LC Oscillations, Qualitatively — Question 10

Question 10

Reversing current reconstructs the earlier charge history

At some instant an ideal LC oscillator has state $(q_*,i_*)$. An idealized external operation reverses current without changing its magnitude or the capacitor charge. Ignore the switching implementation.

1. Compare the energy immediately before and after reversal.

2. Using the time-reversal symmetry of $\ddot q=-q/(LC)$, write the subsequent charge in terms of the original trajectory.

Question 10 - Solution

Reversing current reconstructs the earlier charge history

(a) Method and calculation

$$U_- =q_*^2/(2C)+Li_*^2/2,\quad U_+=q_*^2/(2C)+L(-i_*)^2/2.$$$$\boxed{U_+=U_-}.$$

(b) Interpretation and further calculation

$$q_{\mathrm{new}}(t)=q_{\mathrm{old}}(-t)\Rightarrow q_{\mathrm{new}}(0)=q_*,\quad\dot q_{\mathrm{new}}(0)=-i_*.$$$$\boxed{\ddot q_{\mathrm{new}}=-q_{\mathrm{new}}/(LC)}.$$

Physical check

Current reversal mirrors the initial state across the charge axis; subsequent charge values retrace the earlier charge history.