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Problem 8: 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 onefourth
its initial value. Determine L if C and T are known.
The energy stored in the capacitor is given by
Thus,
o 1 2 0 COHO 1 '
c 2C 2C 2C 0
U (I) =.d...L = ( )'
_ 0 0 = Q o cos' W 1
cos' woT cos' woT
=---"-
cos' (O) 4
I
=> cos woT = -
2
which implies that
woT = tr rad = 60° . Therefore, with Wo = ~ ,
3 '\fLC
we obtain
(b) A capacitor in a series LC circuit has an initial charge Q o and is being discharged.
The inductor is a solenoid with N turns. Find, in terms of Land C, the flux through each
of the N turns in the coil at time I, when the charge on the capacitor is Q(t).
We can do this two ways, either is acceptable. First,we can make the explicit assumption that
Q(/) = Q o COSWol and the total flu x through the inductor is L1 = L ~7 = -LwoQo sin Wol
Therefore the flux through one turn of the inductor at time 1 is C!>o"'t"m = - LwoQ o sin Wol
N
or in terms of Land C, (j).,tot"m = -~ ~ sin Wol. Or second, we can simply leave Q(I)
as an unspecified function of time and write (using the same arguments as above) that
C!>
one lum
=.!:... dQ
N dl
(c) An LC circuit consists of a 20.0-mH inductor and a 0.500-,uF capacitor. If the
maximum instantaneous current is 0.100 A, what is the greatest potential difference
across the capaci tor?