Question

The electric field of a plane electromagnetic wave is given by $\vec{E} = E_0 \frac{\hat{i}+\hat{j}}{\sqrt{2}} cos(kz + \omega t)$. At $t=0$, a positively charged particle is at the point $(x,y,z) = (0,0,\frac{\pi}{k})$. If its instantaneous velocity at $t=0$ is $v_0 \hat{k}$, the force acting on it due to the wave is

Answer 1

$\vec{F}=-\frac{qE_0}{\sqrt2}\left(1+\frac{v_0}{c}\right)(\hat{i}+\hat{j})$

Answer 2

Evaluate the electric field at the given time and position to get the electric force. Determine the magnetic field using $\vec{B}=\frac{1}{c}\,\hat{n}\times\vec{E}$. Calculate the magnetic force $q\vec{v}\times\vec{B}$ and add to the electric force.