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{BA}=\frac{W{BA}}{q0} where W{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\epsilon0}\int\tau \frac{\rho\,d\tau'}{|\vec r-\vec r'|}
- Surface charge density \sigma over area S : \displaystyle V=\frac{1}{4\pi\epsilon0}\int{S}\frac{\sigma\,dS'}{|\vec r-\vec r'|}
- Line charge density \lambda along length L : \displaystyle V=\frac{1}{4\pi\epsilon0}\int{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 q0 slowly (no acceleration): \vec F{ext}=q_0\vec E.
Work done (external) from A to B:
- dW{ext}=\vec F{ext}\cdot d\vec r=q_0\vec E\cdot d\vec r
Also, dW{ext}=q0\bigl(VA-VB\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{ext}: U=qV{ext}.
Two point charges q1,q2 separated by r without external field: U=\dfrac{1}{4\pi\epsilon0}\dfrac{q1q_2}{r} (sign shows repulsion/attraction).
With external field, total U =\sum qiV{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

Net \vec E=0 inside a conductor in electrostatic equilibrium (induced field cancels external field).
Just outside, \vec E is normal to the surface (no tangential component; else charges would move).
Excess charge resides on the outer surface; interior net charge = 0 (Gauss’s law with interior Gaussian surface gives zero flux).
Potential is constant throughout the volume and surface.
Surface field magnitude relates to surface charge density: E=\dfrac{\sigma}{\epsilon_0} (for locally flat surface; pill-box argument).
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}{\epsilon0}=\dfrac{Q}{A\epsilon0}.
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{eq}}=\sum{i=1}^{n}\frac{1}{C_i}.
Equivalent capacitance less than smallest individual C.

Parallel

Potential same; total charge additive.
C{eq}=\sum{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{\epsilon0A}{d}\right)E^2dA=\frac{1}{2}\epsilon0E^2(Ad).
Energy per unit volume: \boxed{u=\frac{1}{2}\epsilon0E^2} (general expression for any electric field in vacuum or linear dielectric \epsilon=\kappa\epsilon0 → u=\tfrac12\epsilon E^2).

Redistribution of Charge Between Connected Conductors

Two isolated conductors A,B with C1,C2 and initial charges Q1,Q2, potentials V1=Q1/C1,\ V2=Q2/C2.
Connect via thin wire (negligible capacitance): charges flow until common potential V_f.
- Total charge conserved: Q1+Q2=Q'1+Q'2.
- Vf=Q'1/C1=Q'2/C_2.
- Combined capacitance: C1+C2 (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.
  O2,H2), 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=\epsilon0\chie\vec E where \chi_e = electric susceptibility.
Dielectric constant \kappa=1+\chie; relates to permittivity \epsilon=\kappa\epsilon0.
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 C0=\dfrac{\epsilon0A}{d}.
Original field E0=\dfrac{Q}{A\epsilon0}.
Dielectric inserted: induced surface charges create opposing field EP=\dfrac{P}{\epsilon0}=\left(1-\frac{1}{\kappa}\right)E_0.
Net field in dielectric: E=E0/\kappa; in remaining gap E0.
Potential difference: V=E0(d-t)+\frac{E0}{\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{\epsilon0A}{d-t}} (always > original C0).

Applications of Capacitors

Create uniform fields (e.g.
Millikan oil-drop experiment).
Tuning circuits in radios (variable capacitors adjust resonance).
Smooth ripple in rectifier power supplies (filter capacitors).
Phase shifting & starting torque in induction motors.
Tank circuits of oscillators.
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{sphere}=4\pi\epsilon0R.
C{parallel}=\dfrac{\epsilon0A}{d}; with dielectric slab C=\dfrac{\epsilon_0A}{d-t+t/\kappa}.
C{spherical\,cap}=4\pi\epsilon0\dfrac{ab}{b-a} ; C{cyl}=\dfrac{2\pi\epsilon0L}{\ln(b/a)}.
Series: 1/C{eq}=\sum 1/Ci ; Parallel: C{eq}=\sum Ci.
Stored energy: U=\tfrac12 CV^2=\tfrac12 QV=\tfrac12 \dfrac{Q^2}{C}.
Energy density: u=\tfrac12 \epsilon E^2.