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Physics / Introduction to Electrodynamics 4 / Chapter 3 / Problem 30P

Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths

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In Ex. 3.9, we derived the exact potential for a spherical

Chapter 3, Problem 30P

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QUESTION:

Problem 30P

In Ex. 3.9, we derived the exact potential for a spherical shell of radius R, which carries a surface charge σ = k cos θ.

(a) Calculate the dipole moment of this charge distribution.

(b) Find the approximate potential, at points far from the sphere, and compare the exact answer (Eq. 3.87). What can you conclude about the higher multipoles?

Reference equation 3.87

Reference example 3.9

A specified charge density σ0(θ) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere.

Questions & Answers

QUESTION:

Problem 30P

In Ex. 3.9, we derived the exact potential for a spherical shell of radius R, which carries a surface charge σ = k cos θ.

(a) Calculate the dipole moment of this charge distribution.

(b) Find the approximate potential, at points far from the sphere, and compare the exact answer (Eq. 3.87). What can you conclude about the higher multipoles?

Reference equation 3.87

Reference example 3.9

A specified charge density σ0(θ) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere.

ANSWER:

a.)

Step 1 of 2

We have to calculate the dipole moment of the given charge distribution, that is

By symmetry is clearly in the $z$ direction

Therefore, the dipole moment of the given charge distribution is .

b.)

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