MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2007
Problem Set 12 Solutions
Problem 1:
LC
Circuit
(a) Initially, the capacitor in a series
LC
circuit is charged.
A switch is closed, allowing
the capacitor to discharge, and after time
T
the energy stored in the capacitor is one-
quarter its initial value.
Determine
L
if
C
and
T
are known.
If the energy is at ¼ of it’s initial value then the voltage has fallen in half.
That is,
(
)
(
)
2
0
0
2
1
9
cos
2
3
3
V
T
V T
V
T
T
L
T
C
LC
π
π
ω
ω
ω
π
=
=
⇒
=
⇒
=
=
⇒
=
(b) A capacitor in a series
LC
circuit has an initial charge
and is being discharged.
Find, in terms of
L
and
C
, the flux through each of the
N
turns in the coil at time
t
, when
the charge on the capacitor is
Q
(
t
).
0
Q
The flux depends on the current and inductance:
LI
N
=
Φ
.
So we just need to write the
current in terms of the charge.
By conservation of energy
2
2
2
1
0
2
2
2
Q
C
LI
Q
C
+
=
.
So,
( )
(
)
2
2
0
Q
Q t
LI
L
N
N
LC
−
Φ =
=
(c) An
LC
circuit consists of a 90.0-mH inductor and a
1.000- F
μ
capacitor.
If the
maximum instantaneous current is 0.100 A, what is the greatest potential difference
across the capacitor?
By conservation of energy,
2
2
1
1
max
max
2
2
LI
CV
=
, so
(
)
(
)
(
)
max
max
90 mH
0.1 A
30 V
1.0
μ
F
L
V
I
C
=
=
=
Problem 2: Solenoid
Consider a long solenoid of length
D
and radius
a
with
N
turns, placed along the z-axis (it
is centered at z=0).
It is hooked in series with a capacitor
C
, a resistor
R
, and a power
supply that is driving the circuit on resonance.
At time t = 0 the current flowing through
the solenoid is a maximum,
I
0
.
Problem Set 12 Solutions
p. 1 of 7

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