Characteristics of an LC circuit using Equipartition

An LC circuit is one which has a capacitor and an inductor connected to each other. It exhibits oscillations just like a mass on a spring (a harmonic oscillator). In fact, the analogy is quite accurate with the capacitor playing the role of the spring and the inductor playing the role of the mass inertia.

Just like any harmonic oscillator, we can use equipartition to estimate the energy and frequency of oscillations using equipartition.

The average energy in the capacitor is: $$ E_C \sim \left\langle {Q^2 \over C} \right\rangle = {\left\langle { Q^2} \right\rangle \over C} . $$ On the other hand, the average energy in the inductor is: $$ E_L \sim \left\langle { L I^2} \right\rangle = { L \left\langle d Q \over dt\right\rangle^2}. $$ The next thing to note is that if there are oscillations with a period $\omega$, then $\left\langle { dQ / dt } \right\rangle \sim \omega^2 \left\langle {Q } \right\rangle$. In fact, because the oscillations are harmonic, it will be an exact relation, but we don’t necessarily know that without further investigation, which we wish to avoid!

The last step is to compare the two energy components to each other, and find that $$ E_C \sim E_L ~~\Rightarrow~~ {\left\langle { Q^2} \right\rangle \over C} \sim {L \omega^2 \left\langle { Q^2} \right\rangle }, $$ giving that $$ \omega^2 \sim {1 \over LC}. $$ It turns out that because both energy components have the same quadratic behavior, the equipartition is not approximate but infact exact, and for this reason, the $\omega$ we find is accurate as well. But again, we cannot know that without further investigation.