A Circulating Charged Particle — Question 5

Question 5

Opposite charge signs produce mirror orbits with a predictable separation

Two particles of equal mass m and charges +q and -q start together at the origin with velocity $v\hat x$. A uniform field $-B\hat z$ fills the plane. Define $r=mv/(qB)$ and $\omega=qB/m$, with q positive.

1. Find the position of each particle as a function of time.

2. Find their separation and velocity directions at one quarter of the common orbital period.

Question 5 - Solution

Opposite charge signs produce mirror orbits with a predictable separation

(a) Method and calculation

$$x_+(t)=x_-(t)=r\sin\omega t.$$$$y_+(t)=r(1-\cos\omega t),\quad y_-(t)=-r(1-\cos\omega t).$$

(b) Interpretation and further calculation

$$t=T/4:\quad\vec r_\pm=r\hat x\pm r\hat y,\quad|\vec r_+-\vec r_-|=2r.$$$$\vec v_+=v\hat y,\quad\vec v_-=-v\hat y.$$

Physical check

Changing charge sign reverses curvature while leaving the radius and period magnitudes unchanged.