Maxwell's Equations

Maxwell's equations unify all electromagnetics and predict wave propagation — the conceptual heart of the section. GATE EC tests the four equations, displacement current and their wave consequence (1–2 marks).

Key results

• Gauss (E): ∇·D = ρ_v. Gauss (B): ∇·B = 0.
• Faraday: ∇×E = −∂B/∂t (time-varying B induces E). Ampere-Maxwell: ∇×H = J + ∂D/∂t (∂D/∂t = displacement current density).
• Displacement current resolves charging-capacitor continuity; it makes EM waves possible.
• In free space (source-free), the equations give the wave equation ∇²E = με ∂²E/∂t², wave speed c = 1/√(μ₀ε₀) = 3×10⁸ m/s.
• Continuity equation: ∇·J = −∂ρ/∂t (charge conservation).

Equation	Differential form	Physical meaning
Gauss (E)	∇·D = ρ	charge is E-flux source
Gauss (B)	∇·B = 0	no magnetic monopole
Faraday	∇×E = −∂B/∂t	changing B → E
Ampere-Maxwell	∇×H = J + ∂D/∂t	current/changing E → H

Exam Tricks & Tips

• 🎯 Displacement current density = ∂D/∂t; it's what Maxwell added to Ampere's law to enable EM waves.
• 🎯 Faraday's law: only a TIME-VARYING magnetic field induces an electric field (∇×E = −∂B/∂t).
• 🎯 In statics, the ∂/∂t terms vanish, decoupling E and B — that's why electro/magnetostatics are separate.
• 🎯 Free-space wave speed c = 1/√(μ₀ε₀); intrinsic impedance η₀ = √(μ₀/ε₀) = 120π ≈ 377 Ω.
• 🎯 The continuity equation ∇·J = −∂ρ/∂t follows from taking divergence of Ampere-Maxwell.
• ❌ Common mistake: forgetting the displacement-current term in Ampere-Maxwell for time-varying fields.

Expected exam pattern

1–2 marks: identify which Maxwell equation applies, compute displacement current, or reason about induced fields / wave-equation consequence.

Quick recap

∇·D=ρ, ∇·B=0, ∇×E=−∂B/∂t, ∇×H=J+∂D/∂t. Displacement current ∂D/∂t enables waves. Statics: ∂/∂t=0 decouples E,B. c=1/√(μ₀ε₀)=3×10⁸ m/s, η₀=377 Ω. Continuity: ∇·J=−∂ρ/∂t.

Cover the answer, recall, then check. 12 cards on Maxwell's equations for GATE EC.

Q1. Write Gauss's law for electric and magnetic fields (differential).
A1. ∇·D = ρ_v and ∇·B = 0.

Q2. Write Faraday's law (differential).
A2. ∇×E = −∂B/∂t (a time-varying B induces E).

Q3. Write the Ampere-Maxwell law.
A3. ∇×H = J + ∂D/∂t.

Q4. What is displacement current density?
A4. ∂D/∂t — the term Maxwell added so Ampere's law holds for time-varying fields (e.g. a charging capacitor).

Q5. What does ∇·B = 0 signify?
A5. No magnetic monopoles exist; magnetic field lines form closed loops.

Q6. What condition induces an electric field per Faraday?
A6. A time-varying magnetic field (∂B/∂t ≠ 0).

Q7. Speed of EM waves in free space?
A7. c = 1/√(μ₀ε₀) = 3×10⁸ m/s.

Q8. Intrinsic impedance of free space?
A8. η₀ = √(μ₀/ε₀) = 120π ≈ 377 Ω.

Q9. State the continuity equation.
A9. ∇·J = −∂ρ/∂t (conservation of charge).

Q10. Why do electrostatics and magnetostatics decouple?
A10. In static conditions the time derivatives ∂/∂t vanish, so E and B are no longer linked.

Q11. Which two equations have zero on the right for source-free regions?
A11. ∇·D = 0 and ∇·B = 0 (no free charge, never any monopoles).

Q12. Which Maxwell equation directly predicts that changing E creates H?
A12. The Ampere-Maxwell law (via the ∂D/∂t displacement-current term).

Maxwell's four equations unify electricity and magnetism and predict electromagnetic waves — the crown of the electromagnetics syllabus. Adding the displacement current term let Maxwell show that changing fields propagate as light. GATE tests the equations in differential/integral form, the displacement current, and the wave equation they imply.

Core concept: Maxwell's equations relate E and B to charges and currents; together they yield the wave equation, showing that time-varying fields propagate at the speed of light.

The theory

Beginner — the four equations (differential form)

• Gauss (electric): ∇·D = ρ (charge is the source of D).
• Gauss (magnetic): ∇·B = 0 (no magnetic monopoles).
• Faraday: ∇×E = −∂B/∂t (a changing B induces E).
• Ampère-Maxwell: ∇×H = J + ∂D/∂t (current and changing D induce H).
  Constitutive relations: D = εE, B = µH, J = σE.

Intermediate — integral form and displacement current

The integral forms (via Gauss/Stokes theorems):

• ∮D·dS = Q_enc; ∮B·dS = 0.
• ∮E·dl = −dΦB/dt (Faraday's EMF law).
• ∮H·dl = I_enc + dΨD/dt.
  Maxwell's key addition is the displacement current Jd = ∂D/∂t: it makes Ampère's law consistent (current can "flow" through a capacitor's dielectric as changing D) and is essential for wave propagation. In a good conductor Jd ≪ Jc; in a good dielectric Jc ≪ Jd — the ratio σ/(ωε) is the loss tangent.

Advanced — the wave equation

Taking the curl of Faraday's law and substituting Ampère-Maxwell (in a source-free, lossless medium) gives:
∇²E = µε·∂²E/∂t².
This is the wave equation with propagation speed v = 1/√(µε); in vacuum v = 1/√(µ0ε0) = c ≈ 3×10⁸ m/s. So a changing field self-propagates as an EM wave. The equations also give continuity (∇·J = −∂ρ/∂t, charge conservation) and the Poynting vector S = E × H (power flow density).

Worked example

A parallel-plate capacitor with area A carries a displacement current. If E between the plates changes at dE/dt = 10⁹ V/m/s with εr = 2, A = 1 cm², find the displacement current.
Jd = ∂D/∂t = ε·dE/dt = ε0εr·dE/dt = (8.854×10⁻¹²)(2)(10⁹) = 1.77×10⁻² A/m².
Id = Jd·A = 1.77×10⁻² × 10⁻⁴ = 1.77 µA.
This displacement current equals the conduction current in the wires — the reason Ampère's law needs Maxwell's term for consistency.

GATE relevance

1–2 mark MCQ/NAT: identify/interpret each Maxwell equation, displacement current calculation, loss tangent σ/ωε (conductor vs dielectric), wave speed 1/√(µε), continuity equation, and Poynting-vector power flow. Conceptual "which equation says X" items are common.

Exam tricks and shortcuts

• Four equations: ∇·D=ρ, ∇·B=0, ∇×E=−∂B/∂t, ∇×H=J+∂D/∂t.
• Displacement current Jd = ∂D/∂t = ε·dE/dt; enables wave propagation.
• Wave speed v = 1/√(µε); c = 1/√(µ0ε0).
• Loss tangent = σ/(ωε): ≫1 conductor, ≪1 dielectric.
• Mnemonic: "Faraday: changing B makes E; Ampère-Maxwell: changing D makes H."

Key formulas (point form)

• ∇·D = ρ (Gauss, electric).
• ∇·B = 0 (Gauss, magnetic — no monopoles).
• ∇×E = −∂B/∂t (Faraday).
• ∇×H = J + ∂D/∂t (Ampère–Maxwell).
• Constitutive: D = εE, B = μH, J = σE.
• Wave speed: v = 1/√(με); c = 1/√(μ₀ε₀).

Maxwell's equations unify electricity and magnetism and predict self-propagating electromagnetic waves at speed 1/√(με).

Worked Example

Problem: A parallel-plate capacitor in free space has a uniform electric field E(t) = E₀·sin(ωt) between its plates, with E₀ = 10⁶ V/m and frequency f = 1 MHz. Using the Ampère–Maxwell law, find the displacement current density and its peak value. (ε₀ = 8.85 × 10⁻¹² F/m.)

Solution:
Maxwell added the displacement current term to Ampère's law; in a source-free dielectric (free space) the displacement current density is:
J_d = ε₀ ∂E/∂t.

Differentiate the field:
∂E/∂t = E₀·ω·cos(ωt), with ω = 2πf = 2π × 10⁶ ≈ 6.283 × 10⁶ rad/s.

So:
J_d(t) = ε₀·E₀·ω·cos(ωt).

Peak displacement current density:
J_d(peak) = ε₀·E₀·ω = (8.85 × 10⁻¹²)(10⁶)(6.283 × 10⁶)
= 8.85 × 10⁻¹² × 6.283 × 10¹²
≈ 55.6 A/m².

Answer: J_d(t) = ε₀E₀ω·cos(ωt), with peak value ≈ 55.6 A/m².