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Question

# The nuclear charge (Ze) is non-uniformly distribute within a nucleus of radius R. The charge density ρ(r) [charge per unit volume] is dependent on the radial distance r from the centre of the nucleus as shown in figure. Select correct alternative/s.

A
Electric field at r=R is independent of b
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B
Electric potential at r=R is proportional to b
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C
Electric field at r=R is proportional to a
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D
Electric potential at r=R is proportional to a
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Solution

## The correct option is D Electric field at r=R is independent of b According to Gauss I aw ∮E⋅dS=qinϵ0⟶ (i) also, qin =∫ρdV⟶(ii)ρ=Charge density dV → small wlume From graph, ρ=dar(0⩽r⩽a)ρ=d(a⩽r⩽R and also a<b<R)V=43πr3 differentiating dV=4πr2drqin =∫ρdV=∫a0dra4πr2dr+∫Radyπr2dr=4πd×(a4−0)a×4+4πdR3−a33=4xd(a34+R33−a33)qin=4πd(R33−a312)3from eq(i)E×43πR3=qin/ε0E=d(1−a34R3)⟶( independent of b)

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