Physics · Ch 5 — Electrostatic Potential and Capacitance

Summary

Electrostatic Potential is the work done per unit charge to bring a test charge from infinity to a point in an electric field: $V = \frac{W}{q_0}$ . It is a scalar quantity.
Potential due to a point charge $Q$ at distance $r$ : $V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$ .
Potential due to a dipole at a point far away: $V = \frac{1}{4\pi\varepsilon_0} \frac{p \cos\theta}{r^2}$ , where $p = q \cdot 2a$ is the dipole moment.
Equipotential surfaces are surfaces where potential is constant. No work is done in moving a charge along them; electric field lines are perpendicular to them.
Relation between field and potential: $E = -\frac{dV}{dr}$ (in one dimension) or $\vec{E} = -\nabla V$ (in three dimensions).
Potential energy of a system of two point charges: $U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}$ .
Capacitance of a conductor: $C = \frac{Q}{V}$ . Unit: farad (F).
Parallel plate capacitor capacitance: $C = \frac{\varepsilon_0 A}{d}$ (with vacuum). With a dielectric of constant $K$ , $C = \frac{K\varepsilon_0 A}{d}$ .
Dielectric constant $K = \frac{\varepsilon}{\varepsilon_0}$ ; it reduces the effective field inside the dielectric by a factor $K$ .
Capacitors in series: $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ ; charge same on each, voltage divides.
Capacitors in parallel: $C_{\text{eq}} = C_1 + C_2 + \dots$ ; voltage same across each, charge divides.
Energy stored in a capacitor: $U = \frac{1}{2} CV^2 = \frac{1}{2} QV = \frac{Q^2}{2C}$ . …

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