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Question

The electric potential at the centre of hemisphere of radius R having uniform surface charge density σ is

A
σRϵ0
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B
2σRϵ0
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C
σR2ϵ0
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D
σR3ϵ0
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Solution

The correct option is C σR2ϵ0
Consider a very small shaded element on the hemisphere which is subtended by an angle dθ with centre as shown in figure,

The area of shaded element is
dA=2πrRdθ

From figure, r=Rsinθ, so
dA=2π(Rsinθ)Rdθ
dA=2πR2sinθ dθ

Charge on this element,
dq=σ×2πR2sinθ dθ

Potential at the centre C due to this element,
dV=14πϵ0dqR

Substituting the value of dq, we get
dV=14πϵ0σ2πR2sinθ dθR

dV=σR2ϵ0sinθ dθ

Now the potential at C due to whole hemisphere,
dV=σR2ϵ0π20sinθ dθ

V=σR2ϵ0

Hence, option (c) is the correct answer.

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