How Does a Dipole Inside a Conducting Shell Affect the Electric Field?

[ShayanJ]

Homework Statement

There is a uniform electric field with strength $E_0$.Then we place a conducting spherical shell with a dipole at the center of it,inside the field,somehow that the dipole moment is prependicular to the initial field.Find the potential/electric field of the new configuration.(It should be solved in spherical coordinates $(r,\theta,\phi)$ and the answer is independent of $\phi$.)

Homework Equations

$\phi(r,\theta)=\Sigma _{l=0} ^∞ (A_l r^l+\frac{B_l}{r^{l+1}})P_l(cos \theta)$
Where $A_l$ and $B_l$ are constants to be determined and $P_l(x)$ are the legendre polynomials.
I'm not sure,but maybe it can be solved by the method of image charges too.

The Attempt at a Solution

I tried to use the formula above but I got so confused about the boundary conditions and also how to use them to calculate As and Bs.

 

[Mindscrape]

A good start is to look at the potential inside and outside the sphere. On the outside, for example, you can look at r>>R, and on the inside you know that the only thing allowed to blow up is the dipole.

 

[ShayanJ]

Thanks
I tried that,but I thought r>>R means $r \rightarrow \infty$ because otherwise it wouldn't help much.We know that the potential at infinity is $-E_0 r cos\theta + A_0$ and so I got $A_1=-E_0$ and $A_{l>1}=0$.
Then we know that the spherical shell is an equipotential and we can assume it to be zero.
From here,I calculated Bs from l=0 to l=2,using the orthogonality formula for legendre polynomials.They were zero so I suspect all Bs are zero(is it right?). But this means that the potential function is just $-E_0 r cos\theta + A_0$,which obviously is not.
What have I done wrong?

Can we say that inside the shell,the uniform field is canceled by the rearrangement of charges on the sphere and the only field inside is the field of dipole?

Sorry,I know it makes the problem really different,but what if the sphere was dielectric?

Thanks

 

[Mindscrape]

Not quite, what is the potential term of a dipole? Is there a B_l that will match that term? Hmm, I honestly kind of forget how these problems work with dielectrics.

 

[ShayanJ]

I guess the presence of dipole only changes the boundary conditions which are used to determine the constants and there doesn't need to be sth we call the term associated to the dipole.

Thanks