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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.
dt
dN B
L
E
BB
IB
dt
dIL E
dt
dIL
dt
dN B
L
E
I
NL B
dtdIL LE
(Henry=H=V.s/A)
Il
NnIB 00
Il
NABAB 0
l
AN
I
NL B
2
0
VnL 2
0
AlV
nlN
l N
A
Inductors are circuit elements with large self-induction.
A circuit with an inductor will generate some back-emf 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.
dt
dILL E
0dt
dILIRE
/1 teR
I E
R
L
When S2 is at a
0dt
dILIR
/t
ieII R
L
When S2 is at b
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.
dt
dILIRII 2E
0dt
dILIRE
Battery
Power
Resistor
Power
Inductor
Power
The energy stored in the inductor
dt
dILI
dt
dU
II
IdILLIdIdUU00
2
2
1LIU
For energy density,
consider a solenoid
AlnL 2
0
nIB 0
AlB
n
BAlnLIU
0
22
0
2
0
2
22
1
2
1
0
2
2
B
Al
UuB
BdAB
a
bIl
r
drIlldr
r
IBdA
b
a
b
a
B ln222
000
ldrdA
a
bl
IL B ln
2
0
2
2
1LIU
a
blIU ln
4
2
0
A change in the current of one circuit can induce an emf in a nearby circuit
1
12212
I
NM
dt
dIM
N
IM
dt
dN
dt
dN 1
12
2
1122
1222
E
dt
dIM 2
211 E
It can be shown that: MMM 2112
dt
dIM 2
1 Edt
dIM 1
2 E
Il
NB B
0
l
ANN
I
BAN
I
NM BHHBHH
0
Provide a transfer of energy between the capacitor and the inductor.
Analogous to a spring-block 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.
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.
At the same time, the current increases the stored energy in the
magnetic field of the inductor.
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.
The current continues to flow and now starts to charge the
capacitor again, this time with the opposite polarity.
Then the cycle
completes itself in
reverse.
At some arbitrary time, t:
22
2
1
2LI
C
QUUU LC
0dt
dU
02
2
dt
QdL
C
Q
LC
1 Natural frequency of oscillation
tQQ cosmax
tItQI sinsin maxmax
tLI
tC
QUUU LC 2max
22
2
max sin2
cos2
22
max22
max LI
C
Q
ttC
QU 22
2
max sincos2
C
QU
2
2
max
a) f = ? Hz
LCf 6
12310
1091081.22
1
2
1
2
b) Qmax = ? and Imax = ?
CCQ 1012
max 1008.112109 E
AfQQI 4
maxmaxmax 1079.62
c) Q(t) = ? and I(t) = ?
tCtQQ 610
max 102cos1008.1cos
tAtII 64
max 102sin1079.6sin
d) U = ? JC
QU 10
2
max 1048.62
RLC Circuit - A more realistic circuit
The resistor represents the losses in the system.
Can be oscillatory, but the amplitude decreases.
212
2
max
2
1
cos
L
R
LC
teQQ
d
d
LRt
CLR 4
Reading Assignment
Chapter 33 - Alternating Current Circuits
WebAssign: Assignment 10