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Q3.27P

Expert-verified
Found in: Page 154

### Introduction to Electrodynamics

Book edition 4th edition
Author(s) David J. Griffiths
Pages 613 pages
ISBN 9780321856562

# A sphere of radius R, centered at the origin, carries charge densitywhere k is a constant, and r, are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

The approximate potential for points on the z axis, far from the sphere of radius R, centered at the origin, carries charge density R is

.

See the step by step solution

## Step 1: Given data

There is a sphere of radius R, centered at the origin carrying a charge density

R .

where k is a constant.

## Step 2: Volume element in spherical coordinates

The volume element in spherical polar coordinates is

## Step 3: Potential far from the sphere

The monopole term is

Here, is the permittivity of free space.

Substitute form of charge density and use equation (1),

The dipole term is

Substitute form of charge density and use equation (1),

Substitute form of charge density and use equation (1),

Thus, the approximate potential is

Thus, the potential is .