Electrostatics

In short

Electrostatics is the study of charges at rest: the forces between them, the electric field and potential they set up, and how conductors and dielectrics respond. It rests on Coulomb's law and its field-theoretic equivalent, Gauss's law, and supplies the first two of Maxwell's equations in their static form. In B.Sc "Electricity and Magnetism" papers it typically carries the largest single block of marks, and it dominates GATE/JAM/NET electromagnetism questions on Gauss's law, image charges and boundary-value problems. The same physics underlies capacitors, insulation design, high-voltage testing to Indian Standards, electrostatic precipitators in thermal plants and MEMS/semiconductor device work at ISRO, BARC, BHEL, CSIR labs and semiconductor employers.

Explained

Charge and Coulomb's law

• Charge is quantised (q = ne, e = 1.602 × 10⁻¹⁹ C) and conserved; SI unit coulomb (C), 1 C = 1 A·s.
• Coulomb's law: force between two point charges in vacuum is along the line joining them, proportional to q₁q₂ and inversely proportional to r². Valid for point/spherically symmetric charges at rest.
• 1/4πε₀ = 8.988 × 10⁹ N·m²/C²; ε₀ = 8.854 × 10⁻¹² C²/N·m² (F/m).
• Superposition: total force on a charge is the vector sum of pairwise Coulomb forces — this linearity is what makes the whole subject tractable.
• Inverse-square law is verified to extremely high precision (Plimpton–Lawton type experiments); any deviation would give the photon a mass.
• Electric field E = F/q₀ for a vanishingly small test charge; units N/C = V/m. Field of a point charge: radial, 1/r².
• Continuous distributions: replace q by λ dl (line), σ dA (surface) or ρ dV (volume) and integrate.
• Field lines: start on +ve, end on −ve charge, never cross, density ∝ |E|. Flux Φ = ∮E·dA, units N·m²/C = V·m.

Gauss's law and its applications

• Statement: ∮E·dA = q_enclosed/ε₀ over any closed surface. Differential form ∇·E = ρ/ε₀ (Maxwell's first equation).
• A direct consequence of the inverse-square law plus superposition; it is exact, but *useful* only when symmetry (spherical, cylindrical, planar) lets you take E out of the integral.
• Standard results (derive these — they are examined every year):
  1. Point charge / outside a uniformly charged sphere: E = q/4πε₀r².
  2. Inside a uniformly charged solid sphere (charge density ρ): E = ρr/3ε₀ = qr/4πε₀R³ — rises linearly, zero at centre.
  3. Inside a uniform spherical shell: E = 0 (shielding).
  4. Infinite line charge: E = λ/2πε₀r, radial.
  5. Infinite plane sheet: E = σ/2ε₀, independent of distance.
  6. Just outside a conductor: E = σ/ε₀, normal to the surface.
  7. Between parallel plates with ±σ: E = σ/ε₀; outside, zero.
• Procedure: identify symmetry → choose a Gaussian surface on which E is constant and either parallel or perpendicular to dA → compute enclosed charge → solve for E.
• Limitation: cannot give E for a finite rod, a disc or a square plate — there integration (or numerical methods) is needed.

Potential, Poisson and Laplace

• The electrostatic field is conservative: ∮E·dl = 0, so ∇×E = 0 and E = −∇V.
• Potential V(r) = work per unit charge to bring a test charge from infinity; unit volt (V) = J/C. Potential of a point charge V = q/4πε₀r.
• Potential difference V_A − V_B = ∫ from A to B of E·dl. Only differences are physical; the reference (usually infinity, or earth at 0 V) is a choice.
• Equipotential surfaces are everywhere perpendicular to E; no work is done moving a charge along one. A conductor's surface is an equipotential.
• Combining ∇·E = ρ/ε₀ with E = −∇V gives Poisson's equation ∇²V = −ρ/ε₀; in charge-free regions Laplace's equation ∇²V = 0.
• Uniqueness theorem: a solution of Laplace's equation satisfying the given boundary conditions is *the* solution — this licenses clever guesses such as image charges.
• Mean-value property / Earnshaw's theorem: V has no local maximum or minimum in a charge-free region, so a charge cannot be held in stable equilibrium by electrostatic forces alone.
• Solution methods: separation of variables (Cartesian, spherical with Legendre polynomials, cylindrical with Bessel functions), method of images, multipole expansion, relaxation/numerical methods.

Conductors, capacitance and energy

• In electrostatic equilibrium inside a conductor: E = 0, ρ = 0, V is constant throughout, all excess charge resides on the surface, and E just outside is σ/ε₀ along the normal.
• Electrostatic shielding: a cavity in a conductor with no enclosed charge has E = 0 — the basis of Faraday cages, shielded cables and HV test enclosures.
• Force per unit area on a conductor's surface (electrostatic pressure) = σ²/2ε₀, always outward.
• Capacitance C = Q/V, unit farad (F = C/V); practical units µF, nF, pF. Depends only on geometry and the dielectric.
• Standard capacitances: parallel plate ε₀A/d; isolated sphere 4πε₀R; concentric spheres 4πε₀ab/(b−a); coaxial cable 2πε₀L/ln(b/a).
• Series: 1/C = Σ1/Cᵢ (same charge, voltages add). Parallel: C = ΣCᵢ (same voltage, charges add).
• Energy stored U = ½QV = ½CV² = Q²/2C; energy density u = ½ε₀E² (in a dielectric, ½D·E), i.e. the energy resides in the field.
• Self-energy vs interaction energy: U = ½∑qᵢVᵢ for a set of charges counts each pair once; the self-energy of a point charge diverges — a known limitation of the classical point-charge model.

Dielectrics and polarization

• A dielectric has no free charge; an applied field produces polarization P (dipole moment per unit volume, C/m²) by (i) electronic distortion, (ii) ionic/atomic displacement, (iii) orientation of permanent dipoles (polar molecules such as H₂O).
• Bound charges: σ_b = P·n̂ on the surface, ρ_b = −∇·P in the volume.
• Displacement vector D = ε₀E + P, with ∇·D = ρ_free and ∮D·dA = q_free — the form used in all practical capacitor and cable work.
• Linear isotropic media: P = ε₀χ_e E, D = εE, ε = ε₀ε_r, ε_r = 1 + χ_e (dielectric constant, dimensionless).
• Inserting a dielectric of ε_r at constant charge reduces E and V by ε_r and raises C by ε_r; at constant voltage (battery connected) charge and stored energy rise by ε_r and the slab is pulled in.
• Typical ε_r: air ≈ 1.0006, transformer oil ≈ 2.2, paper ≈ 3.5, mica 5–7, water ≈ 80, barium titanate ceramics in the hundreds to thousands.
• Clausius–Mossotti relation links ε_r to molecular polarizability α: (ε_r − 1)/(ε_r + 2) = Nα/3ε₀.
• Dielectric strength / breakdown: air ≈ 3 kV/mm (3 MV/m), transformer oil ≈ 10–15 kV/mm, mica and polymer films far higher. Exceeding it causes breakdown — ionisation, arcing, permanent damage.
• Indian practice: breakdown-voltage testing of insulating oil and HV test techniques follow the relevant Indian Standards (IS 6792 for insulating-oil electric strength; IS 2071 series for high-voltage test techniques); edition numbers are revised periodically, so check the current edition before quoting one.
• Boundary conditions at an interface between two media: the tangential component of E is continuous (E_t1 = E_t2); the normal component of D jumps by the free surface charge (D_n1 − D_n2 = σ_free; equal if none). Hence field lines refract: tan θ₁/tan θ₂ = ε_r1/ε_r2.

Dipoles, multipoles and images

• Electric dipole moment p = q·d (C·m), directed from −q to +q. Potential V = p·r̂/4πε₀r², field falls as 1/r³.
• On the axis E = 2p/4πε₀r³; on the equatorial line E = p/4πε₀r³ (opposite in direction).
• In a uniform field: net force zero, torque τ = p × E, energy U = −p·E (minimum when aligned). In a non-uniform field there is also a net force F = ∇(p·E).
• Multipole expansion: V = (1/4πε₀)[q/r + p·r̂/r² + quadrupole/r³ + …]; the leading non-vanishing term dominates at large r. Monopole term vanishes for a neutral body, so a neutral molecule's far field is dipolar.
• Method of images: replace a conductor by fictitious charges that reproduce its boundary condition.
  • Point charge q at distance d from an infinite earthed plane → image −q at −d; force = −q²/4πε₀(2d)² (attractive), induced surface charge integrates to −q.
  • Point charge q at distance d from an earthed sphere of radius R → image −qR/d at R²/d from the centre.
• Polarizability and induced dipole: a neutral atom in a field acquires p = αE, giving the 1/r⁵ attraction of an atom to a point charge.

Key points

• Coulomb's law — F = q₁q₂/4πε₀r², the experimental foundation; inverse-square and along the line of centres.
• Superposition — fields and potentials from many charges add vectorially/algebraically; all of electrostatics is linear.
• Gauss's law — flux through any closed surface = enclosed charge/ε₀; exact always, practical only with symmetry.
• Conservative field — ∇×E = 0, so a single-valued potential V exists and E = −∇V.
• Shell results — field zero inside a uniform shell; outside, as if all charge were at the centre.
• Plane sheet — E = σ/2ε₀, uniform and distance-independent; two opposite sheets give σ/ε₀ between, zero outside.
• Conductor in equilibrium — E = 0 inside, charge on the surface, surface is an equipotential, E_outside = σ/ε₀.
• Poisson/Laplace — ∇²V = −ρ/ε₀ in charge, ∇²V = 0 outside; uniqueness theorem justifies images.
• Earnshaw's theorem — no stable electrostatic trap for a charge in free space; V has no interior extremum.
• Capacitance C = Q/V depends only on geometry and ε_r, never on the applied voltage.
• Energy density u = ½ε₀E² (½D·E in a dielectric) — energy is stored in the field itself.
• Polarization P = dipole moment per unit volume; bound charges σ_b = P·n̂, ρ_b = −∇·P.
• D-field — ∇·D = ρ_free lets you ignore bound charge entirely in linear media.
• Dielectric constant ε_r = 1 + χ_e ≥ 1; inserting a dielectric always increases capacitance.
• Boundary conditions — E tangential continuous, D normal jumps by σ_free; field lines bend at interfaces.
• Dipole — τ = p × E, U = −p·E, V ∝ 1/r², E ∝ 1/r³.
• Images — earthed plane: image −q at mirror position; earthed sphere of radius R: −qR/d at R²/d.
• Dielectric strength — air ≈ 3 kV/mm sets practical clearances in all HV design.

Formulas and examples

Example 1 — Coulomb force and the field of two charges. Charges q₁ = +4 µC at x = 0 and q₂ = −2 µC at x = 0.30 m in air.

• Force on q₂: F = 8.988 × 10⁹ × (4 × 10⁻⁶ × 2 × 10⁻⁶)/(0.30)² = 8.988 × 10⁹ × 8 × 10⁻¹²/0.09 = 0.0719/0.09 ≈ 0.799 N, attractive (towards q₁).
• Field at the midpoint x = 0.15 m: from q₁, E₁ = 8.988 × 10⁹ × 4 × 10⁻⁶/0.0225 = 35,952/0.0225 ≈ 1.598 × 10⁶ V/m along +x; from q₂, E₂ = 8.988 × 10⁹ × 2 × 10⁻⁶/0.0225 ≈ 0.799 × 10⁶ V/m, also along +x (towards the negative charge). Total ≈ 2.40 × 10⁶ V/m along +x.
• Potential at the midpoint: V = 8.988 × 10⁹ × (4 × 10⁻⁶ − 2 × 10⁻⁶)/0.15 = 8.988 × 10⁹ × 2 × 10⁻⁶/0.15 ≈ 1.20 × 10⁵ V. Note E ≠ 0 where V is non-zero, and V can be zero where E is not — the two are independent questions.

Example 2 — Capacitor with a dielectric, charge held constant. Parallel plates of area A = 200 cm² = 0.02 m², separation d = 1.0 mm, air.

• C₀ = ε₀A/d = 8.854 × 10⁻¹² × 0.02/10⁻³ = 1.771 × 10⁻¹⁰ F ≈ 177 pF.
• Charged to 500 V: Q = CV = 1.771 × 10⁻¹⁰ × 500 = 8.86 × 10⁻⁸ C ≈ 88.6 nC; U₀ = ½ × 1.771 × 10⁻¹⁰ × 500² = 2.21 × 10⁻⁵ J.
• E₀ = V/d = 500/10⁻³ = 5 × 10⁵ V/m = 0.5 kV/mm — safely below air's ≈ 3 kV/mm.
• Battery disconnected, a mica slab (ε_r = 6) fills the gap: C = 6 × 177 = 1,063 pF; Q unchanged, so V = Q/C = 8.86 × 10⁻⁸/1.063 × 10⁻⁹ ≈ 83.3 V and E drops to 8.33 × 10⁴ V/m.
• New energy U = Q²/2C = (8.86 × 10⁻⁸)²/(2 × 1.063 × 10⁻⁹) ≈ 3.69 × 10⁻⁶ J — one-sixth of U₀. The lost energy did work pulling the slab in.

Example 3 — Gauss's law on a charged solid sphere. Insulating sphere R = 10 cm carries Q = 5 nC uniformly through its volume.

• ρ = Q/(4/3 πR³) = 5 × 10⁻⁹/(4.18879 × 10⁻³) ≈ 1.194 × 10⁻⁶ C/m³.
• At r = 5 cm (inside): E = ρr/3ε₀ = 1.194 × 10⁻⁶ × 0.05/(3 × 8.854 × 10⁻¹²) = 5.97 × 10⁻⁸/2.656 × 10⁻¹¹ ≈ 2,248 V/m. (Check with qr/4πε₀R³: 8.988 × 10⁹ × 5 × 10⁻⁹ × 0.05/10⁻³ = 2,247 V/m.)
• At r = 20 cm (outside): E = 8.988 × 10⁹ × 5 × 10⁻⁹/0.04 ≈ 1,123 V/m.
• Surface (r = R): E = 8.988 × 10⁹ × 5 × 10⁻⁹/0.01 ≈ 4,494 V/m — the maximum; V at the centre = 1.5 × V(R) = 1.5 × 449.4 ≈ 674 V.

Example 4 — Image charge near an earthed plane. q = 2 nC held 2 cm from a large earthed metal plate.

• Image: −2 nC at 2 cm behind the plate, separation 2d = 4 cm.
• F = 8.988 × 10⁹ × (2 × 10⁻⁹)²/(0.04)² = 8.988 × 10⁹ × 4 × 10⁻¹⁸/1.6 × 10⁻³ = 3.595 × 10⁻⁸/1.6 × 10⁻³ ≈ 2.25 × 10⁻⁵ N, attractive.
• Work to remove the charge to infinity = −q²/16πε₀d = −8.988 × 10⁹ × 4 × 10⁻¹⁸/(4 × 0.02)… evaluating ½ of the two-point-charge energy: W = q²/16πε₀d = 8.988 × 10⁹ × 4 × 10⁻¹⁸/(4 × 0.02) ≈ 4.49 × 10⁻⁷ J must be supplied. (The factor ½ relative to a genuine charge pair arises because the image is not a real charge.)

Example 5 — Dipole in a field. p = 3 × 10⁻³⁰ C·m in E = 2 × 10⁵ V/m, at 30° to the field.

• τ = pE sin θ = 3 × 10⁻³⁰ × 2 × 10⁵ × 0.5 = 3 × 10⁻²⁵ N·m.
• U = −pE cos θ = −3 × 10⁻³⁰ × 2 × 10⁵ × 0.866 ≈ −5.20 × 10⁻²⁵ J.
• Work to rotate from 0° to 90° = pE(cos 0 − cos 90) = 6 × 10⁻²⁵ J.

Where it is used

• Capacitor and cable manufacture — ceramic, film and electrolytic capacitors; XLPE and paper-oil HV cable insulation design where C, ε_r and dielectric strength decide the geometry (Epcos/TDK India, Finolex, Polycab, KEI).
• High-voltage engineering and testing — clearances, creepage, bushings, insulator design and HV test labs at CPRI Bengaluru, ERDA Vadodara, BHEL, Siemens and ABB India, working to the IS 2071 high-voltage test technique series.
• Pollution control and process industry — electrostatic precipitators on thermal power station and cement kiln stacks (BHEL, Thermax); electrostatic painting and powder coating in automobile plants; electrostatic separation in mineral processing.
• Semiconductor and MEMS work — MOS gate capacitance, depletion-region fields solved from Poisson's equation, ESD protection design; roles at the India Semiconductor Mission fabs, SCL Mohali, Intel/AMD/Qualcomm India design centres.
• Instrumentation and research — capacitive sensors (level, pressure, touchscreens), Van de Graaff and electrostatic accelerators, ion optics and mass spectrometry at BARC, IPR Gandhinagar, IUAC Delhi and ISRO payload labs.
• Teaching and competitive exams — the topic is examined in GATE Physics (EM section), JAM, CSIR-UGC NET, JEST, and in the Engineering Services and state PSC syllabi; the same derivations appear in NCERT Class XII, which many teaching jobs require.

Interview questions

• Q: Why is Gauss's law always true but only sometimes useful? A: It follows from the inverse-square law and superposition for any closed surface, but you can extract E from the integral only when symmetry makes E constant in magnitude and fixed in direction over the surface.
• Q: Why is the field zero inside a conductor in equilibrium? A: Free charges move until they redistribute on the surface; any residual internal field would keep driving current, so equilibrium requires E = 0.
• Q: Why is E = σ/ε₀ just outside a conductor but σ/2ε₀ for an isolated sheet? A: A conductor has no field on its inner side, so all the flux goes out one way; an isolated sheet sends flux out both faces, halving each.
• Q: Can V be zero where E is not, and vice versa? A: Yes — at the midpoint between equal opposite charges V = 0 but E ≠ 0; at the centre of a uniform shell E = 0 but V ≠ 0. E is the gradient of V, not proportional to it.
• Q: What does Earnshaw's theorem rule out? A: Stable equilibrium of a charge in a purely electrostatic field in free space, because Laplace's equation forbids a local minimum of potential energy — hence Penning and RF (Paul) traps use magnetic or time-varying fields.
• Q: What justifies the method of images? A: The uniqueness theorem: if an image configuration reproduces the same boundary conditions in the region of interest, its solution of Laplace's equation is the only solution there.
• Q: What is the difference between E, D and P? A: E is the total field that exerts force, P is the induced dipole moment per unit volume, and D = ε₀E + P is the auxiliary field whose divergence counts only free charge.
• Q: Does inserting a dielectric ever reduce stored energy? A: Yes — at constant charge (battery disconnected) U falls by ε_r because the slab is pulled in and does work; at constant voltage U rises by ε_r, with the battery supplying it.
• Q: Why do field lines bend at a dielectric interface? A: Tangential E is continuous while normal D is continuous (no free charge), so tan θ₁/tan θ₂ = ε_r1/ε_r2.
• Q: Where is the energy of a charged capacitor stored? A: In the field between the plates, with density ½ε₀E² (½D·E in a dielectric) — integrating it over the volume reproduces ½CV².
• Q: Why does the self-energy of a point charge diverge? A: ∫½ε₀E²dV with E ∝ 1/r² diverges as r → 0; classical electrostatics has no consistent model of a point charge, and the problem is handled by renormalisation in quantum field theory.
• Q: A charge is placed inside an uncharged hollow conductor — what are the fields? A: −q is induced on the inner wall and +q on the outer; the field in the conductor is zero and outside it is as if q sat at the centre, so the cavity's contents are shielded but the charge is not hidden.

Common mistakes

• Using Gauss's law on a finite rod, disc or square plate, where no symmetry exists — the answer must come from integration.
• Treating the Gaussian surface's enclosed charge as the *total* charge: charges outside contribute to E at the surface but not to the net flux.
• Confusing E = σ/ε₀ (conductor surface) with σ/2ε₀ (isolated sheet), and forgetting that between two oppositely charged plates the two sheets add to σ/ε₀.
• Writing V as proportional to E, or assuming E = 0 wherever V = 0 — V is a scalar sum, E a vector sum.
• Forgetting the 1/r³ fall-off of a dipole field, or using the axial formula on the equatorial line (they differ by a factor of 2 and in direction).
• Mixing up which quantity stays fixed when a dielectric is inserted — Q is fixed only if the battery is disconnected, V only if it stays connected.
• Unit slips: µC and nC left unconverted, cm used for r, or ε₀ omitted; and quoting ε_r with units (it is dimensionless).

Related topics

• Magnetism and electromagnetic induction · Maxwell's equations and electromagnetic waves · DC circuits and network theorems · Dielectric materials and polarization mechanisms · Boundary-value problems and mathematical physics methods · Semiconductor device physics (p-n junction fields)