Question

A hollow right angle cylinder of radius a and length L has both ends and the sides grounded except for a band in the middle of the side of the cylinder. The band in the middle is held at a potential of V0 and extends from the center of the side of the cylinder for a distance of L/4 in both directions. a) Find the potential everywhere inside the cylinder. b) Find the charge density on the surface of the cylinder.

Answer 1

(a) Inside the hollow cylinder, E = q/2pie₀ r H , for r > R where q is the charge, R is the radius of the cylinder, H is the height of the cylinder, e₀ is epsilon₀ . therefore, potential V will be

V = L [ E. dr] , where L is the integral, dr is the infinitesimal change in the radius

V = L^r₀ [ q/2pie₀r H dr] = ₀^r[q x logr /2pie₀ H] = q x logr/2pie₀H - q/2pie₀H

V = q/2pie₀H [ logr - 1]

(b) Surface charge density, s will be

V = 1/4pie₀ L[s/ r]

also, V = q/2pie₀ H [ log r - 1]

q/2pie₀ H [logr - 1] = 1/4pie₀ L[s/r]

2q/H [logr -1] = L[s/ r]

d[ 2q/H [logr -1]/dr = s/r

2q/H x 1/r = s/r

s = 2q/H .