Electrostatics Essentials in Capacitance

Overview

Chapter covers electrostatic potential, potential energy, equipotential surfaces, conductors, dielectrics, capacitors, their combinations, and energy storage. Emphasis on conservative nature of electrostatic forces and capacitance with/without dielectrics.

Conservative Forces and Potential Energy

• Conservative forces store work as potential energy; sum K + U conserved.
• Electrostatic force is conservative; Coulomb force analogous to gravity.
• Potential energy difference between points R and P: ΔU = U(P) − U(R) = Wext(R→P).
• Only potential differences are physical; zero of potential chosen conveniently (often at infinity).
• With zero at infinity, U(P) = Wext(∞→P) for charge q.

Electrostatic Potential

• Potential V: work per unit positive test charge from infinity to point.
• Potential difference VP − VR = (UP − UR)/q; independent of path.
• Potential is determined up to an additive constant; choose V(∞)=0.

Potential Due to a Point Charge

• V(r) = (1/4πe0)(Q/r); sign follows Q.
• Work is path-independent; potential scales as 1/r.
• Field vs potential radial dependence: E ∝ 1/r², V ∝ 1/r.

Potential Due to an Electric Dipole

• Dipole: charges ±q separated by 2a, dipole moment p = 2qa, from −q to +q.
• Approximate potential (r ≫ a): V(r) = (1/4πe0)(p·r̂)/r².
• On axis: V = ±(1/4πe0)(p/r²); equatorial plane potential is zero.
• Falls as 1/r² and depends on angle between p and position vector.

Potential Due to a System of Charges

• Superposition: V(P) = Σ (1/4πe0)(qi/riP).
• Continuous distribution: integrate density ρ(r) over volume elements.
• Uniform spherical shell: V(r≥R) = (1/4πe0)(q/r); V(r≤R) = (1/4πe0)(q/R) (constant inside).

Equipotential Surfaces and E–V Relation

• Equipotential: surface of constant V; no work to move charge along it.
• E is perpendicular to equipotential surface; points toward decreasing V.
• Relation: |E| = |dV|/dl, where dl is normal displacement between surfaces.
• Uniform field: equipotentials are planes perpendicular to field direction.

Potential Energy of Systems and in External Fields

• Two-point charges: U = (1/4πe0)(q1q2/r12); sign positive (like), negative (unlike).
• Three charges: U = Σ pairwise (1/4πe0)(qiqj/rij).
• In an external potential V(r): single charge U = qV(r).
• Two charges in external field: U = q1V(r1)+q2V(r2)+(1/4πe0)(q1q2/r12).
• Electron-volt: 1 eV = 1.6×10⁻¹⁹ J; extends to keV, MeV, GeV, TeV.

Potential Energy of a Dipole in Uniform Field

• Torque on dipole: τ = p × E; tends to align with E.
• Work to rotate from θ0 to θ1: W = pE[cos θ0 − cos θ1].
• Potential energy (choice with U=0 at θ=90°): U(θ) = −pE cos θ = −p·E.

Electrostatics of Conductors

• E = 0 inside conductor in electrostatic equilibrium.
• At surface, E is normal; no tangential component (else charges move).
• Excess charge resides only on surface in static case.
• Potential is constant throughout conductor and equal on its surface.
• Surface field magnitude: E = σ/e0, directed along outward normal (vector form E = (σ/e0) n̂).
• Electrostatic shielding: field in a charge-free cavity inside a conductor is zero; charges reside on outer surface.

Dielectrics and Polarisation

• Dielectrics: insulators; external field induces dipoles or aligns permanent dipoles.
• Polarisation P (dipole moment per unit volume) for linear isotropic dielectrics: P = e0χe E.
• Polarised slab equivalent to induced surface charges ±σp; reduces internal field.
• Dielectric constant K = e/e0 > 1; permittivity e = e0K.

Capacitors and Capacitance

• Capacitor: two conductors separated by an insulator; charges ±Q, potential difference V.
• Capacitance: C = Q/V; depends on geometry and medium.
• Large C allows large Q at small V; dielectric strength limits electric field.
• Units: farad (F); practical subunits: mF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).

Parallel Plate Capacitor (Vacuum)

• Geometry: plate area A, separation d (d² ≪ A).
• Field between plates: E = σ/e0 = Q/(e0A) (uniform; fringing ignored).
• Potential difference: V = Ed = Qd/(e0A).
• Capacitance: C0 = e0A/d.

Effect of Dielectric on Capacitance

• With linear dielectric fully filling gap: field reduced by K; V = V0/K.
• Capacitance increases: C = KC0 = Ke0A/d = eA/d.
• Dielectric constant K defined generally by C/C0.

Combination of Capacitors

• Series: same charge; voltages add.
• Parallel: same voltage; charges add.

Capacitor Combination Formulas

Energy Stored in a Capacitor and Energy Density

• Work to charge from 0 to Q: U = (1/2)QV = (1/2)CV² = Q²/(2C).
• For parallel plates: U = (1/2)e0E²(Ad) = energy density × volume.
• Electric energy density: u = (1/2)e0E² (general for any field configuration).

Key Terms & Definitions

• Electrostatic potential V: Work per unit positive charge from infinity to point.
• Potential energy U: Stored work in configuration of charges or in external field.
• Dipole moment p: 2qa directed from −q to +q.
• Equipotential surface: Locus of constant potential; E ⟂ surface.
• Polarisation P: Dipole moment per unit volume of dielectric.
• Electric susceptibility χe: Constant relating P = e0χeE (linear isotropic).
• Permittivity e: e0K; measure of medium’s response; K is dielectric constant.
• Capacitance C: Ratio Q/V for a capacitor.
• Dielectric strength: Maximum E before breakdown of medium.
• Electron-volt (eV): Energy gained by charge e across 1 V; 1 eV = 1.6×10⁻¹⁹ J.

Action Items / Next Steps

• Practice applying superposition for potentials with multiple charges.
• Sketch equipotential surfaces and E-field lines for basic configurations.
• Solve problems on energy of assembled charge systems and dipoles in fields.
• Compute capacitance changes with partial and full dielectric insertion.
• Analyze series/parallel capacitor networks and energy redistribution.