Reading assignment:
Read Chapter 32: p.1014-1030
and the summary on p.1033-1034
Homework #10: (due April 23)
Chapter 32: #6,12,19,22,44,49
Today’s topics
• RL circuits
• Mutual induction
• Inductor and energy storage
• Oscillation in an LC circuit
• Quiz #5: Tuesday April 22, 2003 on
Chapter 32
• Quiz #6: Tuesday April 29, 2003 on
Chapter 34
Self-inductance
Definition of self-induced emf:
εL
ε
dΦ B
dI
=−
= −L
dt
dt
• L = self-induced emf
• ΦB = Magnetic flux
• L = self-inductance
For a single coil:
NΦ B
L=
I
For a single solenoid:
B
I
Magnetic field: B = µ 0 nI
Magnetic flux: Φ B = BA
Self-inductance:
NΦB
2
L=
= µ 0 n Al
I
RL Circuits
See Figure 32.3 on p. 1018
R = resistance
L = self-inductance
Using Kirchhoff’s loop rule:
ε
so that:
− IR − LdI / dt = 0
ε
I=
R
(1 − e
−t / τ
)
where time constant τ = L / R
RL Circuits
• Right after closing the switch:
Current I = 0
• Long time after closing the
switch:
Current I = ε / R
• How fast does the current
approach the value of ε / R :
Small τ = L / R : fast
Large τ = L / R : slow
Problem solving exercise:
In an RL circuit, R = 4.0 kΩ, L = 12
mH, and emf = 240 V. The switch
is closed at t = 0 when the current
is zero. When the current I = 15
mA, what is the potential difference
across the inductor?
A. 240 V
B. 60 V
C. 0
D. 180 V
E. 190 V
The correct answer is D.
Question regarding RL circuit:
What is the major difference in
I(t) between an R circuit and an RL
circuit when the switch is closed?
• An R circuit consists of a
resistor, a battery, and a switch
in serial connection;
• An RL circuit consists of a
resistor, an inductor, a battery,
and a switch in serial
connection.
Mutual Induction
Mutual induction depends on the
interaction between two circuits.
That is, the induced emf in one
circuit is generated by the timevarying current in another circuit.
see figure 32.12 on p. 1024
Definition of mutual-inductance:
dI 2
dΦ 21
= −M 21
ε1 = −N1
dt
dt
N1Φ21
with M 21=
I2
• M21 = mutual inductance
• N1 = the number of turns of a
coil in circuit 1.
• I2 = the current in circuit 2
• Φ21 = Magnetic flux through
one turn of the coil in circuit 1
due to the current in circuit 2
N2Φ12
Similarly, M 12 =
.
I1
Mutual induction
(1) In mutual induction, the emf
induced in one coil is always
proportional to the rate at which
the current in the other coil is
changing.
dI 2
dΦ 21
= −M 21
ε1 = −N1
dt
dt
dI 1
dΦ12
= −M12
ε 2 = − N2
dt
dt
(2) Mutual inductance between
two coils are equal:
M 21 ≡ M12 = M
Problem solving exercise
A step-up transformer has an input
voltage of 110 V (rms). There are
100 turns on the primary and 1500
turns on the secondary. What is the
output voltage?
A. 1600 V (max)
B. 1650 V (rms)
C. 3260 V (max)
D. 165 kV (rms)
E. 7.3 V (rms)
Hint: the magnetic flux per turn is the
same in the secondary coil as in the
primary coil.
In mutual induction:
dI1
• ε2 ∝
dt
dI2
• ε1 ∝
dt
Questions:
(1) Does mutual inductance M depend
on I1 or I2?
(2) Can you have mutual inductance
without self-inductance?
(3) Can you have self inductance
without mutual inductance?
Inductor and Energy
An inductor can store energy in
form of magnetic field
Work done to establish a current in
an inductor:
dI
dU
= I ( −ε L ) = LI
dt
dt
So energy stored in an inductor:
U = (1/2)LI
2
2
B
Al
For a solenoid, U =
2µ0
Oscillations in an LC Circuit
(1) An LC circuit is made of an
inductor (L) and a capacitor (C).
(2) Both inductors and capacitors
can store energy:
2
2
U L = (1/2)LI & UC = (1/2)Q / C
(3) The total energy in an LC
circuit is conserved. The
oscillation takes place by
converting energy between the
inductor and the capacitor.
Oscillations in an LC Circuit
At t=0, the switch is closed:
Q = Q0 in the capacitor
I = 0 in the circuit
For t > 0, after the switch is
closed:
Utotal = UC + U L
Q2
=
2C
LI 2
+
2
dUtotal
dQ
Since
,
= 0 and I =
dt
dt
(1)
we have:
2
d Q 1
Q=0
+
2
dt
LC
Solution to equation (2)
(1) Q (t ) = Q0 cos ωt
dQ
(2) I (t ) =
= −ωQ0 sin ωt
dt
where ω = 1 / LC
(2)
Oscillation in energy
• Energy in the capacitor:
2
UC
Q
= 0
2C
cos ωt
2
• Energy in the inductor:
2
LI
2
max
UL =
sin ωt
2
where I max = ωQ 0
• The total energy:
Utotal =
LI
2
2
Q
max = 0
2
2C
Problem solving exercise
At time t = 0 the charge on the 50µF capacitor in an LC circuit is 15
µC and there is no current. If the
inductance is 20 mH the maximum
current is:
A. 15 nA
B. 15 µA
C. 6.7 mA
D. 15 mA
E. none of above
The correct answer is D.