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The curl free nature of the electric field in electrostatics implies existence of a potential,\(\phi(\vec(r))\), from which the electric field can be derived as \(\vec{E}=-\nabla \phi\). The potential at a point is just the work done in moving a unit point charge from infinity to its current position.

Expressions for electrostatic energy of system of point charges is derived.

Starting with the Gauss law and using divergence theorem of vector calculus we derive Maxwell's first equation \(\nabla\cdot \vec{E}= \rho/\epsilon_0\).

An expression for the electric stress tensor is derived for a charge distribution in a volume \(V\). The surface integral of the stress  tensor gives the total electric force on the  charge in the volume \(V\).

Expression for the Lagrangian for a charged particle in electromagnetic field is given. Gauge invariance of the Lagrangian furnishes an example quasi invariance under the gauge transformations.

The large distance expansion of potential due to a localized charge distribution is obtained. The first three terms receiving contributions from the monopole, the dipole moment and the quadrupole moment are explicitly displayed.  Important properties of dipole and quadrupole moment are discussed.

 The electrostatic energy of a uniformly charged solid sphere is computed by computing the energy required to bring infinitesimal quantities and filling up the sphere.

We discuss the path independence of the work done by static electric field. This leads to, as in mechanics, introduction of the electric potential. An expression of the electric potential is derived by an explicit computation of work done by on a unit positive charge by the electric field of a point charge \(q\). For an arbitrary distribution of charges, the electric potential is obtained by making use of the superposition principle.

Maxwell's equation, \(\nabla \times \vec{E}=0\), can be easily proved by direct computation of curl of electric field of a point charge and appealing to the superposition principle.

The derivation of Maxwell's first equation, \(\nabla\cdot\bar{E}=\rho/\epsilon_0\), from from Coulomb's law is outlined using the Green function for the Poisson equation.

Maxwell's equation, \(\text{curl}\vec{E}=0\), is used to prove that the electric field inside an empty cavity in a conductor is zero.

Maxwell's equations for electrostatics are summarized and relation with the known laws is described.

The energy stored in a  charged capacitor, \(\frac{1}{2}CV^2\) is shown to coincide with the expression derived from the energy density,  \(\frac{\epsilon_0}{2}\big(\vec{E}\cdot\vec{E}\big)\), of static fields. 

The electromagnetic contribution to the difference in binding energies of mirror nuclei is computed. The numerical values are compared with the binding energy difference

The  two expressions for electrostatic energy in terms of the electric field for a  systems of point charges and for a  continuous charge distributions are discussed. The computation of electrostatic energy for point charges does not include the self energy. This expression  can be positive or negative and is zero for a single point charge. On the other hand the expression for energy density for continuous charges is always  be positive definite  and becomes infinite when applied to a single point charge.

The electrostatic energy  associated with continuous charge distribution is shown to correspond to  energy \(\frac{\epsilon_0}{2} |\vec E|^2\) per unit volume.

The electric potential due to charge spread uniformly on a finite line segment is computed.The electric potential due to charge spread uniformly on a finite line segment is computed.

We derive expressions for force, potential energy and torque on a dipole in electric field.

A vector calculus proof of Gauss law is given starting from the Maxwell's equation \(\text{div} \vec E=\frac{\rho}{\epsilon_0}\)

An intuitive proof of path independence of work done by electrostatic forces is given following Feynman.