Understand equations of a conducting sphere Can somebody explain to me, when the following two equations (equations 2.48 and 2.50 in this document) are applicable and what $\Phi_s$ and $\Phi$ actually are? The thing is, I want to find general equations that determine the field produced by conducting spherical sphere in an external field and was wondering whether these are the equations I am looking for.
$$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{a}{r} \right )^{l+1}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r>a$$
$$\Phi (r,\theta,\phi)=\sum_{l,m}\left(\frac{r}{a} \right )^{l}Y_{lm}(\theta,\phi)\oint\Phi_{s}(\theta',\phi')Y^*_{lm}(\theta',\phi')d\Omega',\,\,\,\,r<a$$
Or is it rather this equation (equation (17.3) in this document), probably they are one and the same:
$$\Phi (\mathbf{x})=-\frac{1}{4\pi}\int_S \Phi(\mathbf{x}')(\hat n.\nabla')G_D(\mathbf{x},\mathbf{x}')dS'$$
 A: If I understand the situation correctly, you have a metal sphere with a total charge of $Q$ in an external uniform field. ($Q$ can be zero; It's the general case). Assume the field's directions is $\hat z$.

Since we usually choose the center of the sphere as zero potential point, to make the problem simpler we can substitute the charged sphere with a grounded sphere and a charge $Q$ at the origin. Nothing changes.
Now we solve the problem for the grounded sphere without considering the charge at the center and then add the two potentials:
$$\text{boundary conditions: }\,\cases{V=0 \,\,\,\,\,\,\,\,\,\,\ r=R \\ V \to -E_0 r \cos \theta \,\,\,\,\,\,\,\,\,\,\  r\gg R }$$
$$V(r,\theta)=\sum_{l=0}^{\infty}\left(A_lr^l+B_lr^{-(l+1)}\right)\mathrm{P}_l(\cos \theta)  $$
Applying these boundary conditions we will have:
$$V(r,\theta)=-E_0\left( r-\frac{R^3}{r^2}\right)\cos \theta$$
Now we add the effect of the charge $Q$:
$$V_{total}(r,\theta)=-E_0\left( r-\frac{R^3}{r^2}\right)\cos \theta+ \frac{1}{4\pi \epsilon_0}\frac{Q}{r}$$
