## Solving for the electric field using Gauss’ theorem

Bianca Rae B. Sambo

Problem 1.4 (Classical Electrodynamics, 3rd Edition by Jackson)

Each of the three charged spheres of radius a has a total charge Q. One is conducting, one has a uniform charge density within its volume and one having a spherically symmetric charge density that varies radially as where (r>-3). Use Gauss’ theorem to obtain the electric fields both inside and outside the sphere.

SOLUTION:

A. Conducting Sphere

A.1 INSIDE

Note that no charge resides inside a conducting sphere. All charges reside in the outer surface thus

which implies that for r<a

A.2 OUTSIDE

Let r be the distance from the center of sphere a.

where is the solid angle.

where

B. SPHERE WITH UNIFORM CHARGE DENSITY

Since Q is specified to be the total charge then we can get an expression for

so thatB.1 INSIDE SPHERE B

Let b be the distance from the center of the sphere.

b is evaluated from 0 to r where r<a, from 0 to and from o to

but

B.2 OUTSIDE THE SPHERE

Outside the sphere, the total charge enclosed is still Q.

where r is evaluated from o to a

so that

where

C. SPHERE WITH

The charge density has the form where A is a constant. Now the total charge for sphere C should be Q thus

since r is evaluated from o to a

so that

and the charge density can be written as

where r is the distance from the center of sphere C

C.1 INSIDE SPHERE C

where r<a

C.2 OUTSIDE THE SPHERE

where r is evaluated from o to a

where