LC Oscillations, Quantitatively — Question 10

Question 10

Peak energy-transfer power occurs between the field-energy extrema

An ideal LC oscillator starts with maximum capacitor energy U and $q=Q\cos\omega t$.

1. Derive instantaneous power flowing into the capacitor.

2. Find the largest magnitude of transfer power and the charge and current magnitudes at that instant.

Question 10 - Solution

Peak energy-transfer power occurs between the field-energy extrema

(a) Method and calculation

$$P_C=\frac d{dt}(U\cos^2\omega t)=-\omega U\sin2\omega t.$$$$\boxed{|P_C|_{\max}=\omega U}.$$

(b) Interpretation and further calculation

$$|\sin2\omega t|=1\Rightarrow\cos^2\omega t=\sin^2\omega t=1/2.$$$$\boxed{|q|=Q/\sqrt2,\quad|i|=I_{\max}/\sqrt2}.$$

Physical check

Largest transfer power occurs at equal field energies, not when either store alone is maximum.