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The electrostatic  energy of a continuous charge distribution is defined as the energy required to assemble the charges at  infinity into the positions as in the given distribution. For a continuous charge distribution it is shown to be \( \dfrac{\epsilon_0}{2}\iiint(\vec E\cdot\vec E) dV\) . Thus a volume of space having nonvanishing electric field has energy density  \(\dfrac{\epsilon_0}{2}(\vec E\cdot\vec E)\).The expression for the electrostatic energy reduces to the usual answer \(\frac{1}{2}  CV^2\) for a charged parallel plate capacitor. For a  uniformly charged sphere of radius \(R\) the electrostatic energy is proved to be equal to \(\frac{3}{5}\Big(\frac{Q^2}{4\pi\epsilon_0 R^2} \Big)\).   

Large distance expansion of electric potential of a charge distribution in a finite voluem \(V\) is presented. Expressions for dipole and quadrupole moments are obtained.

The flux of electric field through a surface in defined as a surface integral and the statement of Gauss law is given. A few examples of computing the electric field using Gauss law and symmetry of the problem are discussed. A simple and intuitive proof of Gauss law is given following Feynman lectures. The task of proving Gauss law for arbitrary charge distribution is reduced to the problem proving the Gauss law for a single point charge by appealing to the superposition principle.

Work Under Progress

(LUPDate : Nov 24, 2023)

Problem

A solid sphere of radius \(R\) carries a charge density \(\rho(\vec{r})\). Show that the average of the electric field inside the sphere is \[\vec{E}= - \frac{1}{4\pi\epsilon_0} \frac{\vec{p}}{R^3},\] where \(\vec{p}\) is the total dipole moment of the sphere.

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Let \(AB\) be a line charge with charge per unit length $\lambda$, which is given to be uniform. The field is to be computed at a point \(P\),. Let \(p\) be the perpendicular distance of \(P\) from the line \(AB\). let \(\alpha, \beta\) be the angles as shown in the figure. Electric field will be calculated at a general point \(P\), the corner point \(C\) and the middle point \(M\) as shown in figure below.