LC Oscillations, Qualitatively — Question 8

Question 8

A normalized state must lie on its conserved energy ellipse

An LC oscillator is known to have maximum charge Q and peak current Imax. A sensor reports $q=.8Q$ and $i=.8I_{\max}$ simultaneously.

1. Test whether these readings are compatible with the stated energy.

2. If the charge reading is correct, find the permitted current magnitudes and the reported fractional energy excess.

Question 8 - Solution

A normalized state must lie on its conserved energy ellipse

(a) Method and calculation

$$\frac{U_{\mathrm{inferred}}}U=(q/Q)^2+(i/I_{\max})^2=.64+.64=1.28.$$$$\boxed{\text{The reported state is inconsistent with the stated energy.}}$$

(b) Interpretation and further calculation

$$(i/I_{\max})^2=1-.8^2=.36.$$$$\boxed{|i|=.6I_{\max},\quad(U_{\mathrm{inferred}}-U)/U=.28}.$$

Physical check

The energy ellipse tests simultaneous measurements without requiring an oscillation phase.