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Question

# (a) Use Gauss's theorem to find the electric field due to a uniformly charged infinitely large plane thin sheet with surface charge density σ (b) An infinitely large thin plane sheet has a uniform surface charge density +σ . Obtain the expression for the amount of work done in bringing a point charge q from infinity. to a point, distant r, in front of the charged plane sheet.

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Solution

## (a) Let electric charge be uniformly distributed over the surface of a thin non- conducting infinite sheet. Let the surface charge density be σ. According to Gauss theorem : ∫1→E.→ds+∫2→E.→ds+∫3→E.→ds=qϵ0∫1Eds cos0∘+∫2Eds cos0∘+∫3Eds cos90∘=qϵ0 ∫1Eds+∫2Eds =qϵ0 2∫Eds =qϵ0 2E.S=σSϵ0 E=σ2ϵ0 (b) The electric field due to a uniformly charged infinitely large thin sheet with surface charge density σ is, E=σ2ϵ0 ...(i) The amount of work done in bringing a point charge q from infinity to a point, at a distance r is given by W=∫r∞F.dr Where F is the force experienced by charge q. Now, W=q∫r∞−E.dr [∵F=−Eq]W=−qσ2ϵ0∫r∞dr [from (i)]W=qσ2ϵ0[∞−r]=∞

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