Question

in an L-C circuit, current is oscillating with frequency 4x1000000Hz. what is the frequency with which magnetic energy is oscillating?

Answer 1

$\begin{array}{l}\text{Here,}\\ f=4\times {10}^6\text{Hz}\\ \text{Current in an LC circuit is given by}\\ I=-\frac{V_o}{\sqrt{LC}}\sin \omega t\\ \text{magnetic energy in an LC ckt is stored in the inductor}\\ \text{Energy stored in LC ckt is given by}\\ U=\frac{1}{2}L{I}^2\\ =>U=\frac{1}{2}L{(-\frac{V_o}{\sqrt{LC}}\sin \omega t)}^2\\ =>U=\frac{V_o}{4C}2{\sin }^2\omega t=\frac{V_o}{4C}(1-\cos 2\omega t)\\ =>U=\frac{V_o}{4C}(1-\cos 2\left(2\pi f\right)t)\\ =>U=\frac{V_o}{4C}(1-\cos 2\pi \left(2f\right)t)\\ \text{So, U is oscillating with}2f\text{or 8}\times {\text{10}}^6\text{Hz}\text{.}\end{array}$