Problem

A cylindrical capacitor consists of a cylindrical wire of radius , and a coaxial cylindrical shell of radius . Both the wire and cylindrical shell have length . The wire has a total charge of distributed on its surface. The cylindrical shell has a total charge of distributed on its surface.

Calculate the following:

Part 1: Electric field in the region

Part 2: Electric potential in the region

Part 3: Capacitance of the spherical capacitor.

Cylinder capacitor.gif

Solution

Part 1: Electric field

The first thing to calculate is the electric field between the spherical shells. Use Gauss' law

.

The total charge enclosed in a Gaussian surface between the wire and the cylindrical shell is . For cylindrical geometry

.

The electric field from the inner spherical shell emanates radially outward, so

.

Part 2: Electric potential

Part 3: Capacitance

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