Electrostatic Potential & Capacitance – Key Bullet Notes

Potential Energy & Electrostatic Potential

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

  • PE difference: \Delta U = UP-UR = W_{RP}^{(ext)} (path‐independent)

  • Choose U(\infty)=0: UP = W{\infty P}^{(ext)}

  • Electrostatic potential V: work per unit positive test charge; VP-VR = (UP-UR)/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\varepsilon0}\dfrac{\mathbf{p}\cdot\hat{r}}{r^{2}}; on axis V=\pm\dfrac{p}{4\pi\varepsilon0 r^{2}}, equatorial plane V=0.

  • System of discrete charges: V(P)=\sumi \dfrac{1}{4\pi\varepsilon0}\dfrac{qi}{r{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\varepsilon0}\dfrac{q}{r} • Inside ( r\le R ): V=\dfrac{1}{4\pi\varepsilon0}\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\varepsilon0}\dfrac{q1 q2}{r{12}} (positive for like, negative for unlike).

  • Three charges: U=\sum{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}