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Q3.27P

Expert-verifiedFound in: Page 154

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 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

.

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

R .

where k is a constant.

The volume element in spherical polar coordinates is

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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