Wien velocity selector

Problem

A velocity selector has electric and magnetic fields orthogonal to each other and to the velocity of the charge. (a) Derive the relation between $E$, $B$ and the velocity $v$ that allows undeflected passage. (b) What happens to particles faster or slower than $v = E/B$?

Solution

(a) The charge experiences the electric force $F_E=qE$ and the magnetic force $F_B=qvB$, directed in opposite senses. The passage is undeflected when they balance: $qE=qvB \quad\Rightarrow\quad v=\frac{E}{B}$

$\ev{v = \dfrac{E}{B}}$

(b) For other velocities the balance is broken:

• faster particles ($v>E/B$): the magnetic force $qvB$ dominates, the particle deflects to one side;
• slower particles ($v<E/B$): the electric force $qE$ dominates, the particle deflects to the opposite side.

Only particles with $v=E/B$ cross the selector in a straight line; the others end up against the walls. The device therefore selects a single velocity.

Links

Topics: Magnetismo Concepts: Forza di Lorentz · Campo magnetico