LC Oscillations, Quantitatively — Question 8

Question 8

The time spent above an energy threshold is not the energy fraction itself

An ideal LC oscillator has $U_C/U=\cos^2\omega t$. Let $0<\eta<1$.

1. Find the fraction of a full cycle with capacitor energy greater than eta times total energy.

2. Evaluate the fraction for eta=3/4.

Question 8 - Solution

The time spent above an energy threshold is not the energy fraction itself

(a) Method and calculation

$$U_C>\eta U\Rightarrow|\cos\theta|>\sqrt\eta,\quad\theta=\omega t.$$$$\boxed{f=\frac{4\arccos\sqrt\eta}{2\pi}=\frac2\pi\arccos\sqrt\eta}.$$

(b) Interpretation and further calculation

$$\eta=3/4\Rightarrow\arccos(\sqrt3/2)=\pi/6.$$$$\boxed{f=1/3}.$$

Physical check

Threshold occupancy measures phase intervals, not the average share of energy.