• # question_answer A capacitor of capacitance C is having a charge${{Q}_{0}}$. It is connected to a pure inductor of inductance L. The inductor is a solenoid having N turns. Find the magnitude of magnetic flux through each of the N turns in the coil at the instant charge on the capacitor becomes$\frac{{{Q}_{0}}}{2}$. A) $\frac{{{Q}_{0}}}{2N}\sqrt{\frac{3L}{C}}$      B) $\frac{{{Q}_{0}}}{N}\sqrt{\frac{3L}{C}}$ C) $\frac{2{{Q}_{0}}}{N}\sqrt{\frac{L}{C}}$                   D) $\frac{{{Q}_{0}}}{N}\sqrt{\frac{L}{C}}$

[a] $Q={{Q}_{0}}\cos (\omega t)$ where $\omega =\frac{1}{\sqrt{LC}}$ When   $Q=\frac{{{Q}_{0}}}{1};$  $\cos \,(\omega t)=\frac{1}{2}$ Flux linked to the coil is $\phi =LI=L\frac{dQ}{dt}$ $\therefore \,\,\,\,\,\,\,\,\,|\phi |=L{{Q}_{0}}\omega |\sin (\omega t)|$ When   $\cos \,(\omega t)=\frac{1}{2};\,\,\sin (\omega t)=\frac{\sqrt{3}}{2}$ $\therefore \,\,\,\,\,|\phi |=\frac{\sqrt{3}}{2}L{{Q}_{0}}\omega$ Flux through each turn $\frac{|\phi |}{N}=\frac{\sqrt{3}L{{Q}_{0}}\omega }{2N}=\frac{1}{2}\sqrt{\frac{3L}{C}}\frac{{{Q}_{0}}}{N}$