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?

To find the electric field at a radius just outside the sphere, directly over the center of the circular, cut-out hole, you can use Gauss's Law. Gauss's Law states that the total electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of the medium.

In this case, let's consider a Gaussian surface in the shape of a sphere with its center at the center of the hole and its radius just outside the sphere. The Gaussian surface will enclose the entire charge of the spherical shell except for the charge in the cut-out hole.

To calculate the electric field, we need to find the electric flux through the Gaussian surface and then divide it by the surface area of the Gaussian surface. The equation for electric flux is given by:

Flux = E * A

Where E is the electric field and A is the area of the surface. For a sphere, the formula for the surface area is A = 4πr^2, where r is the radius of the sphere.

In this case, the electric field is the same at every point on the Gaussian surface, as the spherical shell has uniform charge density. So we can take E out of the integral notation.

The total charge enclosed by the Gaussian surface is the charge of the spherical shell, which is given by Q = 4/3πR^3σ, where R is the radius of the spherical shell and σ is the charge density.

Using Gauss's Law, we have:

Flux = E * 4πr^2
Charge enclosed = 4/3πR^3σ

According to Gauss's Law, Flux = Charge enclosed / ε0, where ε0 is the permittivity of free space.

Equating the two equations, we get:

E * 4πr^2 = (4/3πR^3σ) / ε0

Simplifying, we find:

E = (1 / (4πε0)) * ((4/3πR^3σ) / (4πr^2))

Canceling out the common factors, we get:

E = (1 / (3ε0)) * (R^3σ / r^2)

Therefore, the electric field at a radius just outside the sphere, directly over the center of the circular cut-out hole, is (1 / (3ε0)) * (R^3σ / r^2).