A uniform surface charge density σ exists over the entire x-y plane except for a circular hole of radius a centred at the origin. Find the expression for the electric field at point p(0,0,z) on the z axis.

Hi Srijit,

Applying gauss law considering there is no hole then the Electric field at (0,0,z) is E1=σ/2ε

let us consider a  disc of radius 'a' centered at origin having charge density -σ and field due to this is E2

then E2 = ?

For ring of radius a ,E= Qz/4πε(a²+z²)^3/2

for disc of radius a,E=∫dE = ∫^a(σ2πaz/4πε(a²+z²)^3/2) da = σ/2ε - (σ/2ε)(z/(a²+z²)^1/2)

therefore E2 =- σ/2ε + (σ/2ε)(z/(a²+z²)^1/2)

therefore electric field at (0,0,z) on z axis is E1+E2= σ/2ε - σ/2ε + (σ/2ε)(z/(a²+z²)^1/2) = σz/2ε√(a²+z²)