LC Oscillations, Qualitatively — Question 4

Question 4

Equal field energies do not require half the maximum charge

An ideal LC circuit has total energy U and maximum charge Q. Consider a state with equal electric and magnetic energies.

1. Find the normalized magnitudes of charge and current.

2. Compare this with the state $|q|=Q/2$.

Question 4 - Solution

Equal field energies do not require half the maximum charge

(a) Method and calculation

$$q^2/(2C)=Li^2/2=U/2.$$$$\boxed{|q|/Q=|i|/I_{\max}=1/\sqrt2}.$$

(b) Interpretation and further calculation

$$|q|=Q/2\Rightarrow U_C/U=1/4,\quad U_L/U=3/4.$$$$\boxed{|i|/I_{\max}=\sqrt3/2}.$$

Physical check

Energy fractions are squares of normalized amplitudes.