A spherical shell of uniform charge density σ has a circular hole cut out of it.What is the Electric Field at a radius just outside the sphere, directly over the center of the circular, cut-out hole? Type "sigma" for σ and "epsilon_o" for ϵo. HINT: the hole is small enough that you can treat it as flat, and the point at which you are calculating the field is so close to the hole that it can be approximated as an infiniate plane.

Answer 1

Without the hole the field would be sigma/epsilon_0 directed in the radial direction. Then instead of cutting a hole, you can put a surface charge of
-sigma there. The field is then sigma/epsilon_0 plus the field due to the additional surface charge densisty of -sigma. That additional field is
-sigma/(2 epsilon_0), so the total field is:

sigma/(2 epsilon_0)

in the radial direction.