Three Simple Circuits — Question 8

Question 8

A pure capacitor returns as much energy as it absorbs each cycle

An ideal capacitor has $v=V_p\cos\omega t$ and $i=C\dot v$. Define $U_{\max}=CV_p^2/2$.

1. Find instantaneous power and its average.

2. Find total positive energy absorbed and total magnitude of returned energy during one full voltage cycle.

Question 8 - Solution

A pure capacitor returns as much energy as it absorbs each cycle

(a) Method and calculation

$$p=vi=-\omega CV_p^2\cos\omega t\sin\omega t=-\omega U_{\max}\sin2\omega t.$$$$\boxed{\langle p\rangle=0}.$$

(b) Interpretation and further calculation

$$U_C=U_{\max}\cos^2\omega t\quad\text{rises from }0\text{ to }U_{\max}\text{ twice per }T.$$$$\boxed{W_+=2U_{\max},\quad |W_-|=2U_{\max}}.$$

Physical check

Zero average power means reversible exchange, not zero instantaneous power.