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Self-Inductance
RL Circuits
Energy in a Magnetic Field
Mutual Inductance
Oscillations in an LC Circuit
When the switch is closed, the battery
(source emf) starts pushing electrons around
the circuit.
The current starts to rise, creating an
increasing magnetic flux through the circuit.
This increasing flux creates an induced emf
in the circuit.
The induced emf will create a flux to oppose
the increasing flux.
The direction of this induced emf will be opposite the source emf.
This results in a gradual increase in the current rather than an instantaneous one.
The induced emf is called the self-induced emf or back emf.
When I changes, an emf is induced in the coil.
If I is increasing (and therefore increasing the flux through the coil), then the
induced emf will set up a magnetic field to oppose the increase in the
magnetic flux in the direction shown.
If I is decreasing, then the induced emf will set up a magnetic field to oppose
the decrease in the magnetic flux.
d B
E L  N
dt
dI
dt
B  B
EL 
BI
d B
dI
E L  N
 L
dt
dt
N B
L
I
L
EL
dI dt
(Henry=H=V.s/A)
A
N
B  0 nI  0 I
l
N
NA
 B  BA  0
I
l
N B
N2A
L
 0
I
l
L  0 n 2V
N  nl
V  Al
l
Inductors are circuit
elements with large selfinduction.
 A circuit with an inductor
will generate some backemf in response to a
changing current.
 This back-emf will act to
keep the current the way
it used to be.
 Such a circuit will act
“sluggish” in its response.

When S2 is at a
E  IR  L
I
dI
0
dt

1 e 
R
E
t / 

L
R

L
R
When S2 is at b
IR  L
E L  L
dI
dt
dI
0
dt
I  I i e  t /
When the switch is closed, the current through the circuit
exponentially approaches a value I = E / R. If we repeat
this experiment with an inductor having twice the number
of turns per unit length, the time it takes for the current to
reach a value of I / 2
1. increases.
2. decreases.
3. is the same.
dI
E  IR  L  0
dt
dI
IE  I R  LI
dt
2
Battery
Power
Inductor
Power
Resistor
Power
For energy density,
consider a solenoid
L  0 n 2 Al
B  0 nI
The energy stored in the inductor
dU
dI
 LI
dt
dt
I
I
0
0
U   dU   LIdI  L  IdI
U
1 2
LI
2
2
 B 
1 2 1
B2
2
 
U  LI  0 n Al 
Al
2
2
20
 0 n 
U
B2
uB 

Al 20
 B   BdA
dA  ldr
0 I
0 Il b dr 0 Il  b 
 B   BdA  
ldr 

ln  

2r
2 a r
2  a 
a
b
 B  0l  b 
L

ln  
I
2  a 
U
1 2
LI
2
0lI 2  b 
U
ln  
4
a
A change in the current of one circuit can induce an emf in a nearby circuit
N 212
M 12 
I1
E2   N2
d12
d M I 
dI
  N 2  12 1    M12 1
dt
dt  N 2 
dt
E1   M 21
dI 2
dt
It can be shown that:
M12  M 21  M
dI 2
dt
dI1
dt
E1   M
E2  M
NB
B  0
I
l
N H  BH N H BA
NH NB A
M

 0
I
I
l
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Provide a transfer of
energy between the
capacitor and the
inductor.
Analogous to a springblock system
Assume the capacitor is fully charged and thus has some stored
energy.
When the switch is closed, the charges on the capacitor leave the
plates and move in the circuit, setting up a current.
This current starts to discharge the capacitor and reduce its stored energy.
At the same time, the current increases the stored energy in the
magnetic field of the inductor.
When the capacitor is fully discharged, it stores no energy, but the
current reaches a maximum and all the energy is stored in the
inductor.
The current continues to flow and now starts to charge the
capacitor again, this time with the opposite polarity.
When the capacitor becomes fully charged (with the opposite
polarity), the energy is completely stored in the capacitor again.
The e-field sets up the current flow in the opposite direction.
Then the cycle
completes itself in
reverse.
At some arbitrary time, t:
Q2 1 2
U  UC  U L 
 LI
2C 2
dU
0
dt
Q
d 2Q
L 2 0
C
dt
Q  Qmax cos t
I  Qmax sin t   I max sin t

1
LC
Natural frequency of oscillation
2
2
Qmax
LI
max
2
U  UC  U L 
cos t 
sin 2 t
2C
2
2
Qmax
LI 2 max

2C
2
2

Qmax
U
cos 2 t  sin 2 t
2C
Qmax
U
2C
2

a) f = ?
f 

1

2 2 LC 2
1
2.8110 9 10 
3
12
 106 Hz
b) Qmax = ? and Imax = ?
Qmax  CE  9  1012 12  1.08  1010C
I max  Qmax  2fQmax  6.79 104 A
c) Q(t) = ? and I(t) = ?

 

Asin2 10 t 
Q  Qmax cos t  1.08 1010 C cos 2 106 t

I   I max sin t   6.79 104
2
d) U = ?
Q
U  max  6.48 1010 J
2C
6
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RLC Circuit - A more
realistic circuit
The resistor represents the
losses in the system.
Can be oscillatory, but the
amplitude decreases.
Q  Qmax e  Rt 2 L cos d t
 1  R 
d  
 
 LC  2 L 
2 12



R  4L C
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Reading Assignment
 Chapter 33 - Alternating Current Circuits
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WebAssign: Assignment 10