Chapter 2: Electrostatic Potential and Capacitance | Summary

The potential at distance r from point charge Q is V(r) = Q/(4πε₀r). For positive Q, V > 0; for negative Q, V < 0. Potential varies as 1/r while electric field varies as 1/r², so potential decreases more slowly with distance than the field.

Work done in bringing charge q from infinity to a point where potential is V equals qV. This work is path-independent, a direct consequence of the conservative nature of electrostatic force.

An electric dipole (charges +q and −q, separation 2a, moment p = 2qa) gives potential V = p cosθ/(4πε₀r²) for r >> a. At axial points (θ = 0 or π): V = ±p/(4πε₀r²). At equatorial plane (θ = π/2): V = 0.

Unlike a single charge (V ∝ 1/r), dipole potential falls as 1/r² and depends on angle θ — axially symmetric but not spherically symmetric. Superposition principle applies: total potential is algebraic sum of potentials due to +q and −q.

For charges q₁, q₂, ..., qₙ at distances r₁P, r₂P, ..., rₙP from point P, the total potential is V = (1/4πε₀) × Σ(qᵢ/rᵢP) — the superposition principle applied to scalars, simpler than vector superposition of fields.

For a uniformly charged spherical shell: outside (r ≥ R), V = q/(4πε₀r); inside (r < R), V = q/(4πε₀R), a constant equal to the surface value. The constant interior potential follows from E = 0 inside the shell.

An equipotential surface has the same potential at every point. For a single charge: concentric spheres. For uniform field along x-axis: planes parallel to y-z plane. For dipole: complex surfaces, with equatorial plane as the V = 0 surface.

The electric field at any point is always perpendicular to the equipotential surface through that point — a field component along the surface would do work, contradicting the definition. Relation: |E| = −dV/dl, so E points in direction of steepest potential decrease.

Potential energy of two charges: U = q₁q₂/(4πε₀r₁₂). If q₁q₂ > 0 (like charges), U > 0 (work done against repulsion). If q₁q₂ < 0 (unlike charges), U < 0 (energy released on assembly).

For three charges: U = (1/4πε₀)[q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃] — sum of all pairwise interactions. This generalises to any number of charges and is independent of assembly order.

Potential energy of charge q in external field at r is U = qV(r). For two charges in external field: U = q₁V(r₁) + q₂V(r₂) + q₁q₂/(4πε₀r₁₂), combining individual interactions with field and mutual interaction.

A dipole in uniform field E experiences torque τ = p × E but no net force. Potential energy U(θ) = −pE cosθ = −p·E (taking U = 0 at θ = 90°). Stable equilibrium at θ = 0° (p parallel to E, U = −pE); unstable at θ = 180°.

Six key results: (1) E = 0 inside conductor. (2) E is normal to surface, magnitude E = σ/ε₀. (3) Excess charge resides only on surface. (4) Potential is constant throughout volume and surface. (5) Cavity in conductor has E = 0 (electrostatic shielding). (6) Charges inside cavity induce inner surface charges; exterior not shielded from interior.

Electrostatic shielding protects sensitive instruments from external fields. Aircraft tyres are conducting to drain static charge. Vehicles carrying inflammable materials use metallic ropes to ground continuously accumulated static.

Dielectrics are non-conducting. Non-polar molecules develop induced dipole moments in external field; polar molecules partially align permanent dipoles. Net dipole moment per unit volume is polarisation P = ε₀χeE, where χe is electric susceptibility.

Polarisation produces induced surface charge densities ±σp, creating opposing field that reduces total field to E = E₀/K (K = dielectric constant > 1). Unlike a conductor, dielectric field is only reduced, not cancelled.

A capacitor (two conductors separated by insulator, charges Q and −Q) has capacitance C = Q/V, unit farad (F = C V⁻¹). C depends on geometry and dielectric nature. Practical units: μF (10⁻⁶ F), nF (10⁻⁹ F), pF (10⁻¹² F).

Large C allows storage of large Q at small V, staying below dielectric breakdown (air: 3 × 10⁶ V/m). Capacitors are critical in AC circuits, filtering, timing, and energy storage.

For plates of area A, separation d (d² << A, vacuum between plates): E = σ/ε₀ = Q/(ε₀A) between plates (zero outside). Potential difference V = Ed = Qd/(ε₀A). Capacitance C₀ = ε₀A/d.

C increases with A and decreases with d. For A = 1 m², d = 1 mm: C ≈ 8.85 nF. A capacitance of 1 F would require plates ~30 km × 30 km for 1 cm separation — illustrating how impractically large the farad unit is.

Dielectric of constant K inserted fully: polarisation reduces net field to E₀/K, potential difference to V₀/K, and capacitance increases to C = KC₀ = Kε₀A/d. Permittivity ε = ε₀K. The ratio K = C/C₀ = ε/ε₀ > 1.

This result holds for any capacitor type. A partially inserted dielectric slab can be treated as series combination of dielectric and air sections.

Series: 1/C = Σ(1/Cᵢ); all carry same charge Q, voltages add; series C is less than smallest individual. Parallel: C = ΣCᵢ; all at same V, charges add; parallel C exceeds largest individual.

Series preferred for higher voltage tolerance or obtaining small C. Parallel preferred for larger C and charge storage. Any network reduces to series/parallel steps.

Energy U = Q²/(2C) = (1/2)CV² = (1/2)QV — work done in charging, stored in the electric field between plates. Energy density: u = (1/2)ε₀E², valid for any electric field configuration.

When two capacitors are connected (charge redistribution), energy is lost as heat and radiation. This loss equals C₁C₂(V₁−V₂)²/[2(C₁+C₂)] and is unavoidable.

Chapter 2 establishes electrostatic potential and capacitance as central tools for analysing electrostatic systems. Key results include V = Q/(4πε₀r) for a point charge, V ∝ 1/r² for a dipole, superposition principle for systems of charges, perpendicularity of E to equipotential surfaces, U = qV in external fields, and U = −p·E for a dipole. Conductors in equilibrium obey six fundamental properties. Dielectrics increase capacitance by factor K through polarisation. Capacitors store energy U = CV²/2 with energy density u = ε₀E²/2.

From an examination perspective, this chapter is among the most heavily tested in CBSE Class 12. Board questions frequently cover: V = Q/(4πε₀r), C = ε₀A/d derivation, dielectric effect C = KC₀, series-parallel combinations, energy stored and energy density, conductor properties and shielding, dipole potential, and numerical problems on potential, potential energy, and capacitance.