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Physics 121: Electricity &
Magnetism – Lecture 12
Induction II & E-M Oscillations
Dale E. Gary
Wenda Cao
NJIT Physics Department
Induction Review
Faraday’s Law: A changing
magnetic flux through a coil of
wire induces an EMF in the wire,
proportional to the number of
turns, N.
 Lenz’s Law: The direction of the
current driven by the EMF is such
that it creates a magnetic field to
oppose the flux change.
 Induction and energy transfer:
The forces on the loop oppose the
motion of the loop, and the power
required to move the loop
provides the electrical power in
the loop.
 A changing magnetic field creates
and electric field.
   N d B

dt
 
P  F  v  Fv
P  i

 
d B
  E  ds   N
dt
November 7, 2007
Induction and Inductance
When we try to run a current
through a coil of wire, the
changing current induces a “backEMF” that opposes the current.
 That is because the changing
current creates a changing
magnetic field, and the increasing
magnetic flux through the coils of
wire induce an opposing EMF.
 We seek a description of this that
depends only on the geometry of
the coils (i.e., independent of theInductance units: henry (H), 1 H = 1 T-m2/A
current through the coil).
 We call this the inductance (c.f.
N B
L

Inductance
capacitance). It describes the
i
proportionality between the
current through a coil and the
q
C

magnetic flux induced in it.
V

November 7, 2007
Inductance of a Solenoid

Consider a solenoid. Recall that the magnetic field inside a solenoid is
B  0in
The magnetic flux through the solenoid is then  B   B  dA  0inA
Number of turns per unit length n = N/l.
 The inductance of the solenoid is then:
N B N0inA
L

 nl 0 nA  0 n 2lA
i
i


Note that this depends only on the geometry. Since N = nl, this can also be
written
 A
0 N 2 A
Compare with capacitance of a capacitor C  0
L
l
l
Can also write 0= 4p ×107 H/m = 1.257 H/m
Compare with 0 = 8.85 pF/m
November 7, 2007
Self-Induction
You should be comfortable with the notion
that a changing current in one loop induces
an EMF in other loop.
 You should also be able to appreciate that
if the two loops are part of the same coil,
the induction still occurs—a changing
current in one loop of a coil induces a backEMF in another loop of the same coil.
 In fact, a changing current in a single loop
induces a back-EMF in itself. This is called
self-induction.
iL  N B
N B
 Since for any inductor L 
then
di
d B
i
L N
dt
dt


But Faraday’s Law says
L
d B
di The self-induced EMF is opposite
 N
 L
to the direction of change of
dt
dt
current
November 7, 2007
Induced EMF in an Inductor
1.
Which statement describes the current through
the inductor below, if the induced EMF is as
shown?
L 
A.
B.
C.
D.
E.
Constant and rightward.
Constant and leftward.
Increasing and rightward.
Decreasing and leftward.
Increasing and leftward.
November 7, 2007
Inductors in Circuits—The RL Circuit
Inductors, or coils, are common in
electrical circuits.
 They are made by wrapping insulated
wire around a core, and their main
use is in resonant circuits, or filter
circuits.
 Consider the RL circuit, where a

battery with EMF  drives a current
around the loop, producing a back

EMF L in the inductor.
Kirchoff’s loop rule gives
  iR  L di  0
dt

Solving this differential equation for i
gives

i  (1  e  Rt / L ) Rise of current
R
November 7, 2007
RL Circuits


When t is large: i 

When t is small (zero), i = 0.

The current starts from zero and
increases up to a maximum of i   / R
with a time constant given by
L 
Compare:


L
R
 C  RC
R
Inductor acts like a wire.
i
 (1  e  Rt / L )
R
Inductor acts like an open
circuit.
Inductor time constant
Capacitor time constant
The voltage across the resistor is VR  iR   (1  e  Rt / L )
The voltage across the inductor is
VL    VR     (1  e  Rt / L )   e  Rt / L
November 7, 2007
Inductive Time Constant
2.
The three loops below have identical inductors,
resistors, and batteries. Rank them in terms of
inductive time constant, L/R, greatest first.
A.
I, then II & III (tie).
II, I, III.
III & II (tie), then I.
III, II, I.
II, III, I.
B.
C.
D.
E.
I.
II.
November 7, 2007
III.






What happens when the switch is
thrown from a to b?
Kirchoff’s Loop Rule was:
  iR  L di  0
dt
di
Now it is:
iR  L  0
dt
The decay of the current, then, is
given by

i  e  Rt / L Decay of current
R
Voltage across resistor:
VR  iR   e  Rt / L
Voltage across inductor:
di
 d e  Rt / L   e  Rt / L
VL  L  L
dt
R dt
VR (V)
RL Circuits
November 7, 2007
What is Happening?





When the battery is removed, and the RL series circuit is shorted, the
current keeps flowing in the same direction it was for awhile. How can this
be?
In the case of an RC circuit, we would see the current reverse as the stored
charge flowed off the capacitor. But in the case of an RL circuit the
opposite happens—charge continues to flow in the same direction.
What is happening is that the current tries to drop suddenly, but this
induces an EMF to oppose the change, causing the current to keep flowing
for awhile.
Another way of thinking about it is that the magnetic field that was stored
in the inductor is “collapsing.”
There is energy stored in the magnetic field, and when the source of
current is removed, the energy flows from the magnetic field back into the
circuit.
November 7, 2007
Make Before Break Switches







The switch in a circuit like the one at right has to be a
special kind, called a “make before break” switch.
The switch has to make the connection to b before
breaking the connection with a.
If the circuit is allowed to be in the state like this…even
momentarily, midway between a and b, then a big
problem results.
Recall that for a capacitor, when we disconnect the
circuit the charge will merrily stay on the capacitor
indefinitely.
Not so on an inductor. The inductor needs current, i.e.
flowing charge. It CANNOT go immediately to zero.
The collapsing magnetic field in the inductor will force
the current to flow, even when it has no where to go.
The current will flow in this case by jumping the air
gap.
Link to video
You have probably
seen this when
unplugging something
with a motor—a spark
that jumps from the
plug to the socket.
November 7, 2007
Example Circuit

This circuit has three identical resistors R = 9
W, and two identical inductors L = 2.0 mH.
The battery has EMF
 = 18 V.
(a)
What is the current i through the battery just
after the switch is closed?
 2A
i

(acts like open wire)
R
(b)
What is the current i through the battery a
long time after the switch is closed?
3
(acts like straight wire) i 
6A
R
(c)
What is the behavior of the current between
these times? Use Kirchoff’s Loop Rule on
each loop to find out.
November 7, 2007
Current Through Battery 1
3.
The three loops below have identical inductors,
resistors, and batteries. Rank them in terms of
current through the battery just after the switch
is closed, greatest first.
A.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
B.
C.
D.
E.
I.
II.
III.
November 7, 2007
Current Through Battery 2
4.
The three loops below have identical inductors,
resistors, and batteries. Rank them in terms of
current through the battery a long time after
the switch is closed, greatest first.
A.
I, II, III.
II, I, III.
III, I, II.
III, II, I.
II, III, I.
B.
C.
D.
E.
I.
II.
III.
November 7, 2007
Energy Stored in Magnetic Field

By Kirchoff’s Loop Rule, we have
  iR  L di
dt
We can find the power in the circuit by
multiplying by i.
i  i 2 R  Li di
dt
power stored in
magnetic field
power provided power dissipated
by battery
in resistor
dU B
di
 Power is rate that work is done, i.e. P 
 Li
dt
dt
1
 So dU B  Li di , or after integration U B  Li 2
Energy in magnetic field
2

q2 1
Recall for electrical energy in a capacitor: U E 
 CV 2
2C 2
November 7, 2007
The LC Circuit





What happens when we make a circuit from
both an inductor and capacitor?
If we first charge the capacitor, and then
disconnect the battery, what will happen to the
charge?
Recall that initially the inductor acts like an open
circuit, so charge does not flow immediately.
However, over longer times the inductor acts
like a simple, straight wire, so charge will
eventually flow off from the capacitor.
As the charge begins to flow, it develops a
magnetic field in the inductor.
November 7, 2007
Electromagnetic Oscillations
q2 1 1
U E U  Li 2CV 2
B2C
2 2
November 7, 2007
Oscillations Forever?
5.
What do you think (physically) will happen to
the oscillations over a long time?
A.
They will stop after one complete cycle.
They will continue forever.
They will continue for awhile, and then suddenly stop.
They will continue for awhile, but eventually die away.
There is not enough information to tell what will happen.
B.
C.
D.
E.
November 7, 2007
Ideal vs. Non-Ideal





In an ideal situation (no resistance in
circuit), these oscillations will go on
forever.
In fact, no circuit is ideal, and all have at
least a little bit of resistance.
In that case, the oscillations
U  mgh get smaller
with time. They are said to be “damped
K  12 mv2   g
oscillations.”
l
This is just like the situation with a
pendulum, which is another kind of
oscillator.
There, the energy oscillation is between
potential energy and kinetic energy.
Spring Animation
Damped Oscillations
U  12 kx2 K  12 mv2  
November 7, 2007
k
m
Derivation of Oscillation Frequency
We have shown qualitatively that LC circuits act like an oscillator.
 We can discover the frequency of oscillation by looking at the
equations governing the total energy.
q2 1 2
U  UE UB 
 Li
2C 2
 Since the total energy is constant, the time derivative should be zero:
dU q dq
di

 Li  0
dt C dt
dt
dq
di d 2 q
d 2q q
i
 2 , so making these substitutions: L 2   0
 But
and
dt
dt dt
dt
C
 This is a second-order, homogeneous differential equation, whose
solution is q  Q cos(t   )
 i.e. the charge varies according to a cosine wave with amplitude Q and
2
frequency . Check by taking
dq
d
q


Q

sin(

t


)
 Q 2 cos(t   )
2
two time derivatives of charge: dt
dt
 Plug into original equation:
1
1
d 2q q
Q
2
2


 L   0
L 2    LQ cos(t   )  cos(t   )  0
LC
C
dt
C
C

November 7, 2007
Example
a)
What is the expression for the voltage change across the capacitor in the
circuit below, as a function of time, if L = 30 mH, and C = 100 F, and the
capacitor is fully charged with 0.001 C at time t=0?
First, the angular frequency of oscillation is
1
1


 577.4 rad/s
2
4
LC
(3 10 H)(10 F)
Because the voltage across the capacitor is proportional to
the charge, it has the same expression as the charge:
q Q cos(t   )
VC  
C
C
At time t = 0, q = Q, so  = 0. Therefore, the full expression for the voltage
across the capacitor is
103 C
VC  6 cos(577t )  1000 cos(577t ) volts
10 F
November 7, 2007
Example, cont’d
b)
What is the expression for the current in the circuit?
The current is i 
dq
 Q sin( t )
dt
i  (103 C)(577 rad/s ) sin( 577t )  0.577 sin( 577t ) amps
c)
How long until the capacitor charge is reversed?
That happens after ½ period, where the period is T 
1 2p

f

T p
  5.44 ms
2 
November 7, 2007
Summary
Inductance (units, henry H) is given by L  N B
i
 Inductance of a solenoid is:
0 N 2 A
L
(depends only on geometry)
l
 EMF, in terms of inductance, is:
 L   N d B   L di
dt
dt


RL circuits Rise of current
i


Decay of current
 (1  e  Rt / L )
R
Energy in inductor:
i
Inductor time constant
 e  Rt / L
L 
R
L
R
1 2
Energy in magnetic field
Li
2
2
q
1 2
LC circuits: total electric + magnetic energy is conserved U  U  U 

Li
E
B
2C 2
UB 
Charge equation
Current equation
q  Q cos(t   )
i  Q sin( t   )
Oscillation frequency

1
LC
November 7, 2007