Electrostatic Potential and Capacitance – Comprehensive Notes

Electrostatic Potential & Potential Difference

• Potential difference (PD) between two points = external work needed to move a unit positive charge from one point to the other against electrostatic forces.
  • Mathematical form: $V<em>{BA}=\frac{W</em>{BA}}{q<em>0}$ where $W</em>{BA}$ is work done in carrying test charge $q_0$ from A to B.
  • SI unit = volt (V) $\bigl(1\,\text{V}=1\,\text{J}\,\text{C}^{-1}\bigr)$.
• Electric potential at a point: work done in bringing a unit positive charge from infinity (potential taken as zero) to the point, against electrostatic forces.

Electric Potential Due to Continuous Charge Distributions

• Treat the body as a collection of micro-elements $dq$ at position $\vec r_i$; field point $P$ has position $\vec r$.
• Generic expression:
  • $V(\vec r)=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{|\vec r-\vec r'|}$
• Three standard density forms
  • Volume charge density $\rho$ over volume $\tau$ :$\displaystyle V=\frac{1}{4\pi\epsilon<em>0}\int</em>\tau \frac{\rho\,d\tau'}{|\vec r-\vec r'|}$
  • Surface charge density $\sigma$ over area S : $\displaystyle V=\frac{1}{4\pi\epsilon<em>0}\int</em>{S}\frac{\sigma\,dS'}{|\vec r-\vec r'|}$
  • Line charge density $\lambda$ along length L : $\displaystyle V=\frac{1}{4\pi\epsilon<em>0}\int</em>{L}\frac{\lambda\,dl'}{|\vec r-\vec r'|}$
• Superposition: Net potential at P = algebraic sum/integral of contributions.

Relation Between Electric Field & Potential

• Adjacent points A and B separated by $dr$; $E$ assumed uniform over small element.
• External force to move charge $q<em>0$ slowly (no acceleration): $\vec F</em>{ext}=q_0\vec E$.
• Work done (external) from A to B:
  • $dW<em>{ext}=\vec F</em>{ext}\cdot d\vec r=q_0\vec E\cdot d\vec r$
• Also, $dW<em>{ext}=q</em>0\bigl(V<em>A-V</em>B\bigr)=q_0(-dV)$.
• Equating: $-dV = \vec E\cdot d\vec r \implies \boxed{\vec E = -\nabla V}$
• Potential gradient $\displaystyle -\frac{dV}{dr}$ gives magnitude of $E$ along the line of greatest decrease.
• Properties:
  • Field direction is the direction of steepest potential decrease.
  • $|\vec E| = \left|\frac{dV}{dn}\right|$ normal to an equipotential surface.

Equipotential Surfaces

• Definition: surface with equal potential everywhere; work to move charge along it = 0.
• Properties
  1. No work in moving a test charge over it.
  2. $\vec E$ always perpendicular to the surface.
  3. Surfaces closer together where $|\vec E|$ is large, farther apart where $|\vec E|$ is small.
  4. Two equipotential surfaces never intersect.
• Visual patterns
  • Single positive charge → concentric spheres.
  • Electric dipole ((+q),(-q)) → surfaces compressed between charges.
  • Two equal like charges → sparse between charges, dense outside.
  • Uniform field → parallel planes perpendicular to field lines.

Electric Potential Energy (EPE)

• For a system of charges: work needed to assemble them from infinity.
• Single charge in external potential $V<em>{ext}$: $U=qV</em>{ext}$.
• Two point charges $q<em>1,q</em>2$ separated by $r$ without external field: $U=\dfrac{1}{4\pi\epsilon<em>0}\dfrac{q</em>1q_2}{r}$ (sign shows repulsion/attraction).
• With external field, total $U =\sum q<em>iV</em>{ext}(\vec r_i)+$ mutual terms.

Free vs Bound Charges & Behaviour of Conductors

• Metals: loosely bound outer electrons → free charges (electrons) + fixed positive ion cores (bound charges).
• Insulators: electrons tightly bound → essentially no free charges.
• Electrolytes: mobile ions of both signs; motion limited by field & mutual forces.

Electrostatic Properties of Conductors

1. Net $\vec E=0$ inside a conductor in electrostatic equilibrium (induced field cancels external field).
2. Just outside, $\vec E$ is normal to the surface (no tangential component; else charges would move).
3. Excess charge resides on the outer surface; interior net charge = 0 (Gauss’s law with interior Gaussian surface gives zero flux).
4. Potential is constant throughout the volume and surface.
5. Surface field magnitude relates to surface charge density: $E=\dfrac{\sigma}{\epsilon_0}$ (for locally flat surface; pill-box argument).
6. Field in cavity of a hollow charged conductor = 0 (electrostatic shielding).

Electrostatic Shielding – Applications

• Metal enclosures around sensitive electronics.
• Coaxial cables: grounded outer conductor shields central conductor.

Capacitance

• Definition: $C=\dfrac{Q}{V}$ – charge required to raise potential by one volt.
• SI unit: farad (F).
• Depends on geometry, surroundings’ permittivity, nearby conductors.

Isolated Spherical Conductor

• Potential at surface: $V=\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}$.
• Capacitance: $\boxed{C=4\pi\epsilon_0R}$ → proportional to radius.

Principle of a Capacitor

• Bringing an uncharged plate near a charged one induces opposite charge, lowering potential of the first plate and increasing its ability to store charge.
• Earthing the far face of induced plate removes like charge, further enhancing capacitance → two-plate system = capacitor.

Parallel-Plate Capacitor (Vacuum)

• Geometry: plates area $A$, separation $d$ (edge effects neglected when $A\gg d^2$).
• Field between plates: $E=\dfrac{\sigma}{\epsilon<em>0}=\dfrac{Q}{A\epsilon</em>0}$.
• Potential difference: $V=Ed=\dfrac{Qd}{A\epsilon_0}$.
• Capacitance: $\boxed{C=\dfrac{\epsilon_0A}{d}}$.
• Factors: area ↑ ⇒ $C$ ↑; separation ↑ ⇒ $C$ ↓; higher permittivity medium ⇒ $C$ ↑.

Spherical Capacitor (Concentric Shells)

• Inner radius $a$, outer $b$ ($+Q$ on outer, $-Q$ on inner).
• Field in region a<r<b: $E=\dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2}$ (depends only on inner charge).
• Potential difference: $V=\dfrac{Q}{4\pi\epsilon_0}\left(\dfrac{1}{a}-\dfrac{1}{b}\right)$.
• Capacitance: $\boxed{C=4\pi\epsilon_0\dfrac{ab}{b-a}}$.

Cylindrical Capacitor (Coaxial)

• Radii a<b, length $L\ (!!\gg b$) ensures negligible edge effect.
• Linear charge density on inner: $\lambda$.
• Field for a<r<b: $E=\dfrac{\lambda}{2\pi\epsilon_0 r}$.
• Potential difference: $V=\dfrac{\lambda}{2\pi\epsilon_0}\ln\frac{b}{a}$.
• Capacitance: $\boxed{C=\dfrac{2\pi\epsilon_0 L}{\ln(b/a)}}$.

Combination of Capacitors

Series

• Charge same on each: $Q$.
• Equivalent PD: $V=\sum V_i$.
• Formula: $\displaystyle \frac{1}{C<em>{eq}}=\sum</em>{i=1}^{n}\frac{1}{C_i}$.
• Equivalent capacitance less than smallest individual $C$.

Parallel

• Potential same; total charge additive.
• $C<em>{eq}=\sum</em>{i=1}^{n} C_i$.
• Equivalent $C$ exceeds largest individual $C$.

Energy Stored in a Capacitor

• Work to transfer charge incrementally: $U=\int_0^{Q} \frac{q}{C}\,dq=\frac{1}{2}\frac{Q^2}{C}=\frac{1}{2}CV^2=\frac{1}{2}QV$.
• Series / Parallel: total energy = sum of energies of individual elements (no hidden interactions if connection leads ideal).

Energy Density in Electric Field

• For uniform field in parallel plates: $U=\frac{1}{2}CV^2=\frac{1}{2}\left(\frac{\epsilon<em>0A}{d}\right)E^2dA=\frac{1}{2}\epsilon</em>0E^2(Ad).$
• Energy per unit volume: $\boxed{u=\frac{1}{2}\epsilon<em>0E^2}$ (general expression for any electric field in vacuum or linear dielectric $\epsilon=\kappa\epsilon</em>0$ → $u=\tfrac12\epsilon E^2$).

Redistribution of Charge Between Connected Conductors

• Two isolated conductors A,B with $C<em>1,C</em>2$ and initial charges $Q<em>1,Q</em>2$, potentials $V<em>1=Q</em>1/C<em>1,\ V</em>2=Q<em>2/C</em>2$.
• Connect via thin wire (negligible capacitance): charges flow until common potential $V_f$.
  • Total charge conserved: $Q<em>1+Q</em>2=Q'<em>1+Q'</em>2$.
  • $V<em>f=Q'</em>1/C<em>1=Q'</em>2/C_2$.
  • Combined capacitance: $C<em>1+C</em>2$ (wire effect neglected).
• Resulting energy < initial energy; the difference appears as Joule heat in wire (energy loss on redistribution).

Dielectrics

• Non-conductors that permit electrostatic forces but no macroscopic charge flow.
• Molecules
  • Non-polar: charge centres coincide (e.g.
    $O<em>2,H</em>2$), zero permanent dipole.
  • Polar: intrinsic dipole (e.g.
    $H_2O,\ HCl$).

Polarisation & Susceptibility

• In external $\vec E$, induced dipole moments align → polarisation density $\vec P =\text{dipole moment per unit volume}$.
• Surface bound charge density on dielectric faces: $\sigma_P=\vec P\cdot \hat n$.
• For linear isotropic dielectric: $\vec P=\epsilon<em>0\chi</em>e\vec E$ where $\chi_e$ = electric susceptibility.
• Dielectric constant $\kappa=1+\chi<em>e$; relates to permittivity $\epsilon=\kappa\epsilon</em>0$.
• Dielectric strength: maximum sustainable field before breakdown (typ.
  $3\times10^6\,\text{V m}^{-1}$ for air).

Parallel Plate Capacitor with Dielectric Slab (thickness t<d)

• Vacuum capacitance $C<em>0=\dfrac{\epsilon</em>0A}{d}$.
• Original field $E<em>0=\dfrac{Q}{A\epsilon</em>0}$.
• Dielectric inserted: induced surface charges create opposing field $E<em>P=\dfrac{P}{\epsilon</em>0}=\left(1-\frac{1}{\kappa}\right)E_0$.
• Net field in dielectric: $E=E<em>0/\kappa$; in remaining gap $E</em>0$.
• Potential difference: $V=E<em>0(d-t)+\frac{E</em>0}{\kappa}t = E_0\bigl[d-t(1-1/\kappa)\bigr]$.
• New capacitance: $\boxed{C=\dfrac{\epsilon_0A}{d-t+t/\kappa}}$ (increases with $\kappa$).

Parallel Plate Capacitor with Conducting Slab (thickness t<d)

• Field inside conductor zero; effective separation reduces to $d-t$.
• New capacitance: $\boxed{C=\dfrac{\epsilon<em>0A}{d-t}}$ (always > original $C</em>0$).

Applications of Capacitors

1. Create uniform fields (e.g.
   Millikan oil-drop experiment).
2. Tuning circuits in radios (variable capacitors adjust resonance).
3. Smooth ripple in rectifier power supplies (filter capacitors).
4. Phase shifting & starting torque in induction motors.
5. Tank circuits of oscillators.
6. Storage of energy for pulsed power systems.

Van de Graaff Generator

• Goal: generate $\sim10^7\,\text{V}$ (few MV) for particle acceleration.
• Principles
  1. Corona discharge at sharp points: ionisation enables charge spray onto moving belt.
  2. Charge transfer to hollow conductor: any charge on an electrode inside a conducting shell migrates to outer surface regardless of pre-existing potential.
• Construction Highlights
  • Large hollow metal sphere S mounted high on insulating column.
  • Endless insulating belt over lower pulley P₁ (motor-driven, ground level) and upper pulley P₂ inside sphere.
  • Spray comb B₁ near lower pulley at +10 kV wrt ground → deposits positive ions on belt.
  • Collecting comb B₂ inside sphere picks negative charge, leaves positive on sphere.
• Working Cycle
  1. Belt carries + charge up, deposits onto sphere via induction at B₂.
  2. Belt returns neutral, resprayed by B₁.
  3. Potential of sphere builds up to several megavolts; limited by leakage & corona.
• Usage: High-energy beams (protons, deuterons, (\alpha)s) for nuclear physics and material analysis.

Ethical & Practical Considerations / Real-World Links

• Electrostatic shielding critical for safety of human operators and to prevent EMI in medical devices.
• High-voltage generators (e.g.
  X-ray machines, particle accelerators) require robust insulation – breakdown can endanger personnel.
• Capacitors in power grids (power factor correction) reduce energy wastage; environmental benefit.
• Dielectric materials selection involves trade-offs: high $\kappa$ vs dielectric strength vs toxicity / recyclability (e.g.
  PCB oils now banned).

Quick Reference Formulas

• $V=\dfrac{W}{q}$ ; $\vec E=-\nabla V$.
• $C<em>{sphere}=4\pi\epsilon</em>0R$.
• $C<em>{parallel}=\dfrac{\epsilon</em>0A}{d}$; with dielectric slab $C=\dfrac{\epsilon_0A}{d-t+t/\kappa}$.
• $C<em>{spherical\,cap}=4\pi\epsilon</em>0\dfrac{ab}{b-a}$ ; $C<em>{cyl}=\dfrac{2\pi\epsilon</em>0L}{\ln(b/a)}$.
• Series: $1/C<em>{eq}=\sum 1/C</em>i$ ; Parallel: $C<em>{eq}=\sum C</em>i$.
• Stored energy: $U=\tfrac12 CV^2=\tfrac12 QV=\tfrac12 \dfrac{Q^2}{C}$.
• Energy density: $u=\tfrac12 \epsilon E^2$.