Worked exercise — ion deflector

Problem

A charge $q = 10^{-3}\;\text{C}$ of mass $m = 0{,}2\;\text{kg}$ is fired horizontally rightwards with velocity $v = 10\;\text{m/s}$. It enters a region of width $L = 2\;\text{m}$ in which there is a magnetic field pointing out of the page $\vec{B} = (0,0,B)$. What is the minimum value of $B$ for which the charge does a U-turn and comes back without leaving the region?

Solution

The charge traces a semicircle inside the region and comes back only if the diameter of the circle does not exceed the width $L$, i.e. if the radius is less than half the width. Otherwise the charge exits the other side before reversing.

Condition: $r = \frac{mv}{qB} < \frac{L}{2} = 1\;\text{m}$

Substituting the data: $\frac{0{,}2\cdot 10}{10^{-3}\cdot B} < 1 \quad\Longleftrightarrow\quad B > \frac{2}{10^{-3}}$

$\ev{B > 2000\;\text{T}}$

A field of at least $2000\;\text{T}$ is needed: an enormous value (the most powerful laboratory magnets reach $\sim 50\;\text{T}$). The charge in this problem is too heavy to be easily deflected: the exercise shows how the formula works, but also how much orders of magnitude matter.

Links

Topics: Magnetism Concepts: Lorentz force · Uniform circular motion