Electrostatic Potential and Capacitance - Electrostatic Potential

Electrostatic Potential ($V$) at a point is defined as the work done per unit test charge in bringing a small positive test charge from infinity to that point against the electrostatic force: $V = \frac{W}{q_0}$.

The SI unit of electric potential is the Volt ($V$), where $1\text{ V} = 1\text{ J/C}$. It is a scalar quantity.

Potential due to a point charge $q$ at a distance $r$ is given by $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$. Unlike the electric field, it follows an inverse relationship with distance ($V \propto \frac{1}{r}$).

Equipotential Surfaces are surfaces that have the same electric potential at every point. No work is done in moving a charge between two points on an equipotential surface ($W = q\Delta V = 0$).

The electric field $\vec{E}$ is always perpendicular to the equipotential surface at every point and points in the direction of decreasing potential.

Potential Gradient: The electric field is the negative gradient of the electric potential, expressed as $E = -\frac{dV}{dr}$.

Electrostatic Potential Energy ($U$) of a system of charges is the work done in assembling the charges from infinity to their respective locations. For two charges $q_1$ and $q_2$: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.