An Angular Method to Calculate the Electric Field of a Long Cylindrical Shell

Abstract

The electric field inside a uniformly charged long cylindrical shell is a standard example discussed in introductory university physics. The field is zero inside the cylinder; outside of the cylinder, the field behaves like a long wire at the center carrying the same amount of charge per unit length, analogous to that of a uniformly charged spherical shell, typically referred to as Newton’s shell theorem. It is usually proven with the use of Gauss’s law and cylindrical symmetry of the electric field. A more direct proof by integration is lacking at this level. In fact, students often have difficulties visualizing intuitively why the field at off-center points is cancelled out completely and behaves like a line charge at the center on the outside.