Question #43093

A particle of mass $m$ and charge $q$ is embedded in a viscous medium that exerts a drag force $\vec{F}_d = -b\vec{v}$, where $b$ is a positive constant and $\vec{v}$ is the particle's velocity. A plane electromagnetic wave propagates through the medium in the $+z$ direction. At the particle's location, the electric and magnetic fields are given by $\vec{E} = E_0 \cos(\omega t) \hat{i}$ and $\vec{B} = \frac{E_0}{c} \cos(\omega t) \hat{j}$, where $c$ is the speed of light in the medium. Assume the particle's displacement is much smaller than the wavelength of the EM wave, so the spatial variation of the fields can be neglected. Furthermore, assume the velocity induced in the $z$-direction is negligible compared to that in the $x$-direction. After a long time, the particle reaches a steady state. What is the time-averaged magnetic Lorentz force $\langle F_z \rangle$ acting on the particle in the $+z$ direction?