At time `t = 0`, terminal A in the circuit shown in the figure is connected to B by a key and an alternating current `I(t) = I_(0)cos (omega t)`, with `I_(0) = 1 A and (omega) = 500 rads^(-1)` starts flowing in it with the initial direction shown in the figure. At `t = (7pi//6omega)`, the key is switched from B to D. Now onwards only A and D are connected. A total charge Q flows from the battery to charge the capacitor fully. If `C=20(mu)F, R = 10(Omega) and the battery is ideal with emf of 50 V, identify the correct statement(s).
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Options

A. Magnitude of the maximum charge on the capacitor before `t=(7 pi)/(6 omega) is 1 xx 10^(-3)C`
B. The current in the left part of the circuit just before `t=(7 pi)/(6 omega)` is clockwise.
C. Immidiately after A is connected to D, the current in R is 10 A.
D. `Q=2 xx 10^(-3)C`

Answer

C, D

Explanation

If q represents the charge on capacitor's upper plate:
`I(t)=(I_0)cos (omega t)=(dq)/(dt) implies q(t)=(I_0)/(omega) sin (omega t)`
Max charge `=(1 A)/(500 rad s^(-1)) = 2 xx 10^(-3) C`
Charge on upper plate at `t=(7 pi)/(6 omega)=(I_0)/(omega) sin ((7 pi)/(6 omega))=-(I_0)/(2 omega)`
When capacitor is fully charged, charge on upper plate `=50V`
`xx20 (mu)F=1 xx 10^(-3)C`
`:. Q= 1 xx 10^(-2)C-(-(I_0)/(2 omega)) = 2 xx 10^(-3)C`
Voltage across capacitor when A and D are connected
`=(1 xx 10^(-3)C)/(20 xx 10^(-6)mu F)=50 V`
Total voltage resistor `= 100 V`
`implies `Current `=(100 V)/(10 Omega)` 10 A
Current is anticlockwise.