Two charged cylindrical shells share same axis. Shells have negligible thickness

Question

Two charged cylindrical shells share same axis. Shells have negligible thicknesses. The inner shell has constant surface charge density -sigma (sigma > 0) and radius r1. The outer shell has constant surface charge density sigma and radius r2.

a) What is the potential difference between two shells (V = V(r2 ) V(r1))?

b) A particle that has mass m and positive charge q (q > 0) is placed at r = r1 (right inside the outer shell but not touching) with zero initial velocity. The particle will be repelled by the outer shell. Find the veolcity of the particle when it reaches the inner shell.

Explanation / Answer

a)

electric field: E = sigma/(2 pi r e0)

thus, the potential difference:

    V = -int{E dr}

       = -sigma/(2 pi e0) ln(r)

integratio from r1 to r2:

    dV = -sigma/(2 pi e0) (ln(r2) - ln(r1))

        = sigma/(2 pi e0) ln(r1/r2)

b)

the velocity is,

    0.5 m v^2 = q sigma/(2 pi e0) ln(r1/r2)

   v = sqrt(2 q sigma/(2 pi m e0) ln(r1/r2))

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