LC Oscillations, Quantitatively — Question 3

Question 3

A specified signed charge selects different times on the outgoing and returning paths

An ideal loop starts with q=Q and i=0, so $q=Q\cos\omega t$. Consider the first complete cycle.

1. Find both times at which $q=-Q/2$ and give the current signs.

2. Find the duration for which q is below -Q/2.

Question 3 - Solution

A specified signed charge selects different times on the outgoing and returning paths

(a) Method and calculation

$$\cos\omega t=-1/2\Rightarrow\boxed{t_1=T/3,\quad t_2=2T/3}.$$$$i=-\omega Q\sin\omega t\Rightarrow\boxed{i(t_1)=-\sqrt3 I_{\max}/2,\ i(t_2)=+\sqrt3 I_{\max}/2}.$$

(b) Interpretation and further calculation

$$q<-Q/2\quad\text{for}\quad2\pi/3<\omega t<4\pi/3.$$$$\boxed{\Delta t=T/3}.$$

Physical check

The same signed charge can belong to either direction of current.