A solid sphere of radius R contains a total charge Q distributed uniformly throu

Question

A solid sphere of radius R contains a total charge Q distributed uniformly throughout its volume. If the charged sphere is allowed to disperse, this "self-energy" (energy needed to assemble this charge) will appear as kinetic energy of the individual charges. Assume the sphere is composed of copper, except that all of the positive ions have been removed, leaving only the conduction electrons. Let the sphere have radius R=1mm. What is the amount of "self-energy", in Joules, will be converted to kinetic energy when this charge is dispersed?

(Copper has one conduction electron per atom. Its mass density is 8.96g/cm^3, and its atomic mass is 63.546g/mol)

Explanation / Answer

You don't need a double or a triple with this one. Uniform volume charge density ?: k = 1 / 4p*epsilon_0 dUe = k q(r)*dq(r) / r dUe = k (4/3)pr^3?*4pr^2?*dr / r dUe = k (16/3) p^2 ?^2 r^4 dr Ue = k (16/3) p^2 ?^2 ? r^4 dr [0,r] Ue = k (16/15) p^2 ?^2 r^5 *(r/r) ==>q=(4/3)pr^3*? Ue = (3/5) k q^2 / r = (3/20) q^2 / (p*epsilon_0*r)

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