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Q3.27P

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Introduction to Electrodynamics
Found in: Page 154
Introduction to Electrodynamics

Introduction to Electrodynamics

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

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Short Answer

A sphere of radius R, centered at the origin, carries charge density

where 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.

Answer

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 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),

The quadrupole term is

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

Thus, the approximate potential is

Thus, the potential is .

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