Question

A condenser of capacity `C` is charged to a potential difference of `V_(1)`. The plates of the condenser are then connected to an ideal inductor of inductance `L`. The current through the inductor wehnn the potential difference across the condenser reduces to `V_(2)` is
A. `((C(V_(1)-V_(2))^(2))/(L))^(1/2)`
B. `(C(V_(1)^(2)-V_(2)^(2)))/(L)`
C. `(C(V_(1)^(2)+V_(2)^(2)))/(L)`
D. `((C(V_(1)^(2)-V_(2)^(2)))/(L))^(1/2)`

Answer 1

Correct Answer - D
In case of oscillatory discharge of a capacitor through an inductor, charge at instant t is given by
`q = q_(0)cos omega t`
where, `omega = (1)/(sqrt(LC))`
`therefore cos omega t=(q)/(q_(0))=(CV_(2))/(CV_(1))=(V_(2))/(V_(1)) " "(because q=CV)` .....(i)
Current through the inductor
`I=(dq)/(dt)=(d)/(dt)(q_(0)cos omega t)=-q_(0)omega sin omega t`
`|I|=CV_(1)(1)/(sqrt(LC))[1-cos^(2)omega t]^(1//2)`
`=V_(1)sqrt((C)/(L))[1-((V_(2))/(V_(1)))^(2)]^(1//2)=[(C(V_(1)^(2)-V_(2)^(2)))/(L)]^(1//2)` (using (i))