A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the holes is `(sigma//2 in_(0)` `hat(n)`, where `hat(n)` is the unit vector in the outward normal direction, and `sigma` is the surface charge density near ther hole.

Explanation

Let us first assume that there is no hole on the surface of the charged conductor. Now the magnitude electric field intensity everywhere on its surface can be written as `E=sigma/epsilon_0` , and this electric field is directed along the normal to the surface.
Now consider the location where we intend to cut the hole. Before cutting the hole we have zero net electric field intensity just inside the hollow conductor and `sigma//epsilon_0`, just outside the conductor in the vicinity of location of the hole. Just inside the hollow conductor, we can say that the electric field intensity due to the part of the conductor that has to be taken out to make the hole and the rest of the conductor gets cancelled, and outside the conductor these two fields are added. To make the net feld zero inside the expected location of the hole, the electric field intensity due to the material of the hole and the rest of the sphere must be the same in magnitude. Therefore, after we cut the hole, only the half electric field intensity will be there due to the rest of the system. Thus at the location of the hole, the electric field intensity will be `sigma//2epsilon_0`