Initially, the capacitor is charged to a potential of `5 V` and then connected to position 1 with the shown polarity for `1 s`. After `1s` it is connected across the inductor at position 2

(a) Find the potential across the capacitor after `1 s` of its connection to position 1.
(b) Find the maximum current flowing in the `L- C` circuit when capacitor is connected across the inductor. Also, find the frequency of `LC` oscillations.

Options

A. `(20//pi) Hz`
B. `(2//pi) Hz`
C. `(40//pi) Hz`
D. `100//pi Hz`

Answer

D

Explanation

Initial charge `= CV_(0) = Q_(0)`
`= 10 xx 10^(-3) xx 5 = 50 mC`
When capacitor is connected at position 1.
`E - IR - (q)/(C ) = 0`
`int_(0)^(t) (1)/(RC) dt = int_(Q_(0))^(q) (dq)/(EC - q)` or `q = 50 [2 - e^(-t)]mC`
At `t = 1 s, q = 50 [2 - e^(-t)]`
Voltage across the capacitor at the time
`V = (q)/(C ) = (50[(2 - 1//e)])/(10 xx 10^(-3)) = 5xx 10^(-3) [2 - (1//e)]V`
`(1)/(2) Li^(2) = (1)/(2) CV^(2) rArr i = (2 - (1)/(e)) xx 10^(4) A`
Frequency `= (1)/(2pi sqrt(LC)) = (100)/(pi) Hz`