Maxwell's equations and displacement current

Why it matters

Maxwell's four equations summarise everything in this subject — electrostatics, magnetostatics and induction — in one consistent set, and the displacement current that Maxwell added predicts electromagnetic waves. They explain why capacitors pass AC, why cables and PCB traces behave as transmission lines at high frequency, how antennas radiate, and how to decide whether a material behaves as a conductor or a dielectric at a given frequency.

Key ideas

The four laws (general, time-varying form):

1. Gauss's law: electric flux out of a closed surface equals the free charge enclosed. Electric field lines start and end on charges.
2. Gauss's law for magnetism: magnetic flux out of any closed surface is zero. There are no magnetic monopoles; B-lines close on themselves.
3. Faraday's law: a time-varying B produces a circulating E (∇ × E = −∂B/∂t).
4. Ampère–Maxwell law: H circulates around conduction current and around a time-varying D (∇ × H = J + ∂D/∂t).

They are completed by the constitutive relations D = εE, B = μH, J = σE, and the Lorentz force F = Q(E + u × B).

Why displacement current was needed. Take the divergence of the original ∇ × H = J: the left side is always zero (div of a curl), so ∇·J would have to be zero — but the continuity equation says ∇·J = −∂ρv/∂t, non-zero whenever charge accumulates (for example on a charging capacitor plate). Adding Jd = ∂D/∂t fixes this, because ∂(∇·D)/∂t = ∂ρv/∂t. Physically: around the wire to a charging capacitor, an Amperian loop gives the same ∮H·dl whether its surface is pierced by the wire (conduction current) or passes between the plates (displacement current). The two currents are equal: I_d = C dV/dt.

Conduction versus displacement current. For sinusoidal fields in a material, |J_c|/|J_d| = σ/(ωε), the loss tangent.

• σ/(ωε) ≫ 1 (say > 100): good conductor — displacement current negligible.
• σ/(ωε) ≪ 1 (say < 0.01): good dielectric — conduction current negligible. The same material can be either, depending on frequency.

Waves. In a source-free region, combining the two curl equations gives the wave equation, with speed u = 1/√(με). In free space c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s, and the ratio E/H for a plane wave is the intrinsic impedance η₀ = √(μ₀/ε₀) ≈ 377 Ω (120π Ω). Without displacement current there would be no waves.

Static limits. When ∂/∂t = 0 the equations split into electrostatics (∇·D = ρv, ∇ × E = 0) and magnetostatics (∇·B = 0, ∇ × H = J) — the earlier topics of this subject.

Phasor form. For sinusoidal fields, replace ∂/∂t by jω: ∇ × E = −jωB and ∇ × H = J + jωD.

Formulas

• ∇·D = ρv ↔ ∮ D·dS = Q_enc — Gauss's law.
• ∇·B = 0 ↔ ∮ B·dS = 0 — no magnetic monopoles.
• ∇ × E = −∂B/∂t ↔ ∮ E·dl = −d/dt ∫ B·dS — Faraday's law.
• ∇ × H = J + ∂D/∂t ↔ ∮ H·dl = I_enc + d/dt ∫ D·dS — Ampère–Maxwell law.
• Jd = ∂D/∂t = ε ∂E/∂t (A/m²); I_d = C dV/dt for a capacitor (A).
• ∇·J = −∂ρv/∂t — continuity equation (conservation of charge).
• |Jc| / |Jd| = σ / (ωε) — ω = 2πf (rad/s), σ in S/m, ε in F/m.
• c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s, η₀ = √(μ₀/ε₀) ≈ 377 Ω, u = 1/√(με).

Worked examples

Example 1 (standard). A parallel-plate capacitor (A = 0.01 m², d = 1 mm, air) has its voltage changing at 10⁶ V/s. Find the displacement current.

1. dE/dt = (dV/dt)/d = 10⁶ / 10⁻³ = 10⁹ V/(m·s).
2. Jd = ε₀ dE/dt = 8.854 × 10⁻¹² × 10⁹ = 8.854 × 10⁻³ A/m².
3. I_d = Jd A = 8.854 × 10⁻³ × 0.01 = 88.5 μA.
4. Check: C = ε₀A/d = 88.54 pF, and C dV/dt = 88.54 × 10⁻¹² × 10⁶ = 88.5 μA — equal to the conduction current in the leads, as Maxwell requires.

Example 2 (GATE level, conductor or dielectric?). Sea water has σ = 4 S/m and εr = 81. Find the ratio of conduction to displacement current density at 1 MHz and at 1 GHz, and the frequency at which they are equal.

1. At 1 MHz: σ/(ωε) = 4 / (2π × 10⁶ × 81 × 8.854 × 10⁻¹²) = 4 / 4.506 × 10⁻³ = 888 — a good conductor.
2. At 1 GHz (1000 times higher ω): ratio = 0.888 — conduction and displacement currents are comparable; it is a lossy dielectric.
3. Equal when f = σ/(2πε) = 4 / (2π × 81 × 8.854 × 10⁻¹²) = 888 MHz.

Example 3 (plane wave). In free space E = 100 cos(2π × 10⁸ t − βz) a_x V/m. Find β, the wavelength, H and the peak displacement current density.

1. β = ω/c = 2π × 10⁸ / 3.0 × 10⁸ = 2.09 rad/m; λ = 2π/β = 3.0 m.
2. H = E/η₀ = 100 / 376.7 = 0.265 A/m. For propagation along +z with E along a_x, H is along a_y: H = 0.265 cos(2π × 10⁸ t − βz) a_y A/m.
3. Peak Jd = ωε₀E = 2π × 10⁸ × 8.854 × 10⁻¹² × 100 = 0.556 A/m².

Common mistakes

• Writing ∇ × H = J for time-varying fields, forgetting ∂D/∂t.
• Thinking displacement current is a flow of charge — it is a time-varying electric field that has the same magnetic effect as a current.
• Using ∇ × E = 0 when B varies with time (E is then not conservative).
• Sign errors in Faraday's law: ∇ × E = −∂B/∂t (minus), ∇ × H = +∂D/∂t (plus).
• Using f instead of ω = 2πf in σ/(ωε).
• Calling a material a conductor or insulator without stating the frequency.

For GATE IN

Typical items: identifying which Maxwell equation expresses a given law or physical fact; displacement current in capacitors and dielectrics; conduction-to-displacement current ratio and the crossover frequency; checking whether given E and H fields satisfy Maxwell's equations (find the missing field or the constant); continuity equation; c, η₀ and wave parameters from a field expression. Practise curl in Cartesian form and phasor substitution ∂/∂t → jω.

Quick check

1. Which Maxwell equation says there are no magnetic monopoles?
2. What is the displacement current in a 10 μF capacitor whose voltage rises at 200 V/s?
3. For σ/(ωε) = 0.001, is the material a good conductor or a good dielectric?
4. What is the intrinsic impedance of free space?
5. Without displacement current, what goes wrong when Ampère's law is applied to the wire of a charging capacitor?

Answers: 1. ∇·B = 0. 2. 2 mA. 3. A good dielectric. 4. About 377 Ω (120π Ω). 5. The same loop gives ∮H·dl = I for a surface pierced by the wire but zero for a surface passing between the plates — a contradiction.