# An insulating solid sphere of the radius R is charged in a non - uniform manner such that the volume charge density ρ=Ar, where A is a positive constant and r is the distance from the centre. The potential difference between the centre and surface of the sphere is

A

AR8ε0

B

AR4ε0

C

ARε0

D

AR2ε0

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## To find the potential difference between the center and the surface of an insulating solid sphere with a non-uniform charge density given by ρ=Ar, we can follow these steps:Step 1: Understand the Charge DensityThe charge density ρ is given as ρ=Ar, where A is a constant and r is the distance from the center of the sphere. This indicates that the charge density decreases as we move away from the center.Step 2: Calculate the Charge ElementTo find the electric field, we need to calculate the charge dq in a thin spherical shell of radius r and thickness dr. The volume of this shell is:dV=4πr2drThus, the charge dq in this shell can be expressed as:dq=ρ⋅dV=(Ar)⋅(4πr2dr)=4πArdrStep 3: Calculate the Total Charge EnclosedTo find the electric field at a distance r from the center, we need to consider the total charge Qenc enclosed within a sphere of radius r:Qenc=∫r0dq=∫r04πAr′dr′=4πA((r′)22)r0=2πAr2Step 4: Apply Gauss's LawUsing Gauss's law, we know that the electric field E at a distance r from the center is given by:Φ=E⋅A=Qencϵ0where A=4πr2 is the surface area of the sphere. Thus, we have:E⋅4πr2=2πAr2ϵ0From this, we can solve for E:E=2A4ϵ0=A2ϵ0Step 5: Calculate the Potential DifferenceThe potential difference V between the center and the surface of the sphere can be calculated using the relationship:ΔV=−∫0REdrSubstituting the expression for E:ΔV=−∫0RA2ϵ0dr=−(A2ϵ0r)0R=−(0−AR2ϵ0)=AR2ϵ0Final AnswerThus, the potential difference between the center and the surface of the sphere is:ΔV=AR2ϵ0

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