Self Energy of Spherical Shell and Solid Sphere

Self Energy of a Uniformly Charged Spherical Shell and Solid Sphere

Hollow Sphere or Spherical Shell

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Spherical shell, so all the charge is added to the same radius R, so Small amount of work done or energy dU to bring the small elemental charge dq to the hollow sphere of radius r and charge q is, \[dU=k \frac{q \, dq}{R}\] \[U= \frac{k}{R}\int_0^Q q \, dq\] \[U=\frac{k}{R}\left[\frac{q^2}{2}\right]_0^Q\]

\[\boxed{U=k \frac{Q^2}{2R}}\]

Solid Sphere

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Uniform Volume Charge density, so \[\rho = \frac{Q}{\frac{4}{3}\pi R^3}=\frac{q}{\frac{4}{3}\pi r^3}=\frac{dq}{4\pi r^2 dr}\] Small amount of work done or energy dU to bring the small elemental charge dq to the sphere of radius r and charge q is, \[dU=k \frac{q \, dq}{r}\] \[dU=k \left(\frac{Q\,r^3}{R^3}\right) \left(\frac{3Q \, r^2 \, dr}{R^3}\right) \frac{1}{r}\] \[U=k \frac{3Q^2}{R^6}\int_0^R r^4dr\] \[U=k\frac{3Q^2}{R^6}\left[\frac{r^5}{5}\right]_0^R\]

\[\boxed{U=k \frac{3}{5}\frac{Q^2}{R}}\]

Self Energy of two Charged Solid Spheres

Self Energy of Two Concentric Charged Shells

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\[\boxed{U=k \frac{Q_1^2}{2R_1} + k \frac{Q_2^2}{2R_2} + k\frac{Q_1 Q_2}{R_2} } \]

Self Energy of Two Charged Solid Spheres

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\[\boxed{ U = k \frac{3}{5}\frac{Q_1^2}{R_1} + k \frac{3}{5}\frac{Q_2^2}{R_2} + k\frac{Q_1 Q_2}{r} }\]