A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε0) ˆn , where ˆn is the unit vector in the outward normal direction, and σ is the surface charge density near the hole.

Question 1.29 A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε0) ˆn , where ˆn is the unit vector in the outward normal direction, and σ is the surface charge density near the hole.

Answer:

Method -1

Let us consider a conductor with a cavity or a hole. Electric field inside the cavity is zero

Let E is the electric field just outside the conductor, q is the electric charge,

According to Gauss’s law, 

 

Therefore, the electric field just outside the conductor is

 These fields are equal and opposite inside the conductor, and equal in magnitude and direction outside the conductor

 

 Hence, proved.

Method -2