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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 x
Volume 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.dA = 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),
ExHence, option (c) is correct answer.

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