Suppose that you have a sphere of radius $R$ and uniform charge density $\rho$; a cylindrical hole with radius $a$ ($a\ll R$) is drilled through the center of the sphere, leaving it like a "necklace bead".
I would like to find a function for the electric field (1) very far away from the sphere ($r\gg R$) and (2) inside the hole, near the center of the bead $r\ll R$.
In case (1), I simply treat it as a point charge and calculating the electric field is trivial.
However, I am uncertain how to approach part (2) and would appreciate any assistance. The combination of spherical and cylindrical geometries seems to make this quite tricky. I am unsure what approximation or simplification to make from the knowledge that $r\ll R$.
Would it perhaps be correct to find the electric field from (1) a complete, uniformly charged sphere and (2) a cylinder of charge density $-\rho$? Summed together, the charge densities would result in our original "bead" system, so then I can just add together the expressions for the electric field. Doing case (1) is quite easy, but (2) is nontrivial for positions that are not along the axis of the cylinder, but perhaps due to our condition that $r\ll R$ and $a\ll R$, we can assume that the field from the cylinder along the $z$-axis is a good enough approximation.