 0
2
dt
1
2
Angular frequency of LC oscillator
) ! =
LC
Q0 ,
dt2
+ ⇥2Q = 0
are constants derived from initial conditions
1
Angular frequency
2
dQ
⇥
=
(Two initial conditions, e.g. Q(t = 0)LCand i(t
are required)
of the
=LC0)oscillator
=
dt t=0
2
2
Q the initial
Q0 conditions.
Also, Q0stored
, are constants
derived from
(Two initial condiEnergy
in
C = dQ =
cos2 (!t + )
tions, e.g. Q(t = 0), and i(t = 0) 2C
= dt
t=0
are
2Crequired.)
12
Q2 1
2
2Q20
2 2
Energy stored
in
Energy stored in L
C ==
=
cos (⇥t
+ )Q0 sin (!t +
Li
=
L!
2C
2C
2
)
2
1
1
Energy stored in L = Li2 =
L⇥ 2 Q20 sin2 (⇥t + )
2
2
1
Q0 2 2
2
* L! = 2 1 =
sin
+ )
Q20 (!t
2
C =
⇥ L⇥
=
sin (⇥t + )
2C
C
2C
Q20 Q20
Total energy
stored
= =
Total
energy stored
2C 2C
= Initial energy stored in capacitor
)
= Initial energy stored in capacitor
Energy oscillations in LC system and mass-spring system
LC Circuit
Mass-spring System
Energy
9. 6 RLC Circuit (Damped Oscillator)
In real life circuit ☛ there’s always resistance
energy stored in LC oscillator is NOT conserved
and
11.8 The RLC Series Circuit
We now consider a series RLC circuit that contains a res
dU
2 11.8.1.
capacitor, as shown in Figure
=
i
R
= Power dissipated in resistor
dt
Negative sign shows that energy
U is decreasing
i
)
)
z}|{
Q dQ
di
Li
+
·
=
dt
C
dt
d2 Q
R dQ
1
+
·
+
Q = 0
2
dt
L
dt
LC
RLC circuit
FigureJoule’s
11.8.1 A series
heating
i2 R
This is similar to equation of motion of a damped harmonic oscillator
~ = b~v )
(e.g. if a mass-spring system faces a frictional force F
Solution to equation is of form
R
2L t
Q(t) = Q0 e| {z } cos(! 0 t + )
|
{z
}
exponential
decay term
0
! =
!0 =
s
s
1
LC
!02
✓
oscillating term
◆2
R
2L
✓ ◆2
R
2L
R
damping factor
=
2L
There are three possible scenarios depending on the relative values of
and !0
n is less than the undamped oscillation, ! ' < ! 0 . The qualitative
Case
Underdamping
arge
on I:the
capacitor as a!function
of time is shown in Figure 11.10.1.
0 >
Figure 11.10.1 Underdamped oscillations
ppose
the initial
condition
is Q(t
= 0) = Q0 . The phase constant is then
Underdamped
oscillator
always
oscillates
at a lower frequency than natural frequency of oscillator
Case II: Overdamping
ts Q1 and Q2 can
!
<
0
be determined
from the initial conditions
Case III:11.10.2
Critical damping
!0 =
Figure
Overdamping
and critical damping
damping