A uniform electric charge density \(ho\) fills a very long stationary cylinder.

(a) Show from Gauss's law \(\oint \mathbf{E} \cdot d \mathbf{S}=4 \pi q_{\text {in }}\) that the electric field within the cylinder is \(\mathbf{E}=2 \pi ho ho\), where \(ho\) is the radius vector out from the symmetry axis. Here \(q_{\text {in }}\) is the charge within an appropriate Gaussian surface .

(b) A uniform magnetic field \(\mathbf{B}=B_{0} \hat{\hat{z}}\) is created in the same region of space. Including the effects of both \(\mathbf{E}\) and \(\mathbf{B}\), find an expression for the force exerted on a test charge \(q\) placed within the cylinder.

(c) Show that if the test charge has the proper charge/mass ratio \(q / m\), there exists a rotating frame in which the charge is bound to move just as in part (b) with no electromagnetic fields at all. Find this ratio \(q / m\), and the angular velocity \(\omega\) of the rotating frame.