Electrostatic Potential and Capacitance

2.1 Core concepts

The Coulomb force between two point charges, like the gravitational force between two masses, is conservative: its inverse-square form guarantees that the work done in moving a charge between two fixed end-points is independent of the path taken. This single mathematical fact lets us define a potential energy function U(r) for a charge in an electrostatic field, in exact analogy with the gravitational potential energy mgh near Earth's surface (NCERT §2.1, p. 45–46). The electrostatic potential energy difference between two points P and R is defined as the work done by an external agency, slowly and without producing any acceleration, in moving a test charge q from R to P:

ΔU = U_P − U_R = W_RP (NCERT Eq. 2.2, p. 46).

Dividing through by the test charge q (and taking q to be a unit positive charge as a limiting procedure) gives the electrostatic potential V — work per unit positive test charge brought from infinity to the point in question, with V at infinity chosen to be zero (NCERT §2.2, p. 47, Eq. 2.4). Only potential differences are physically meaningful; the absolute V depends on the (conventional) choice of zero.

For a point charge Q at the origin, integrating the Coulomb force along any path from infinity to a point at distance r gives

V(r) = Q/(4πε₀ r) (NCERT Eq. 2.8, p. 48).

The potential is positive for positive Q and negative for negative Q; it falls off as 1/r, while the electric field falls off as 1/r² (NCERT Fig. 2.4, p. 49). Note carefully: V is a scalar, so for a system of charges it is added algebraically with the signs of the charges included, not as a vector — a crucial simplification compared with the field calculation.

Dipole potential (NCERT §2.4, p. 49–50). An electric dipole consists of two equal and opposite charges +q and −q separated by a small distance 2a, with dipole moment p = 2qa directed from −q to +q. For a point at distance r from the centre with r ≫ a, geometry yields

V = (1/4πε₀) (p·r̂)/r² = p cos θ/(4πε₀ r²) (NCERT Eq. 2.15, p. 50),

where θ is the angle between p and the position vector. Two observations stand out. First, V falls off as 1/r² for a dipole — faster than the 1/r decay of a point charge — because of the partial cancellation between the +q and −q contributions. Second, on the axial line (θ = 0 or π) V = ±p/(4πε₀ r²) (maximum magnitude), while on the equatorial plane (θ = π/2) V = 0 even though E is non-zero. This is a common conceptual trap: V = 0 does not imply E = 0.

For a system of n point charges, superposition gives V = (1/4πε₀) Σ q_i/r_iP. For a uniformly charged thin spherical shell of total charge q and radius R, integration (or Gauss's law combined with V = −∫E·dl) yields V = q/(4πε₀ r) outside and V = q/(4πε₀ R) inside (constant) — the same as the surface value, even though the field inside is zero (NCERT §2.5, p. 51–52, Eq. 2.19).

Equipotential surfaces (NCERT §2.6, p. 54). An equipotential surface is a surface on which V has the same value at every point. Several consequences follow immediately. (i) The electric field E must be perpendicular (normal) to the equipotential surface at every point, because any tangential component would do work moving a charge along the surface — contradicting ΔV = 0. (ii) No work is done in moving a test charge along an equipotential. (iii) Two equipotentials cannot intersect (else E would have two directions at the intersection). For a single point charge equipotentials are concentric spheres; for a uniform field they are parallel planes perpendicular to the field; for a dipole they look like distorted figure-eights; for two equal positive charges they form a characteristic dumbbell pattern.

Field–potential relation (NCERT §2.6.1, p. 55). If two closely spaced equipotentials have potentials V and V + δV separated by perpendicular distance δl, the work done moving a unit test charge perpendicular to the surface is qE δl = −δV; cancelling and taking the limit,

E = −dV/dl (NCERT Eq. 2.21, p. 55).

The minus sign says the field points in the direction of steepest decrease of V. Equivalently, in vector form E = −∇V — the field is the negative gradient of the potential.

Potential energy of a system of charges (NCERT §2.7, p. 55–56). Bringing two charges q₁, q₂ from infinity to separation r₁₂ requires external work

U₁₂ = (1/4πε₀)(q₁q₂/r₁₂) (Eq. 2.22).

For three charges, U = (1/4πε₀)(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃) — one term per distinct pair (Eq. 2.26). U is positive for like charges (work has to be done against repulsion) and negative for unlike (system gains energy when assembled).

For a charge q sitting in an external field that produces potential V(r), the potential energy is U = qV(r) (Eq. 2.27, p. 58). One electron volt (eV) is defined as the energy gained by an electron crossing a potential difference of 1 V: 1 eV = 1.6 × 10⁻¹⁹ J (NCERT §2.8.1, p. 59).

Dipole in external field (NCERT §2.8.3, p. 60). A dipole of moment p in a uniform external E experiences zero net force but a torque τ = p × E. The work done rotating the dipole from θ₀ = π/2 to θ is

U(θ) = −pE cos θ = −p·E (NCERT Eq. 2.32, p. 60),

minimum (= −pE, stable) at θ = 0 (p parallel to E) and maximum (= +pE, unstable) at θ = π (anti-parallel).

Electrostatics of conductors (NCERT §2.9, p. 61–63). Five key results follow from the fact that the free electrons in a conductor in equilibrium have zero net drift:

1. E = 0 inside a conductor in electrostatic equilibrium.
2. The electric field at the surface is normal to the surface — any tangential component would drive a surface current.
3. All excess charge resides on the outer surface of a conductor (Gauss's law applied to an interior Gaussian surface gives zero enclosed charge).
4. V is constant throughout the body and surface of a conductor — the entire conductor is one equipotential.
5. The surface field is E = σ/ε₀ n̂ (Eq. 2.35, p. 63), derived using a Gaussian pill-box straddling the surface. Note the σ/ε₀ — not σ/(2ε₀), which is the formula for an isolated infinite plane sheet. A consequence is electrostatic shielding: the field inside a charge-free cavity within a conductor is zero, regardless of how the conductor is charged or what external fields are applied. This is why sensitive electronics are housed in metallic enclosures (Faraday cages). Dielectrics (NCERT §2.10, p. 64–67). A dielectric is an insulator with no free charges, but its bound molecular dipoles can respond to an external field — non-polar molecules (CH₄) acquire an induced dipole moment, polar molecules (H₂O) align their permanent dipoles. Either way, a polarised dielectric carries a dipole moment per unit volume P called the polarisation. For a linear, isotropic dielectric, P = ε₀ χₑ E (NCERT Eq. 2.37, p. 66), where χₑ is the dimensionless electric susceptibility. The induced surface charges on the dielectric reduce the net field inside it by a factor K, the dielectric constant, related to susceptibility by K = 1 + χₑ. Capacitance (NCERT §2.11, p. 67–68). A capacitor is any pair of conductors separated by an insulator. When charge Q is transferred from one conductor to the other (creating +Q on one, −Q on the other), a potential difference V appears between them, proportional to Q. The ratio C = Q/V (Eq. 2.38) is the capacitance. SI unit is the farad (F = C V⁻¹); practical capacitors are rated in μF (10⁻⁶ F) or pF (10⁻¹² F). C depends only on the geometry of the conductors and the dielectric between them — not on the charge stored or voltage applied. The maximum field a dielectric can support without breakdown — the dielectric strength — limits the charge a capacitor can hold; for air it is about 3 × 10⁶ V m⁻¹. Parallel-plate capacitor (NCERT §2.12, p. 68–69). Two large plates of area A separated by d ≪ √A carry charges +Q and −Q. The field between them is E = σ/ε₀ = Q/(ε₀A); the potential difference is V = Ed; hence C = ε₀ A / d (Eq. 2.43, p. 69). If the gap is fully filled with a dielectric of constant K, the field reduces to σ/(K ε₀), V drops by factor K for the same Q, and C = K ε₀ A / d = K C₀ (NCERT §2.13, Eq. 2.51, p. 70). The general definition K = C/C₀ — ratio of capacitance with vs without the dielectric — applies to any capacitor geometry. Combinations of capacitors (NCERT §2.14, p. 71–72). Two rules to remember: - Series: same charge Q on each. The voltages add: V = V₁ + V₂ + … = Q(1/C₁ + 1/C₂ + …). Equivalent 1/C_s = Σ 1/C_i (Eq. 2.60). - Parallel: same voltage V across each. The charges add: Q = Σ Q_i = V Σ C_i. Equivalent C_p = Σ C_i (Eq. 2.67). Notice this is the opposite of the resistor rule — a classic exam trap. Energy stored (NCERT §2.15, p. 73–75). Charging a capacitor amounts to transferring infinitesimal charge δQ′ at the instantaneous voltage Q′/C; integrating, U = Q²/(2C) = ½ CV² = ½ Q V (Eq. 2.73, p. 74). The energy is stored in the field between the plates, with energy density u = ½ ε₀ E² (Eq. 2.73, p. 75) — a general result, valid in vacuum for any electrostatic configuration, not only parallel plates.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig. 2.3 / Fig. 2.4 (p. 48–49): Variation of V (∝ 1/r) and E (∝ 1/r²) with distance from a point charge — useful for distinguishing the two graphs in distractor sets.
• Fig. 2.5 (p. 50): Geometry of potential due to a dipole (r₁, r₂, θ) — basis of V = (p cos θ)/(4πε₀ r²).
• Fig. 2.9–2.11 (p. 54): Equipotential surfaces — concentric spheres for a point charge; parallel planes for a uniform field; characteristic patterns for dipole and two identical positive charges.
• Fig. 2.12 (p. 55): Two closely spaced equipotentials V and V + δV; derivation of E = −δV/δl.
• Fig. 2.17 (p. 63): Pill-box Gaussian surface used to prove E = σ/ε₀ n̂ at the surface of a charged conductor.
• Fig. 2.19 (p. 64): Summary diagram of the five electrostatic properties of a conductor.
• Fig. 2.20–2.23 (p. 65–67): Conductor vs dielectric in an external field; polar/non-polar molecules; bound surface charge ±σ_p reducing the net field.
• Fig. 2.25 (p. 68): Parallel-plate geometry — fields cancel outside, add to σ/ε₀ between the plates.
• Fig. 2.26–2.28 (p. 71–72): Series (same Q) and parallel (same V) combinations.
• Fig. 2.30 (p. 74): Capacitor charging as transfer of infinitesimal charge δQ′ at potential V′ = Q′/C, integrated to U = Q²/(2C).

2.4 Common confusions / NTA trap points

• V varies as 1/r for a point charge but as 1/r² for a short dipole — and E correspondingly as 1/r² and 1/r³. Distractors swap these.
• Potential is a scalar; it is added algebraically (with signs of charges), not as vectors. The hexagon-of-charges and "V at centre of equilateral triangle of charges" type questions exploit this.
• On the equatorial plane of a dipole, V = 0 but E ≠ 0; on the axis E ≠ 0 and V ≠ 0. Students confuse "V = 0 ⇒ E = 0".
• Inside a charged conductor E = 0 but V is constant and non-zero — equals the surface value. Inside a charged spherical shell V is the same as on the surface, V = q/(4πε₀R), not zero.
• When a dielectric is inserted with the battery connected, V is fixed and Q increases by factor K; when the battery is disconnected, Q is fixed and V decreases by factor K (so E inside the dielectric drops). NTA loves this distinction.
• Series combination: 1/C is additive (charge same on each); parallel: C is additive (V same across each). Students often invert this with the resistor rule.
• Capacitance depends only on geometry and the dielectric — not on the charge stored or the voltage applied.
• For a dipole in a uniform field, the net force is zero but the torque (p × E) is not; U = −p·E is minimum when p is aligned with E (stable) and maximum when antiparallel (unstable).
• Surface field of a conductor is σ/ε₀ (not σ/2ε₀ — that is for an isolated infinite plane sheet).
• 1 eV = 1.6 × 10⁻¹⁹ J — a unit of energy, not voltage.
• For a uniform field over a distance l, V = E × l (not E × l²) — option (D) of Q6 below highlights this trap.
• Inserting a slab of thickness t < d reduces the effective gap to d − t + t/K, a frequent CUET tweak.

2.5 Key formulas table