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For a spherically symmetric charge distribution, the electric field at distance r from the centre is E = k*r⁷ (directed radially), where k is a constant. Find the volume charge density rho at distance r from the centre.

1. rho = 9*k*epsilon0*r⁶
2. rho = 5*k*epsilon0*r³
3. rho = 3*k*epsilon0*r⁴
4. rho = 7*k*epsilon0*r⁶

Correct answer: rho = 9*k*epsilon0*r⁶

Solution

By Gauss's law div E = rho/epsilon0. For a radial field with spherical symmetry, div E = (1/r²) d(r² E)/dr. With E = k r⁷, r² E = k r⁹, and d(k r⁹)/dr = 9 k r⁸. Dividing by r² gives 9 k r⁶, so rho = 9 epsilon0 k r⁶.

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