Maxwell's equations and electromagnetic waves

In short

Maxwell's four equations gather all of electrostatics, magnetostatics and induction into one scheme, and adding the displacement current term makes them self-consistent — the consequence being that fields can propagate through empty space as waves at c = 1/√(μ₀ε₀). Everything from a 50 Hz power line to a 5G handset, an optical fibre and an X-ray tube is one phenomenon seen at different frequencies. For a B.Sc student this is the climax of the electricity-and-magnetism paper, routinely carrying the long-answer question, and it reappears in GATE PH (Section: Electromagnetic Theory), JAM, JEST, TIFR, NET-JRF and in interviews at ISRO, DRDO, BARC, SAMEER, BEL, antenna and RF-design teams, and optics/photonics roles.

Explained

The four equations and what each says

• Gauss's law for electricity: ∇·D = ρ (free charge), or ∇·E = ρ/ε₀ in vacuum. Electric field lines begin and end on charge; the flux out of a closed surface measures the enclosed charge.
• Gauss's law for magnetism: ∇·B = 0. No magnetic monopole has ever been found, so B lines close on themselves and net magnetic flux through any closed surface is zero.
• Faraday's law: ∇×E = −∂B/∂t. A changing magnetic field makes a non-conservative circulating electric field — the minus sign is Lenz's law.
• Ampère–Maxwell law: ∇×H = J + ∂D/∂t. Both conduction current and a changing electric field produce a magnetic field.
• Supplementary relations needed to solve them: D = ε₀E + P = εE, B = μ₀(H + M) = μH, J = σE (Ohm), and the Lorentz force F = q(E + v×B), which is how fields are detected but is *not* one of the four.
• Integral forms (preferred for symmetric problems): ∮D·dS = Q_enc; ∮B·dS = 0; ∮E·dl = −dΦ_B/dt; ∮H·dl = I_enc + dΨ_D/dt.

Displacement current — why Maxwell had to add it

• Take the divergence of the uncorrected Ampère law: ∇·(∇×H) = 0 forces ∇·J = 0, which contradicts charge conservation ∇·J + ∂ρ/∂t = 0 whenever charge accumulates (a charging capacitor).
• Adding J_d = ∂D/∂t repairs it exactly: ∇·(J + ∂D/∂t) = ∇·J + ∂ρ/∂t = 0. So the displacement term is *demanded* by continuity, not fitted to data.
• Physical check: apply the integral law to a charging capacitor. Through a surface cutting the wire you count I; through a surface bulging between the plates there is no charge flow, but dΨ_D/dt = ε₀ A d(E)/dt = I. Both give the same B, so the law becomes path-independent.
• In a dielectric the term splits as ∂(ε₀E)/∂t (vacuum part, needs no matter) + ∂P/∂t (polarisation current, real charge motion).
• Magnitude: in copper at 50 Hz, J_d/J_c = ωε/σ ≈ 5 × 10⁻¹⁷ — negligible; at optical frequencies in a dielectric it is the *only* current. The crossover frequency where they are equal is ω = σ/ε.

Wave equation in free space and its solution

Derivation outline (the standard 8-mark answer). In a source-free vacuum (ρ = 0, J = 0) take the curl of Faraday's law: ∇×(∇×E) = −∂(∇×B)/∂t = −μ₀ε₀ ∂²E/∂t². Use ∇×(∇×E) = ∇(∇·E) − ∇²E and ∇·E = 0 to get

• Comparing with ∇²ψ = (1/v²)∂²ψ/∂t² gives v = 1/√(μ₀ε₀) = 2.998 × 10⁸ m/s — Maxwell's own argument that light is electromagnetic.
• Plane-wave solution: E = E₀ exp[i(k·r − ωt)], B = B₀ exp[i(k·r − ωt)], with ω = kc (free space), k = 2π/λ.
• Transversality: ∇·E = 0 → k·E₀ = 0; ∇·B = 0 → k·B₀ = 0. Both fields are perpendicular to the direction of travel; EM waves in free space have no longitudinal component (unlike waves in a hollow waveguide).
• Mutual orientation: Faraday's law gives k×E₀ = ωB₀, so B₀ = (k̂×E₀)/c. E, B, k̂ form a right-handed set, E and B are in phase, and E₀/B₀ = c.
• In a linear medium: v = 1/√(με) = c/n with refractive index n = √(εᵣμᵣ) ≈ √εᵣ for non-magnetic matter; frequency is unchanged on entering a medium, λ shrinks by n.
• Intrinsic impedance: η = E/H = √(μ/ε); η₀ = √(μ₀/ε₀) = 376.73 Ω ≈ 377 Ω (often quoted 120π Ω).

Energy, momentum and the Poynting vector

• Energy densities: u_E = ½ε₀E², u_B = B²/2μ₀. For a plane wave in vacuum these are equal at every instant, so u = ε₀E².
• Poynting vector S = E×H = (1/μ₀)E×B, in W/m², gives the instantaneous power flow per unit area along k̂.
• Poynting's theorem (energy conservation): −∂u/∂t = ∇·S + J·E. The fall in stored field energy equals power radiated out plus ohmic work done on charges.
• Intensity (time-averaged, sinusoidal): I = ⟨S⟩ = ½ε₀cE₀² = E_rms²/η₀ = E₀²/(2η₀).
• Momentum density = S/c²; radiation pressure P = I/c on a perfect absorber, 2I/c on a perfect reflector at normal incidence. This drives solar sails and gives the Sun's ≈1,361 W/m² solar constant a pressure of only ~4.5 µPa.
• Angular momentum is carried by circularly polarised light (spin ±ħ per photon); quantum link E = hν = hc/λ, p = h/λ.

Waves in conducting and dielectric media; skin depth

• In a conductor, substituting J = σE gives ∇²E = μσ ∂E/∂t + με ∂²E/∂t²; the first-order term makes the wave diffusive and attenuated.
• Complex propagation constant γ = α + jβ with k complex, so E = E₀e^−αzcos(ωt − βz).
• Good conductor (σ ≫ ωε): α = β = √(ωμσ/2), and the skin depth δ = 1/α = √(2/ωμσ) = 1/√(πfμσ) — the depth at which amplitude falls to 1/e (37%), power to 13.5%.
• E and H are no longer in phase: in a good conductor H lags E by 45°, and η = √(ωμ/σ)∠45°.
• Consequences used in practice: skin effect in power conductors (hence ACSR, stranded and hollow conductors, Litz wire), EMI shielding, microwave-oven mesh, eddy-current losses.
• Dispersion and absorption: ε becomes complex and frequency dependent (ε = ε′ − jε″); loss tangent tan δ = ε″/ε′ = σ/ωε. Normal dispersion dn/dλ < 0; anomalous near a resonance.
• Plasma: propagation only above the plasma frequency ω_p = √(ne²/ε₀m); below it the wave is reflected — the basis of ionospheric short-wave reflection used by AIR and ham operators.

Boundary conditions, reflection, polarisation

• At an interface: tangential E continuous; tangential H differs by surface current K; normal D differs by surface charge σ_s; normal B continuous. (E₁t = E₂t, D₁n − D₂n = σ_s, B₁n = B₂n, H₁t − H₂t = K.)
• Normal incidence between two dielectrics: reflection coefficient Γ = (η₂ − η₁)/(η₂ + η₁) = (n₁ − n₂)/(n₁ + n₂); transmission τ = 2η₂/(η₂ + η₁). Power reflectance R = Γ², T = 1 − R.
• On a perfect conductor η₂ = 0 → Γ = −1: total reflection with a 180° phase reversal in E, giving a standing wave with a node of E at the surface (basis of ground-plane antennas).
• Oblique incidence: Snell's law n₁sinθ₁ = n₂sinθ₂ follows from phase matching; Fresnel equations give parallel and perpendicular components; Brewster angle θ_B = tan⁻¹(n₂/n₁) gives zero reflection for parallel polarisation; total internal reflection for θ > θ_c = sin⁻¹(n₂/n₁) when n₁ > n₂ — the mechanism of optical fibre.
• Polarisation types: linear, circular, elliptical, and unpolarised (random). Produced by Polaroid (dichroism), reflection at θ_B, double refraction in calcite, scattering (why the sky is polarised).
• Potentials and gauge: B = ∇×A, E = −∇V − ∂A/∂t. In the Lorenz gauge ∇·A + μ₀ε₀ ∂V/∂t = 0 both potentials satisfy wave equations with sources, giving retarded potentials and the radiation field of an accelerating charge (Larmor power P = q²a²/6πε₀c³).

Spectrum, radiation and Indian practice

• Spectrum in increasing frequency: radio (3 kHz–300 MHz), microwave (300 MHz–300 GHz), infrared, visible (≈400–700 nm, i.e. 430–750 THz), ultraviolet, X-ray, gamma.
• A static charge gives no radiation, a uniformly moving one none in its rest frame; only accelerating charge radiates — hence dipole antennas, synchrotron radiation (RRCAT Indore's Indus-1/Indus-2 sources), bremsstrahlung in X-ray tubes.
• A half-wave dipole is λ/2 long with radiation resistance ≈73 Ω, which is why 75 Ω coaxial feeds are common; 50 Ω is the instrumentation standard.
• Indian spectrum is allotted by WPC (Department of Telecommunications) under the Indian Telegraph Act, 1885 and the Indian Wireless Telegraphy Act, 1933, now being superseded by the Telecommunications Act, 2023 as its provisions are notified; TRAI regulates tariffs and service quality. Indian 5G uses 700 MHz, 3.3–3.67 GHz and 26 GHz bands.
• Human-exposure limits: DoT enforces EMF limits for base stations at one-tenth of the ICNIRP general-public levels (since 2012), verified by TERM cells; appliance EMC/EMI testing follows IS/CISPR-series standards under BIS, and medical/industrial safety follows AERB rules for ionising radiation.
• Conventions to note: Gaussian units (used in Jackson's later chapters and in much astrophysics literature, while Griffiths is SI throughout) drop ε₀ and μ₀ and put 4π and c explicitly — check the system before quoting a formula in an exam.

Key points

• Maxwell's equations - ∇·D = ρ, ∇·B = 0, ∇×E = −∂B/∂t, ∇×H = J + ∂D/∂t; four equations plus constitutive relations describe all classical EM.
• Displacement current - ∂D/∂t; added to make Ampère's law consistent with charge conservation, and it makes wave propagation possible.
• Charge continuity - ∇·J + ∂ρ/∂t = 0 is contained in Maxwell's equations, not an extra assumption.
• No monopoles - ∇·B = 0 is why B can be written as ∇×A.
• Wave equation - ∇²E = μ₀ε₀∂²E/∂t²; speed 1/√(μ₀ε₀) = c, identifying light as an EM wave.
• Transverse - E ⊥ B ⊥ k, in phase, right-handed triad, with E₀ = cB₀ in vacuum.
• Impedance of free space - η₀ = √(μ₀/ε₀) = 377 Ω ≈ 120π Ω; E/H everywhere in a plane wave.
• Equal energy - electric and magnetic energy densities are equal in a vacuum plane wave; total u = ε₀E².
• Poynting vector - S = E×H (W/m²), direction of power flow; intensity I = ½ε₀cE₀².
• Radiation pressure - I/c absorbed, 2I/c reflected; momentum density = S/c².
• Refractive index - n = √(εᵣμᵣ); on entering a medium f is fixed, λ and v fall by n.
• Skin depth - δ = 1/√(πfμσ); higher f, σ or μ confine current nearer the surface.
• Boundary conditions - tangential E and normal B always continuous; E_t = 0 on a perfect conductor.
• Brewster angle - tan θ_B = n₂/n₁ gives perfectly polarised reflection.
• Only acceleration radiates - Larmor P = q²a²/6πε₀c³; the basis of every antenna.
• Plasma cut-off - below ω_p = √(ne²/ε₀m) a wave is reflected, as by the ionosphere.
• Gauge freedom - A and V are not unique; the Lorenz gauge gives symmetric wave equations for both.

Formulas and examples

1. Displacement current in a charging capacitor. A parallel-plate capacitor has plates of area 150 cm² separated by 2 mm in air, and the voltage rises at 4 × 10⁵ V/s.

• dE/dt = (dV/dt)/d = 4 × 10⁵ / 2 × 10⁻³ = 2 × 10⁸ V/m·s.
• A = 150 cm² = 1.5 × 10⁻² m².
• I_d = ε₀ A dE/dt = 8.854 × 10⁻¹² × 1.5 × 10⁻² × 2 × 10⁸ = 2.66 × 10⁻⁵ A ≈ 26.6 µA.
• Cross-check through the wire: C = ε₀A/d = 8.854 × 10⁻¹² × 1.5 × 10⁻²/2 × 10⁻³ = 6.64 × 10⁻¹¹ F; I = C dV/dt = 6.64 × 10⁻¹¹ × 4 × 10⁵ = 2.66 × 10⁻⁵ A — the two currents are equal, as the Ampère–Maxwell law requires.

2. Field amplitudes and pressure from intensity. A laser pointer delivers 5 mW into a spot of diameter 2 mm.

• Area = π(1 × 10⁻³)² = 3.1416 × 10⁻⁶ m²; I = 5 × 10⁻³/3.1416 × 10⁻⁶ = 1,592 W/m².
• E₀ = √(2I/ε₀c) = √(2 × 1592 / (8.854 × 10⁻¹² × 2.998 × 10⁸)) = √(3184/2.654 × 10⁻³) = √(1.1998 × 10⁶) = 1,095 V/m.
• B₀ = E₀/c = 1095/2.998 × 10⁸ = 3.65 × 10⁻⁶ T (3.65 µT).
• H₀ = E₀/η₀ = 1095/376.7 = 2.91 A/m; check I = ½E₀H₀ = ½ × 1095 × 2.91 = 1,593 W/m² ✓.
• Radiation pressure on a black surface = I/c = 1592/2.998 × 10⁸ = 5.31 × 10⁻⁶ Pa; on a mirror, 1.06 × 10⁻⁵ Pa.

3. Skin depth in copper at three frequencies. Copper: σ = 5.8 × 10⁷ S/m, μ = μ₀ = 4π × 10⁻⁷ H/m. δ = 1/√(πfμσ).

• πμσ = 3.1416 × 1.2566 × 10⁻⁶ × 5.8 × 10⁷ = 228.9. So δ = 1/√(228.9 f) = 0.0661/√f metres.
• At 50 Hz: δ = 0.0661/7.071 = 9.35 mm.
• At 1 MHz: δ = 0.0661/1000 = 66.1 µm.
• At 1 GHz: δ = 0.0661/31623 = 2.09 µm — hence silver-plating and surface finish dominate microwave losses.

4. Wave entering glass, with reflection. A 600 nm wave in air meets glass (n = 1.5, non-magnetic) at normal incidence.

• f = c/λ = 2.998 × 10⁸/6 × 10⁻⁷ = 5.00 × 10¹⁴ Hz, unchanged in glass.
• In glass v = c/n = 1.999 × 10⁸ m/s and λ = 600/1.5 = 400 nm.
• εᵣ = n² = 2.25; η₂ = η₀/n = 376.7/1.5 = 251.1 Ω.
• Γ = (251.1 − 376.7)/(251.1 + 376.7) = −125.6/627.8 = −0.20; R = 0.04, so 4% of power is reflected, 96% transmitted (the familiar single-surface loss that anti-reflection coatings remove).
• Brewster angle for air→glass: θ_B = tan⁻¹(1.5) = 56.3°.

5. Solar constant at the Earth. Take I = 1,361 W/m².

• E_rms = √(I η₀) = √(1361 × 376.7) = √(512,588) = 716 V/m; E₀ = √2 × 716 = 1,013 V/m.
• B₀ = E₀/c = 1013/2.998 × 10⁸ = 3.38 µT.
• u = I/c = 1361/2.998 × 10⁸ = 4.54 × 10⁻⁶ J/m³; radiation pressure on an absorbing surface = 4.54 µPa (≈9.1 µPa on a reflecting solar sail).
• Power on a 1 kW-rated 2 m² rooftop panel at 18% efficiency and normal incidence ≈ 1361 × 2 × 0.18 ≈ 490 W (less in practice: atmospheric attenuation brings surface irradiance to ~1,000 W/m², the STC rating condition).

Where it is used

• Antenna and RF design: dipole, patch and horn antennas for 5G, radar and satellite links — SAMEER, BEL, Astra Microwave, Tata Advanced Systems; impedance matching to 50 Ω uses exactly the Γ and η formulas.
• Space and remote sensing: ISRO's SAR (RISAT, NISAR) and microwave radiometers depend on polarisation, reflectance and atmospheric transmission windows; telemetry link budgets use I = P/4πr².
• Optical fibre and photonics: total internal reflection, dispersion and attenuation in fibre built and deployed under BharatNet; laser and detector work at CGCRI, IIT photonics labs.
• Power engineering: skin and proximity effects set conductor design (ACSR, bundled EHV lines), eddy-current and hysteresis loss in transformer cores, and corona/EMI compliance under CEA regulations.
• EMI/EMC and shielding: enclosure design, gaskets and cable screening tested to IS/CISPR standards for BIS certification; shielding effectiveness computed from skin depth.
• Medical and industrial: MRI gradient and RF coils, microwave diathermy, industrial heating, X-ray tubes (bremsstrahlung) regulated by AERB; research roles at BARC, RRCAT and SAMEER list EM theory explicitly.

Interview questions

• Q: Why did Maxwell add the displacement current term? A: Without it, taking the divergence of Ampère's law forces ∇·J = 0, which contradicts charge conservation when charge accumulates, as on a capacitor plate; ∂D/∂t fixes this exactly and also permits waves in vacuum.
• Q: Which of Maxwell's equations tells you magnetic monopoles do not exist, and what follows from it? A: ∇·B = 0; it allows B to be written as the curl of a vector potential A.
• Q: How do you show from the equations that light is an electromagnetic wave? A: Taking the curl of Faraday's law in a source-free vacuum gives ∇²E = μ₀ε₀∂²E/∂t², a wave equation with speed 1/√(μ₀ε₀) = 3 × 10⁸ m/s, which equals the measured speed of light.
• Q: Why are electromagnetic waves transverse in free space? A: ∇·E = 0 and ∇·B = 0 give k·E₀ = k·B₀ = 0, so neither field has a component along the propagation direction.
• Q: What is the impedance of free space and where is it used? A: η₀ = √(μ₀/ε₀) = 377 Ω; it is the ratio E/H in a plane wave and is used in antenna matching, reflection coefficients and shielding calculations.
• Q: In a vacuum plane wave, is more energy in the electric or magnetic field? A: Exactly equal — ½ε₀E² = B²/2μ₀ at every instant, since B = E/c.
• Q: What does the Poynting vector represent, and what is its direction for a plane wave? A: Instantaneous power flow per unit area in W/m²; it points along k̂, the direction of propagation, since S = E×H.
• Q: Define skin depth and say how it varies with frequency. A: The depth at which the field amplitude falls to 1/e of its surface value, δ = 1/√(πfμσ); it falls as 1/√f, so high-frequency current flows in a thin surface layer.
• Q: When light enters glass, which of frequency, wavelength and speed change? A: Frequency is fixed by the source; speed and wavelength both fall by the factor n.
• Q: Why is the reflected E field reversed in phase at a conducting surface? A: The tangential electric field must vanish on a perfect conductor, so the reflected wave must cancel the incident one there — Γ = −1.
• Q: What is the Brewster angle and why is reflected light polarised there? A: θ_B = tan⁻¹(n₂/n₁); at that angle the parallel-polarised component has zero reflection coefficient, so the reflection is purely perpendicular-polarised.
• Q: What kind of charge motion radiates? A: Only accelerating charge; radiated power follows the Larmor formula P = q²a²/6πε₀c³, which is the basis of all antennas and of synchrotron and X-ray sources.

Common mistakes

• Treating the Lorentz force law as a fifth Maxwell equation, or omitting the constitutive relations and then trying to solve in matter.
• Writing ∇×B = μ₀J in a time-varying problem — the displacement term is what makes waves exist.
• Saying the magnetic energy of a wave is negligible compared with the electric; numerically B is small (µT) but B²/2μ₀ equals ½ε₀E².
• Mixing the conventions: using √(2I/ε₀c) for E_rms instead of E₀, or forgetting the factor ½ in I = ½ε₀cE₀² for a sinusoidal wave.
• Using radiation pressure I/c for a mirror (it is 2I/c) or forgetting the cosine factor at oblique incidence.
• Claiming frequency changes inside a dielectric, or applying ω = kc instead of ω = kc/n inside a medium.
• Unit slips: δ formula needs μ in H/m and σ in S/m; k is in rad/m not m⁻¹ of cycles; η in ohms not ohm-metres.

Related topics

Electrostatics · Magnetism and electromagnetic induction · Wave optics: interference, diffraction and polarisation · Transmission lines and waveguides · Special relativity and field transformations · Antenna and microwave engineering