Electrostatic Potential & Capacitance – Key Bullet Notes

Potential Energy & Electrostatic Potential

• Conservative forces: work done against electrostatic force stores as potential energy (PE); $W<em>{RP}= -\int</em>{R}^{P} \mathbf{F}_e\cdot d\mathbf{l}$

• PE difference: $\Delta U = U<em>P-U</em>R = W_{RP}^{(ext)}$ (path‐independent)

• Choose $U(\infty)=0$: $U<em>P = W</em>{\infty P}^{(ext)}$

• Electrostatic potential $V$: work per unit positive test charge; $V<em>P-V</em>R = (U<em>P-U</em>R)/q$; with $V(\infty)=0$, $V(r)=W_{\infty P}^{(ext)}/q$.

Potentials of Common Charge Configurations

• Point charge $Q$ at origin: $V(r)=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r}$ (sign follows $Q$).

• Electric dipole (point dipole, $\mathbf{p}$ at origin, $r\gg a$): $V(r)=\dfrac{1}{4\pi\varepsilon<em>0}\dfrac{\mathbf{p}\cdot\hat{r}}{r^{2}}$; on axis $V=\pm\dfrac{p}{4\pi\varepsilon</em>0 r^{2}}$, equatorial plane $V=0$.

• System of discrete charges: $V(P)=\sum<em>i \dfrac{1}{4\pi\varepsilon</em>0}\dfrac{q<em>i}{r</em>{iP}}$; continuous distribution: $V(P)=\int \dfrac{1}{4\pi\varepsilon_0}\dfrac{\rho(\mathbf{r^\prime})\,dV^\prime}{|\mathbf{r}-\mathbf{r^\prime}|}$.

• Uniformly charged spherical shell (total charge $q$, radius $R$):
  • Outside ( $r\ge R$ ): $V=\dfrac{1}{4\pi\varepsilon<em>0}\dfrac{q}{r}$ • Inside ( $r\le R$ ): $V=\dfrac{1}{4\pi\varepsilon</em>0}\dfrac{q}{R}$ (constant).

Equipotential Surfaces & Field Relation

• Equipotential: $V = \text{const}$; $\mathbf{E}\perp$ surface everywhere.

• Magnitude: $|\mathbf{E}| = -\dfrac{dV}{dl}$ (normal derivative).

• Examples: concentric spheres around a point charge; planes normal to uniform $\mathbf{E}$.

Potential Energy of Charge Systems

• Two point charges: $U=\dfrac{1}{4\pi\varepsilon<em>0}\dfrac{q</em>1 q<em>2}{r</em>{12}}$ (positive for like, negative for unlike).

• Three charges: $U=\sum<em>{i0}\dfrac{qi qj}{r_{ij}}$; extendable to $n$ charges.

• Charge $q$ in external potential $V(\mathbf{r})$: $U=qV$.

• Dipole in uniform field: $U=-\mathbf{p}\cdot\mathbf{E} = -pE\cos\theta$.

Conductors in Electrostatics

• $\mathbf{E}=0$ inside conductor (static).

• Field at surface: $\mathbf{E}=\dfrac{\sigma}{\varepsilon_0}\hat{n}$, normal outward.

• Excess charge resides on outer surface; potential constant throughout conductor.

• Cavity inside conductor: $\mathbf{E}=0$ (electrostatic shielding).

Dielectrics & Polarisation

• Polarisation \mathbf{P}=\varepsilon0\chie\mathbf{E} (linear isotropic).

• Effective permittivity: $\varepsilon=\varepsilon0 K$, K=1+\chie (dielectric constant >1).

• Induced surface charge density: $\sigma_p=\mathbf{P}\cdot\hat{n}$; reduces net field inside dielectric.

Capacitors & Capacitance

• Definition: $C=\dfrac{Q}{V}$; depends only on geometry & medium.

• Parallel-plate (vacuum): $C0=\dfrac{\varepsilon0 A}{d}$; with dielectric $C = KC_0$.

• Series: $\dfrac{1}{C{eq}}=\sum\dfrac{1}{Ci}$.

• Parallel: $C{eq}=\sum Ci$.

Energy Stored in a Capacitor

• $U=\dfrac{1}{2}CV^{2}=\dfrac{Q^{2}}{2C}=\dfrac{1}{2}QV$.

• Energy density in electric field: $u=\dfrac{1}{2}\varepsilon_0 E^{2}$ (general result).

Typical Dielectric Strength & Units

• Air breakdown field $\approx 3\times10^{6}\,\text{V m}^{-1}$.

• Common units: $1\,\mu\text{F}=10^{-6}\,\text{F},\;1\,\text{nF}=10^{-9}\,\text{F},\;1\,\text{pF}=10^{-12}\,\text{F}$.

• Energy unit: $1\,\text{eV}=1.6\times10^{-19}\,\text{J}$.

Quick Reference Equations

• $V{point}=\dfrac{1}{4\pi\varepsilon0}\dfrac{Q}{r}$

• $V{dipole}=\dfrac{1}{4\pi\varepsilon0}\dfrac{\mathbf{p}\cdot\hat{r}}{r^{2}}$

• $U{2\,charges}=\dfrac{1}{4\pi\varepsilon0}\dfrac{q1 q2}{r_{12}}$

• $U_{dipole}=-\mathbf{p}\cdot\mathbf{E}$

• $C{pp}=\dfrac{\varepsilon0 A}{d}\,(\text{vacuum});\;C=KC_{pp}\,(\text{dielectric})$

• $U_{cap}=\dfrac{1}{2}CV^{2}$

• $u=\dfrac{1}{2}\varepsilon_0E^{2}$