Question

An electron is trapped in a magnetic field B, & circulating w/ velocity v @ a radius of r. Find an expression for e/m ratio as a function of the frequency of its circular motion.

Answer 1

$$\begin{array}{l} F = q B v \sin 90{}^{\circ} = q v B \\ \frac{m v^{2}}{R} = e v B \gg \frac{m v}{R} = e B \gg R = \frac{m v}{e B} \\ T = \frac{2 \pi R}{v} = \frac{2 \pi m}{e B} \gg v = \frac{1}{T} = \frac{e B}{2 \pi m} \gg \frac{e}{m} = \frac{2 \pi v}{B} \\ \end{array}$$

Answer: $\frac{e}{m} = \frac{2 \pi v}{B}$