A Charge in a Magnetic Field
A particle of mass $m$ and charge $+q$ enters a region of uniform magnetic field $\vec B$
directed into the page, moving in the plane of the page with speed $v$.
1. Show that the magnetic force does no work, and find the radius $r$ and period $T$ of the
resulting motion. What is surprising about $T$?
2. A uniform electric field $\vec E$ is now added so that the particle can pass through
undeflected. For what speed $v$ does this happen, and why is this useful?
Check your answer
Show answer & solution
Answer
$ r = \dfrac{mv}{qB} $
Solution
(a) No work, then the radius. The magnetic force $\vec F = q\,\vec v \times \vec B$ is
always perpendicular to $\vec v$, so $\vec F\cdot\vec v = 0$ and the speed is constant. The
force supplies the centripetal acceleration:
$$qvB = \frac{mv^2}{r} \;\Rightarrow\; \boxed{r = \frac{mv}{qB}}, \qquad
T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}.$$
The period $T$ is independent of $v$ — the cyclotron's key property.
(b) Velocity selector. Add a uniform electric field $\vec E$ in the plane, perpendicular
to $\vec v$. The particle travels straight only when the electric and magnetic forces cancel,
$qE = qvB$, i.e. for the single speed
$$v = \frac{E}{B},$$
independent of both $q$ and $m$ — so the device selects a velocity, not a particle.