Question

A thin spherical shell of radius R has a uniform surface charge density σ. Using Gauss’ law, deduce an expression for electric field

1. outside and
2. inside the shell.

Answer 1

i. Outside the shell:

Let `vecE` be the electric field at P. The electric field through an elemental surface `vec(dS)` is `dphi = vecE * vec(dS)`

Total electric flux through the gaussian surface is

`phi = oint_S EdS = Eoint_SdS = E(4pir^2)`

Also, `phi = q/epsilon_0`

⇒ `E(4pir^2) = q/epsilon_0`

⇒ `E = 1/(4piepsilon_0) q/r^2`  ...(for r > R)

ii. Inside the shell:

E(4πr²) = 0

E = 0