Question

# A sphere of radius R has electric charge uniformly distributed in its entire volume. At a distance x from its centre for x<R, the electric field is directly proportional to

A
1x2
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B
1x
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C
x
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D
x2
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Solution

## The correct option is C xVolume charge density of sphere is, ρ=QV where, Q is the total charge distributed on the entire volume of sphere, V is the volume of the sphere. ⇒ρ=Q43πR3 Let us construct a spherical surface at a distance x from centre of sphere (x<R). Applying Gauss's law, ϕ=qinϵo here, charge enclosed by the sphere of radius x, qin=(43πx3) and electric flux through the sphere, ⇒ϕ=→E.d→A = ρ(43πx3)ϵo ⇒EA=ρ(43πx3)ϵo ⇒E×4π×x2=ρ(43πx3)ϵo Substituting the value of ρ, ⇒E×4π×x2=(3Q4πR3)(43πx3)ϵo ⇒E=Qx4πϵoR3 Thus, for given radius (R) and charge distribution (Q), E∝xHence, option (c) is correct answer.

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