Electric Field Point Inside Charged Shell

Description:

Take a shell of charge with total charge on its surface be ‘q’ and the areal charge density be ‘σ’. Let  R be the radius of the shell. We need to calculate electric field due to a conducting shell at point ‘P’ which is somewhere inside the shell using gauss theorem.

Let the point ‘p’ be inside the sphere at a distance ‘r’ from its centre where ‘r < R’ and we need to find Electric field at that point.

• Consider a Gaussian surface in form of a concentric sphere of radius ‘r’ so that it passes through ‘P’. Electric field is radial (perpendicular to charged surface) with same distance to Gaussian surface. It is symmetrical at every point and area vector A makes 0. At every point.

• We Know

  ∮ E→.dS→ = q/ε₀

  According to gauss theorem ‘q’ is the charge inside the Gaussian surface. But in this case the charge within the surface is ZERO. It means q = 0.

  So, ∮ E→.dS→ 0

  t means either E = 0 OR dS = 0 but here area integration is not 0

  Since ∮ E.dS ≠ 0 (as dS has a certain finite value)

  So, E = 0

  So Gauss Theorem proves that if there is a metal conductor then Electric field inside it would always be ZERO.