Electrostatic Potential & Capacitance – Comprehensive Notes

Introduction & Conservative Forces

• Potential energy concept revisited
  • Work done $W$ by an external agent against conservative forces (gravity, spring, electrostatics) stores as potential energy $U$.
  • Removing the external force lets the system exchange stored $U$ with kinetic energy $K$ such that $K+U=\text{const}$.
• Conservative forces
  • Gravitational, spring (Hooke) and Coulomb forces are conservative: path–independent work.
  • Coulomb’s law shares the $1/r^{2}$ nature with gravity; charges replace masses.

Work & Potential-Energy Difference (Section 2.1)

• Move a test charge $q$ from $R$ to $P$ against field $\mathbf E$ of a source charge $Q$.
  • External force $\mathbf F<em>{\text{ext}}=-\mathbf F</em>E$ (quasi–static motion, no acceleration).
  • Work done by external force
    $W<em>{RP}= - \int</em>R^P \mathbf F<em>E\cdot d\mathbf l = -\int</em>R^P q\mathbf E\cdot d\mathbf l$
• Potential-energy difference $\Delta U = U<em>P-U</em>R = W_{RP}$
  • Path–independent ⇒ electrostatic force is conservative.
• Zero reference: convenient to set $U=0$ at $R=\infty$, then
  $U(P)= W_{\infty P}$

Electrostatic Potential (Section 2.2)

• Potential $V$ at a point = work per unit charge in bringing $+1$ C from infinity to that point:
  $V=\frac{W}{q}$
• Only potential differences physically meaningful; absolute level arbitrary up to an additive constant.

Potential due to a Point Charge (Section 2.3)

• For a charge $Q$ at origin: $V(r)=\frac{1}{4\pi\varepsilon_0}\;\frac{Q}{r}$
  • V>0 for Q>0, V<0 for Q<0.
• Variation:
  • $V\propto 1/r$ (blue curve).
  • $E\propto 1/r^{2}$ (black curve).
• Example 2.1
  • $Q=4\times10^{-7}\,\text{C},\;r=0.09\,\text{m}\Rightarrow V=4\times10^{4}\,\text{V}$.
  • Bringing $q=2\times10^{-9}\,\text{C}$ gives $W=qV=8\times10^{-5}\,\text{J}$ (path-independent).

Potential due to an Electric Dipole (Section 2.4)

• Dipole: charges $+q, -q$ separated by $2a$, dipole moment $\mathbf p=2aq\,\hat{z}$.
• Approximate potential (for $r\gg a$ or point-dipole): $V(\mathbf r)=\frac{1}{4\pi\varepsilon<em>0}\;\frac{\mathbf p\cdot\hat{r}}{r^{2}}=\frac{1}{4\pi\varepsilon</em>0}\;\frac{p\cos\theta}{r^{2}}$
  • On axis ($\theta=0,\pi$): $V=\pm\dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^{2}}$.
  • In equatorial plane ($\theta=\pi/2$): $V=0$.
  • Falls as $1/r^{2}$ not $1/r$.

Potential of a System of Discrete & Continuous Charges (Section 2.5)

• Discrete: $V(\mathbf P)=\dfrac{1}{4\pi\varepsilon<em>0}\sum</em>{i=1}^{n}\dfrac{q<em>i}{r</em>{iP}}$.
• Continuous: subdivide into $\rho(\mathbf r)\,dV$ elements and integrate.
• Uniform spherical shell:
  V(r)=\begin{cases}\dfrac{1}{4\pi\varepsilon0}\dfrac{q}{r}, & r\ge R\[4pt]\dfrac{1}{4\pi\varepsilon0}\dfrac{q}{R}, & r\le R\end{cases}

Equipotential Surfaces (Section 2.6)

• Surface where $V=\text{const}$. No work in moving a test charge on it.
• Properties
  • $\mathbf E$ is ⟂ to each equipotential surface.
  • Separation between surfaces inversely proportional to field strength: $E = -\dfrac{dV}{dl}$.
• Illustrations
  • Point charge → concentric spheres.
  • Uniform field $\mathbf E=E\hat{x}$ → planes normal to $x$-axis.
  • Dipole → nested “lemniscate-like” surfaces.

Potential Energy of Charge Configurations (Section 2.7)

• Two charges: $U<em>{12}=\dfrac{1}{4\pi\varepsilon</em>0}\dfrac{q<em>1q</em>2}{r_{12}}$
  • U>0 (repulsive) for like charges, U<0 (attractive) for unlike.
• Three charges: sum over pairs
  $U=\dfrac{1}{4\pi\varepsilon<em>0}\Big[\frac{q</em>1q<em>2}{r</em>{12}}+\frac{q<em>1q</em>3}{r<em>{13}}+\frac{q</em>2q<em>3}{r</em>{23}}\Big]$
• Extension to $n$ charges: pairwise summation \displaystyle U=\frac{1}{4\pi\varepsilon0}\sum{i<j}\frac{qiqj}{r_{ij}}.

Charges in an External Potential (Section 2.8)

• Single charge: $U=qV(\mathbf r)$ where $V$ is due to external sources only.
• Electron-volt: $1\,\text{eV}=1.6\times10^{-19}\,\text{J}$.
• Two charges $q<em>1,q</em>2$ in external $V(\mathbf r)$:
  $U=q<em>1V(r</em>1)+q<em>2V(r</em>2)+\dfrac{1}{4\pi\varepsilon<em>0}\dfrac{q</em>1q<em>2}{r</em>{12}}$.
• Dipole in uniform field:
  • Torque: $\boldsymbol\tau=\mathbf p\times\mathbf E$.
  • Potential energy (choosing $U=0$ at $\theta=90^\circ$): $U(\theta)=-\mathbf p\cdot\mathbf E=-pE\cos\theta$.

Conductors in Electrostatics (Section 2.9)

• Inside a conductor: $\mathbf E=0$ (charges rearrange until equilibrium).
• $\mathbf E$ at surface normal: no tangential component; field just outside:
  $\mathbf E=\dfrac{\sigma}{\varepsilon_0}\,\hat{n}$ (outward if \sigma>0).
• Excess charge resides only on outer surface.
• Potential constant throughout conductor, same value on its surface.
• Cavity inside conductor with no internal charges ➔ $\mathbf E=0$; electrostatic shielding.

Dielectrics & Polarisation (Section 2.10)

• Non-polar molecule: centres of $+$ and $-$ coincide; acquires induced dipole $\mathbf p_i=\alpha\mathbf E$ in field.
• Polar molecule: permanent $\mathbf p_0$; external field tends to align dipoles against thermal agitation.
• Polarisation vector $\mathbf P$ (dipole moment per unit volume):
  $\mathbf P=\varepsilon<em>0 \chi</em>e\,\mathbf E$ where $\chi_e$ = electric susceptibility.
• Bound surface charge density due to polarisation: $\sigma_p=\mathbf P\cdot\hat{n}$.
• Net field in dielectric is reduced ⇒ capacitance increases.

Capacitors & Capacitance (Section 2.11)

• Two conductors separated by insulator; charges $\pm Q$, potential diff $V$.
• Capacitance definition: $C=\dfrac{Q}{V}$ (geometric property).
• Units: $1\,\text{F}=1\,\text{C}\,\text{V}^{-1}$; practical sub-units: $\mu\text{F},\,\text{nF},\,\text{pF}$.

Parallel-Plate Capacitor (Section 2.12)

• Plates area $A$, separation $d$ (vacuum):
  $C<em>0=\dfrac{\varepsilon</em>0A}{d}$; uniform field $E=\dfrac{\sigma}{\varepsilon<em>0}=\dfrac{Q}{\varepsilon</em>0A}$.
• Fringing neglected for $d^2\ll A$.

Dielectric Slab between Plates (Section 2.13)

• Full insertion of dielectric with constant $K$:
  $C=KC<em>0 \quad\text{or}\quad C=\dfrac{\varepsilon</em>0KA}{d}=\dfrac{\varepsilon A}{d}$ with $\varepsilon=K\varepsilon_0$.
• Dielectric constant $K=\varepsilon/\varepsilon<em>0>1$, equivalently $C/C</em>0$.

Combination of Capacitors (Section 2.14)

• Series (same $Q$):
  $\dfrac{1}{C<em>{\text{eq}}}=\sum</em>{i}\dfrac{1}{C_i}$.
• Parallel (same $V$):
  $C<em>{\text{eq}}=\sum</em>{i}C_i$.

Energy Stored in a Capacitor (Section 2.15)

• Work required to charge: $U=\tfrac12 QV=\tfrac12 CV^{2}=\dfrac{Q^{2}}{2C}$.
• Viewed as field energy; for parallel-plate volume $Ad$:
  $U=\tfrac12 \varepsilon<em>0E^{2}(Ad)$ ⇒ energy density $u=\tfrac12\varepsilon</em>0E^{2}$ (general result).

Summary of Key Formulae

• $V(r)=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r}$ (point charge)
• $V<em>{\text{dip}}=\dfrac{1}{4\pi\varepsilon</em>0}\dfrac{\mathbf p\cdot\hat r}{r^{2}}$
• $\mathbf E= -\nabla V$; in 1-D, $E=-dV/dx$.
• $U<em>{12}=\dfrac{1}{4\pi\varepsilon</em>0}\dfrac{q<em>1q</em>2}{r_{12}}$
• $U_{\text{dip}}=-\mathbf p\cdot\mathbf E$
• Surface field of conductor $\mathbf E=\sigma\hat{n}/\varepsilon_0$.
• Parallel-plate $C=\varepsilon A/d,\;C<em>{\text{with dielectric}}=K\varepsilon</em>0A/d$.
• Series/parallel combination laws (above).
• Energy in capacitor $U=\tfrac12 CV^{2}$; energy density $u=\tfrac12 \varepsilon_0E^{2}$.

Ethical, Practical & Historical Notes

• Alessandro Volta (1745-1827) pioneered the voltaic pile, first battery; unit “volt” honours him.
• Electrostatic shielding critical for protection of sensitive electronics (e.g., Faraday cages, aircraft fuel tanks, high-voltage labs).
• Slightly conducting aircraft tyres & grounding ropes on fuel trucks safely dissipate charge to avoid sparks.

Connections & Real-World Relevance

• Capacitors store energy in camera flashes, pacemakers, defibrillators.
• Dielectrics selected for high $K$ and large dielectric-strength (e.g., mica, ceramics) to maximise $C$ while preventing breakdown.
• Energy density formula parallels magnetic-field energy $u<em>B=\tfrac12 \mu</em>0H^{2}$ (to be studied later).
• Concepts extend to Maxwell’s equations and EM wave propagation where electric and magnetic field energies interchange.

Typical Numerical Constants

• $\varepsilon_0=8.854\times10^{-12}\,\text{C}^{2}\text{N}^{-1}\text{m}^{-2}$.
• Dielectric strength of air ≈ $3\times10^{6}\,\text{V m}^{-1}$.
• Practical sub-units: $1\,\mu\text{F}=10^{-6}\,\text{F}, 1\,\text{nF}=10^{-9}\,\text{F}, 1\,\text{pF}=10^{-12}\,\text{F}$.