Solved Griffiths Problem 3.28 - What to Conclude?

[ehrenfest]

[SOLVED] Griffiths Problem 3.28

Homework Statement

Please stop reading unless you have Griffiths E and M book.

In this problem, I found that the approximation agrees with the exact potential. I am not sure what to conclude about higher multipoles. Are they all identically zero or do they all cancel? Is there something about this charge distribution that makes that happen? Could I have predicted that without calculating it?

Homework Equations

The Attempt at a Solution

 

[pam]

It is a pure dipole.
The cos\theta tells you it is pure P_1(cos\theta).

 

[ehrenfest]

That is definitely not the definition of a pure dipole that they gave me here https://www.physicsforums.com/showthread.php?p=1622972#post1622972
They said that a pure dipole must be a point. Please clarify.

 

[pam]

That post gives the definition of a dipole used in elementary texts.
It describes one simple model of a dipole.
What I meant is that the field outside the sphere has only the dipole term as in the expansion. If you calculate the multipole moments of the sphere, you will find only a diple moment because of the P_1 dilstribution of charge.
Use Griffith's or some other text for definitions, not the web.