In class we calculated the electrostatic self-energy of a uniform-density sphere

Question

In class we calculated the electrostatic self-energy of a uniform-density sphere of charge, to determine the classical radius of the electron. In homework you calculated the self-energy of a spherical shell of charge, viz., the electrostatic self-energy of the Earth. Now find an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution rho(r). Remember that you must justify what you write to get credit. You may not assume that rho(r) represents any point charge, or that it is constant, or that it is piecewise constant, or that it does or does not cut off at any finite radius r. Your expression must cover all possibilities. Your expression may include an integral or integrals which cannot be evaluated without knowing the specific form of rho(r). If so, be sure to distinguish between limits of integration and dummy variables of integration.

Explanation / Answer

dq=pX4piXx^2dx U=1/2 Integration (p4pix^2dx X p(3R^2-x^2)/6epsilon ) limit from 0 to R U= 4piXp^2XR^5/15epsilon

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