LC Oscillations, Quantitatively — Question 9

Question 9

Signed transported charge vanishes over a cycle while total passage does not

An ideal LC loop has $q=Q\cos\omega t$ and $i=\dot q$. Define signed transported charge as $\int i\,dt$ and total passage as $\int|i|dt$.

1. Find both quantities over one full cycle.

2. Find the cycle-average current magnitude and compare it with rms current.

Question 9 - Solution

Signed transported charge vanishes over a cycle while total passage does not

(a) Method and calculation

$$\int_0^Ti\,dt=q(T)-q(0)=\boxed{0}.$$$$\int_0^T|i|dt=Q\int_0^{2\pi}|\sin\theta|d\theta=\boxed{4Q}.$$

(b) Interpretation and further calculation

$$\langle|i|\rangle=4Q/T=2I_{\max}/\pi,\quad I_{\mathrm{rms}}=I_{\max}/\sqrt2.$$$$\boxed{\langle|i|\rangle/I_{\mathrm{rms}}=2\sqrt2/\pi=.900316}.$$

Physical check

Zero signed transfer does not imply that no charge moved through the wire.