Griffiths Example 3.8: Justifications for Claims

[ehrenfest]

[SOLVED] Griffiths Example 3.8

Homework Statement

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

In Example 3.8, Griffiths makes two claims without justification that I want justification for. First, he says that V=0 in the equatorial plane (I assume this means that x-y plane). Second, he says that $$V \to -E_0 r \cos{\theta}$$ for $$r >>R$$. Where does the cosine come from?

Homework Equations

The Attempt at a Solution

 

[Biest]

Homework Statement

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

In Example 3.8, Griffiths makes two claims without justification that I want justification for. First, he says that V=0 in the equatorial plane (I assume this means that x-y plane). Second, he says that $$V \to -E_0 r \cos{\theta}$$ for $$r >>R$$. Where does the cosine come from?

The problem is originally defined in cartesian coordinates. Now that we are using the Legendre Polynomials it has to be in spherical coordinates.

Orginally:
$$V \to -E_0 z$$
goes to
$$V \to -E_0 r \cos{\theta}$$
since
$$z = r \cos{\theta}$$

 

[ehrenfest]

And how do you know V=0 all over th equatorial plane?

 

[Biest]

It is an uncharged metal sphere, so we basically assume it was grounded beforehand

 

[ehrenfest]

I see. Thanks.