Physics Exercises/Electrostatics

1. A very long solid nonconducting cylinder of radius R₀ and length L (R₀ << L) possesses a uniform volume charge density ρ_E (C/m³). For points far from the ends and for which r << L,
   1. Find electric fields for all r, distance from axis of cylinder. You may need separate expressions for r < R₀ and r > R₀. Show that electric field is continuous at r = R₀, i.e. two expressions for electric fields agree at r = R₀.
   2. Describe the motion of a small free charge Q of opposite sign (i.e. Q ρ_E < 0) placed inside the cylinder. Find the frequency of oscillation $\nu$ if the charge oscillates about the axis of cylinder. (Note: $\nu$, frequency, has units of cycles / second, while $\omega$, angular frequency, has units of radians / second.)
   3. Let's change the problem and say we have a cylindrical shell of charge (of radius R₀, and σ_E so that the total charge remains the same). How do the expressions for electric fields change for either regions r < R₀ or r > R₀? Is the electric field still continuous? Give a plausible argument why we might not expect electric field to be continuous.