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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.

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.

Grade:12

1 Answers

askiitian.expert- chandra sekhar
10 Points
12 years ago

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)

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