Crossed Fields: Discovery of the Electron — Question 5

Question 5

A parallel electric-field component prevents exact force cancellation at every speed

Uniform electric and magnetic fields are prescribed with B nonzero. Seek a nonzero charged particle velocity for which $q(\vec E+\vec v\times\vec B)=0$.

1. Derive the necessary condition on E and B and find the full family of velocities when it holds.

2. If $\vec E=E_y\hat y+E_z\hat z$ and $\vec B=B\hat z$, explain why nonzero Ez cannot be compensated by choosing a different velocity.

Question 5 - Solution

A parallel electric-field component prevents exact force cancellation at every speed

(a) Method and calculation

$$(\vec E+\vec v\times\vec B)\cdot\vec B=\vec E\cdot\vec B=0.$$$$\vec v=\frac{\vec E\times\vec B}{B^2}+v_\parallel\hat B\quad\text{when }\vec E\perp\vec B.$$

(b) Interpretation and further calculation

$$\vec E\cdot\vec B=E_zB\ne0\quad\text{if }E_z\ne0.$$$$F_\parallel=qE_z\quad\text{for every }\vec v.$$

Physical check

A magnetic force is always perpendicular to B and cannot cancel electric acceleration along B.