Question #85399

An infinitely extended non-conducting solid slab of thickness $2d$ (in $\text{m}$) has a uniform volume charge density $\rho > 0$ (in $\text{C/m}^3$). The slab is fixed at an angle $\theta$ to the horizontal. A solid conducting sphere of mass $m$ (in $\text{kg}$), radius $R$ (in $\text{m}$), and positive charge $Q$ (in $\text{C}$) is projected directly up the upper surface of the slab with an initial center-of-mass velocity $v_0$ (in $\text{m/s}$). The coefficient of static friction between the sphere and the slab is $\mu$. Assume the charge $Q$ is distributed such that it does not affect the uniform field of the slab, and $\mu$ is sufficiently large for the uncharged sphere to roll. The acceleration due to gravity is $g$ (in $\text{m/s}^2$) and the permittivity of free space is $\epsilon_0$ (in $\text{C}^2/(\text{N}\cdot\text{m}^2)$). Which of the following gives the maximum charge $Q_{\text{max}}$ for which the sphere will roll without slipping during its ascent, and the maximum distance $L$ it travels up the incline before coming to instantaneous rest?