Electrostatic Potential ($V$)
The work done per unit positive charge in bringing it from infinity to a point without acceleration:
$$V(r) = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$
- Relation between Field and Potential:
$$\vec{E} = -\nabla V = -\frac{dV}{dr}\hat{r}$$
- Equipotential Surface: A surface where electric potential is constant everywhere. Electric field lines are always perpendicular to equipotential surfaces at every point.
Capacitance of a Parallel Plate Capacitor
Capacitance is the charge-storing capability per unit potential difference: $C = \frac{Q}{V}$.
Parallel Plate Capacitor with Vacuum:
$$C_0 = \frac{\varepsilon_0 A}{d}$$
With Dielectric Medium of Constant $K$:
$$C = \frac{K\varepsilon_0 A}{d} = K C_0$$
Energy Stored in a Capacitor:
$$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$$
Energy density: $u_E = \frac{1}{2}\varepsilon_0 E^2$.
