Question

A total charge Q is distributed uniformly into a spherical volume of radius R. Find the electrostatic energy of this configuration.

A
Q24πϵ0R
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B
3Q25πϵ0R
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C
3Q220πϵ0R
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D
3Q210πϵ0R
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Solution

The correct option is B 3Q220πϵ0RGiven a total charge Q is distributed uniformly into a spherical volume of radius R. We have to find the electrostatic energy of this configuration.For this we use the formula, the electrostatic energy Vo=∫ε02E2dlwhere, ε0= permeability of free spaceE= electric fielddl= Volume element For this the electric field can be easily found from Gauss' Theorem to be, E=Q4πε0×⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩rR3(r<R)1r2(r>R)⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭The two intergrals needed are:∫R0(rR3)24πr2dr=4π5R∫∞R(1r2)24πr2dr=4πRSo, we obtain Vo=∫∞0ε02E24πr2dr [dτ=4πr2dr]So, Vo=ε02(Q4πε0)2(4π5R+4πR)Vo=3514πε0Q2R=320Q2πε0R

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