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Inside a charged spherical ball the electrostatic potential is phi = a*r² + b, where r is the distance from the centre and a, b are constants. The volume charge density inside the ball is:

1. -24*pi*a*eps0
2. -6*a*eps0
3. -24*pi*a*eps0*r
4. -6*a*eps0*r

Correct answer: -6*a*eps0

Solution

Using Poisson's equation in spherical symmetry: del²(phi) = (1/r²) d/dr(r² dphi/dr). With phi = a*r² + b, dphi/dr = 2ar, so r² dphi/dr = 2a*r³, and d/dr(2a*r³) = 6a*r². Dividing by r² gives del²(phi) = 6a. Then rho = -eps0 * del²(phi) = -6*a*eps0.

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