An infinitely long insulating cylindrical shell has an inner radius a, an outer
ID: 1524363 • Letter: A
Question
An infinitely long insulating cylindrical shell has an inner radius a, an outer radius b, and an unknown uniform positive charge density ? (charge per unit volume) distributed in the region between r = a and r = b.
(a) Using Gauss’s law, find the electric field in the hollow inner region r < a. Begin with a statement of Gauss’s Law and justify all steps leading to your answer.
(b) Suppose the electric field at the outer edge of the cylindrical shell (i.e., at r = b) is measured, and is found to have a magnitude of E0. Use Gauss’s law to express the charge density ? in terms of the quantities E0, a, b, and any fundamental constants you may need. Leave your answer in symbolic form.
Explanation / Answer
a) Imagine a Gaussian cyllinder with radius r(<a) with length L.
charge enclosed by the gaussian cyllinder, Q_in = 0
now use, Gauss'a law,
net elctric flux through the closed surface gaussian surface = Q_in/epsilon
E*pi*r^2*L = 0
==> E = 0
b) Imagine a Gaussian cyllinder with radius r(=b) with length L.
charge enclosed by the gaussian cyllinder, Q_in = charge density*volume of the cyllinder
= rho*pi*(b^2 - a^2)*L
now use, Gauss'a law,
net elctric flux through the closed surface Gaussian surface = Q_in/epsilon
Eo*pi*b^2*L = rho*pi*(b^2 - a^2)*L/epsilon
==> rho = E*b^2*epsilon/(b^2 - a^2) <<<<<<<-------------Answer