Worked exercise — LC circuit, find the natural frequency

Problem

An $LC$ circuit (no resistance) has charge $Q(t)=2\cos(\omega t)$. Find the value of $\omega$ such that Kirchhoff’s equation is satisfied.

Solution

Current and its derivative. With $i=-\dot Q$: $i=-\dot Q=2\omega\sin(\omega t),\qquad \dot i=2\omega^2\cos(\omega t)$

Kirchhoff’s equation with no $R$ and no generator (drop across $L$ plus drop across $C$): $-L\,\dot i-\frac{Q}{C}=0\ \Rightarrow\ -L\cdot 2\omega^2\cos(\omega t)-\frac{2\cos(\omega t)}{C}=0$

Collecting $2\cos(\omega t)\neq 0$: $L\,\omega^2=\frac{1}{C}\ \Rightarrow\ \omega=\frac{1}{\sqrt{LC}}$ $\ev{\omega=\frac{1}{\sqrt{LC}}}$

The natural frequency depends only on $L$ and $C$: it does not depend on the amplitude of the oscillation nor on the initial conditions (simple harmonic motion of the charge).

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Topics: Induzione elettromagnetica Concepts: Oscillazioni LC e RLC · Moto armonico semplice · Risonanza