1
Lehigh University
Physics 21, Spring 2008
March 12, 2008
HW18 Solutions
181.
(HRW 3124)
In an oscillating series RLC circuit, find the time required for the
maximum energy present in the capacitor during an oscillation to fall to half its initial
value. Assume
at
.
Solution:
24. The assumption stated at the end of the problem is equivalent to setting
φ
= 0 in Eq.
3125. Since the maximum energy in the capacitor (each cycle) is given by
q
C
max
/
2
2
,
where
q
max
is the maximum charge (during a given cycle), then we seek the time for
which
2
2
max
max
1
.
2
2 2
2
q
Q
Q
q
C
C
=
⇒
=
Now
q
max
(referred to as the
exponentially decaying amplitude
in §315) is related to
Q
(and the other parameters of the circuit) by
q
Qe
q
Q
Rt
L
Rt
L
max
/
max
ln
.
=
⇒
F
H
G
I
K
J
= −
−
2
2
Setting
q
Q
max
=
/
2 , we solve for
t
:
t
L
R
q
Q
L
R
L
R
= −
F
H
G
I
K
J
= −
F
H
G
I
K
J
=
2
2
1
2
2
ln
ln
ln
.
max
The identities
ln( /
)
ln
ln
1
2
2
2
1
2
= −
= −
were used to obtain the final form of the
result.
182.
(HRW 3129)
(a) At what frequency would a 6.0 mH inductor and a 10
μ
F
capacitor have the same reactance? (b) What would the reactance be? (c) Show that this
frequency would be the natural frequency of an oscillating circuit with the same L and C.
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 Spring '08
 Kim
 Energy, LC circuit, maximum value, HRW, Inductive Reactance

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