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. What is the potential difference between two shells (Delta V = V(r2) - V(r1))? A particle that has mass m and positive charge q (q > 0) is placed at r = r1 (right outside the inner shell but not touching) with zero initial velocity. The particle will be repelled by the inner shell. Find the velocity of the particle when it reaches the outer shell. Two large parallel (infinite planes) non-conducting planes have surface charge densities of sigma and - sigma (sigma > 0). Planes are distance d apart from each other. The origin of the coordinate system was set on the first plane as shown in the figure. Planes are perpendicular to z axis. Find electric potential difference between two planes. Please find the potential difference of Delta V =

Explanation / Answer

a)

E = sigma/(2 pi r epsilon0)

V = -int{E dr} = -sigma/(2 pi epsilon0) Ln(r)

integratio from r1 to r2:

dV = -sigma/(2 pi epsilon0) (Ln(r2) - Ln(r1)) = sigma/(2 pi epsilon0) Ln(r1/r2)

b)

0.5 m v^2 = q sigma/(2 pi epsilon0) Ln(r1/r2)

==> v = sqrt(2 q sigma/(2 pi m epsilon0) Ln(r1/r2))

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a)

E = sigma/2epsilon0 + sigma/2epsilon0 = sigma/epsilon0

V = E d = (d sigma)/(epsilon0)

b)

0.5 m v^2 = q V

==> v = sqrt(2 q d sigma/(m epsilon0))

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