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{\text{ext}}=-\mathbf FE (quasi–static motion, no acceleration).
  • Work done by external force
    W{RP}= - \intR^P \mathbf FE\cdot d\mathbf l = -\intR^P q\mathbf E\cdot d\mathbf l
• Potential-energy difference \Delta U = UP-UR = 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\varepsilon0}\;\frac{\mathbf p\cdot\hat{r}}{r^{2}}=\frac{1}{4\pi\varepsilon0}\;\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\varepsilon0}\sum{i=1}^{n}\dfrac{qi}{r{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{12}=\dfrac{1}{4\pi\varepsilon0}\dfrac{q1q2}{r_{12}}
  • U>0 (repulsive) for like charges, U<0 (attractive) for unlike.
• Three charges: sum over pairs
  U=\dfrac{1}{4\pi\varepsilon0}\Big[\frac{q1q2}{r{12}}+\frac{q1q3}{r{13}}+\frac{q2q3}{r{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 q1,q2 in external V(\mathbf r):
  U=q1V(r1)+q2V(r2)+\dfrac{1}{4\pi\varepsilon0}\dfrac{q1q2}{r{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=\varepsilon0 \chie\,\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):
  C0=\dfrac{\varepsilon0A}{d}; uniform field E=\dfrac{\sigma}{\varepsilon0}=\dfrac{Q}{\varepsilon0A}.
• Fringing neglected for d^2\ll A.

Dielectric Slab between Plates (Section 2.13)

• Full insertion of dielectric with constant K:
  C=KC0 \quad\text{or}\quad C=\dfrac{\varepsilon0KA}{d}=\dfrac{\varepsilon A}{d} with \varepsilon=K\varepsilon_0.
• Dielectric constant K=\varepsilon/\varepsilon0>1, equivalently C/C0.

Combination of Capacitors (Section 2.14)

• Series (same Q):
  \dfrac{1}{C{\text{eq}}}=\sum{i}\dfrac{1}{C_i}.
• Parallel (same V):
  C{\text{eq}}=\sum{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 \varepsilon0E^{2}(Ad) ⇒ energy density u=\tfrac12\varepsilon0E^{2} (general result).

Summary of Key Formulae

• V(r)=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Q}{r} (point charge)
• V{\text{dip}}=\dfrac{1}{4\pi\varepsilon0}\dfrac{\mathbf p\cdot\hat r}{r^{2}}
• \mathbf E= -\nabla V; in 1-D, E=-dV/dx.
• U{12}=\dfrac{1}{4\pi\varepsilon0}\dfrac{q1q2}{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{\text{with dielectric}}=K\varepsilon0A/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 uB=\tfrac12 \mu0H^{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}.