LC Oscillations, Qualitatively — Question 9

Question 9

A small resistance selects energy loss without changing its sign

A nearly ideal LC oscillator has a small series resistance R. Use $L\dot i+Ri+q/C=0$ and $i=\dot q$.

1. Derive the instantaneous total-energy rate.

2. Compare energy loss at maximum charge with energy loss at zero charge during a cycle.

Question 9 - Solution

A small resistance selects energy loss without changing its sign

(a) Method and calculation

$$\dot U=(q/C)i+Li\dot i=i(q/C+L\dot i).$$$$\boxed{\dot U=-Ri^2\le0}.$$

(b) Interpretation and further calculation

$$|q|\ \text{at a turning point}:\ i=0\Rightarrow\dot U=0.$$$$\boxed{q=0,\ i\ne0:\ \dot U<0}.$$

Physical check

A zero instantaneous loss at a turning point does not mean that the full cycle is lossless.