A sphere of radius $R$ carries total charge $Q$ distributed uniformly over its surface. Show that the energy stored in its electric field is $U=k Q^{2} / 2 R$.
Added by Jose Francisco C.
Step 1
The electric field outside a uniformly charged sphere is the same as if all the charge were concentrated at the center of the sphere. Therefore, the electric field at a distance r from the center of the sphere is given by E = kQ/r^2, where k is Coulomb's constant. Show more…
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A sphere of radius $R$ contains charge $Q$ spread uniformly throughout its volume. Use the result of Example $21.3$ to show that the electrostatic energy contained within the sphere is $Q^{2} / 40 \pi \epsilon_{0} R$ (equivalently, $k Q^{2} / 10 R$ ).
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(II) A nonconducting sphere of radius $r_{0}$ is uniformly charged with volume charge density $\rho_{\mathrm{E}} .$ It is surrounded by a concentric metal (conducting) spherical shell of inner radius $r_{1}$ and outer radius $r_{2},$ which carries a net charge $+Q$ . Determine the resulting electric field in the regions (a) $0<r<r_{0},$ (b) $r_{0}<r<r_{1},(c) r_{1}<r<r_{2},$ and $(d) r>r_{2}$ where the radial distance $r$ is measured from the center of the nonconducting sphere.
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