Question #42022

A closed container is completely filled with a viscous incompressible fluid of density $\rho_f$. The container accelerates horizontally along the positive x-axis with a constant acceleration $\vec{a} = g\hat{i}$, where $g$ is the acceleration due to gravity (acting along $-g\hat{j}$). A small spherical particle of mass $m$, charge $+q$, and density $\rho_p = 2\rho_f$ moves in the fluid. A uniform magnetic field $\vec{B} = B_0\hat{k}$ is present in the region. The fluid exerts a drag force $\vec{F}_D = -b\vec{v}$, where $\vec{v}$ is the particle's velocity relative to the container. If the drag coefficient is $b = qB_0$, what is the steady-state terminal velocity $\vec{v}_T$ of the particle relative to the container?