Question #42653

A solid non-conducting sphere of mass $M$, radius $R$, and total positive charge $Q$ is placed on a rough inclined plane that makes an angle $heta = 37^\circ$ with the horizontal. The incline slopes upwards to the right. The sphere has a non-uniform internal charge distribution such that its Effective Center of Charge (ECC) is located at a distance $R/2$ vertically below its Center of Mass (CM). A uniform horizontal electric field of magnitude $E$ is applied in the rightward direction. The incline is sufficiently rough to prevent slipping. If the sphere remains in both translational and rotational equilibrium, what is the required magnitude of the electric field $E$? (Take $\sin 37^\circ = 3/5$ and $\cos 37^\circ = 4/5$)