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Question

In a given L - C circuit, switch S is closed at t = 0, then

A
maximum charge on capacitor can be 2Cε.
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B
when charge on capacitor is half of its maximum charge, current is maximum
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C
Charge on capacitor is εC2 at t=π3LC
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D
Maximum charge on capacitor can be εC
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Solution

The correct options are
A maximum charge on capacitor can be 2Cε.
B when charge on capacitor is half of its maximum charge, current is maximum
C Charge on capacitor is εC2 at t=π3LC

ε=qC+Ldidt
εqC=Ld2qdt2
εCqLC=d2qdt2=didtd2idt2=iLC
d2idt2=w2i=w=1LC&T=2πw
i=i0sinwt or didt=i0wcos wt
At t = 0,
Ldidt=εε=Li0wi0=εLw=εCL& i=dqdt=i0sin wt
q0dq=t0i0sin wt dtq=i0w[cos wt]t0
q=i0w(1cos wt)q=q0(1cos wt)
where q0=i0w=εC
qmax=2q0=2εC
When charge on the capacitor is half of its maximum charge i.e., q0,
q=q0(1cos wt)
q0=q0(1cos wt)coswt=0
Therefore, i=i0sinwt=i0×1=i0, so current takes its maximum value.
At t=π3LC,
q=q0(1cos wt)=εC(1cosπ3)=εC2

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