Applying Gauss' Law: Cylindrical Symmetry — Question 10

Question 10

Circular motion around an infinite line has a radius-independent speed

A particle of mass m and charge $-q$ with $q>0$ moves in the transverse plane of an infinite line with $\lambda>0$. Ignore radiation, gravity and other forces. Define $C=q\lambda/(2\pi\epsilon_0)>0$, so the inward force magnitude is C/r.

1. Find the speed and period of a circular orbit of radius $r_0$.

2. For fixed angular momentum $\ell$, use $m\ddot r=\ell^2/(mr^3)-C/r$ to find the frequency of small radial oscillations about the circular orbit and assess radial stability.

Question 10 - Solution

Circular motion around an infinite line has a radius-independent speed

(a) Method and calculation

$$\frac{mv^2}{r_0}=\frac C{r_0}\Rightarrow v=\sqrt{C/m}=\sqrt{\frac{q\lambda}{2\pi\epsilon_0m}}.$$$$T=2\pi r_0\sqrt{m/C},\quad\Omega=v/r_0.$$

(b) Interpretation and further calculation

$$\ell^2=mCr_0^2,\quad\left.\frac{d}{dr}\left(\frac{\ell^2}{mr^3}-\frac Cr\right)\right|_{r_0}=-\frac{2C}{r_0^2}.$$$$r=r_0+\xi:\quad\ddot\xi+\frac{2C}{mr_0^2}\xi=0,\quad\omega_r=\sqrt2\,\Omega.$$The restoring linear coefficient makes the orbit radially stable within the stated classical model.

Physical check

The line field falls as $1/r$, so the r factors cancel in the circular speed but not in the orbital period.