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Question:

The electrostatic potential inside a charged spherical ball is given by φ=ar² + b where r is the distance from the centre; a, b are constants. Then the charge density inside the ball is :

6aε₀

6aε₀r

4πaε₀r

4πaε₀

Solution:

Potential inside=ar²+b
∴Er=-(dv/dr)=-6ar
Electric field inside uniformly charged solid volume varies with 'r'. So, charge density is constant.
φnet=(6ar)4πr²ε₀/4πr³/3 = σ × 4πr³/3ε₀
σ=6aε₀