The Traveling Electromagnetic Wave, Qualitatively — Question 6

Question 6

A charge moving with the wave sees partially opposing electric and magnetic forces

At an event a +x vacuum wave has $\mathbf E=E_*\hat y$ and $\mathbf B=(E_*/c)\hat z$. A positive test charge q has instantaneous velocity $\mathbf v=\beta c\hat x$, with $0<\beta<1$. Use the laboratory Lorentz force.

1. Find the total force and compare its electric and magnetic contributions.

2. Repeat for velocity -beta c along x and find the ratio of force magnitudes.

Question 6 - Solution

A charge moving with the wave sees partially opposing electric and magnetic forces

(a) Method and calculation

$$\mathbf F_E=qE_*\hat y,\quad\mathbf F_B=q\beta c\hat x\times(E_*\hat z/c)=-q\beta E_*\hat y.$$$$\boxed{\mathbf F_+=qE_*(1-\beta)\hat y}.$$

(b) Interpretation and further calculation

$$\mathbf v=-\beta c\hat x\Rightarrow\boxed{\mathbf F_-=qE_*(1+\beta)\hat y}.$$$$\boxed{|F_-|/|F_+|=(1+\beta)/(1-\beta)}.$$

Physical check

These are instantaneous laboratory forces; the test charge is not assumed to maintain a straight trajectory under the transverse force.