Solution Derivations for Capa #11
Caution: The symbol
E
is used interchangeably for energy and EMF.
1) DATA:
V
b
= 5
.
0
V
,
R
= 155 Ω,
L
= 8
.
400
×
10

2
H
. In the diagram above, what is
the voltage across the inductor in the instant just after the switch is closed?
V
b
= Given
R
= Given
L
= Given
From Kirchhoff’s Loop law, we can get an equation for the voltages around the
circuit.
V
b

IR

E
L
= 0
where
E
L
is the EMF of the inductor. Thus,
E
L
=
V
b

IR
Immediately after the switch is closed, there is no current in the circuit.
The
resistor and inductor are in series, and the inductor opposes the change in current.
Thus, there is no voltage drop across the resistor and the voltage drop across the
inductor is the initial voltage
V
b
.
2) After the switch is closed for a long time, what is the energy stored in the inductor?
The energy stored in an inductor is given by
E
=
1
2
LI
2
The maximum current in this circuit is determined by the battery and the resistor.
From Ohm’s law,
I
=
V
R
Thus,
E
=
1
2
L
V
R
2
3) The switch in the above diagram is closed after being open a long time. The initial
charge on the capacitor is zero. (For each statement select T True, F False).
QUESTION:
A) In the instant after the switch is closed, the voltage across the capacitor equals
the voltage across the battery.
B) In the instant after the switch is closed, the voltage across the inductor equals
the voltage across the battery.
C) A long time after the switch is closed, the voltage across the capacitor equals
1
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the voltage across the battery.
D) A long time after the switch is closed, the current through the resistor is zero.
ANSWER:
A) False, initially capacitors act like wires without resistance, so there is no
voltage drop.
B) True, there is no current elsewhere in the circuit, so there are no other voltage
drops. See (#1).
C) True, after a long time a capacitor acts like an infinite resistor, so no current
can flow. Thus, the voltage drop must be equal to the battery.
D) True, see (C).
CAPA is looking for an answer in the form FTTT
4) The next 6 questions refer to this situation: An LR circuit is hooked up to a battery as
shown in the figure, with the switch initially open. The resistance in the circuit is
R
= 110 Ω,
the inductance is
L
= 3
.
40
H
and the battery maintains a voltage of
E
= 30
.
0
V
. At time
t
= 0 the switch is closed. What is the current through the circuit after the switch has been
closed for
t
= 4
.
57
×
10

2
s
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 Fall '08
 ROGERS
 Physics, Energy, Magnetic Field, long time, Inductor, NC, Energy density

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