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Qwestrum Engineering360 · Electrical & Electronics · Electromagnetic Theory

Maxwell Equations

Maxwell’s four equations unify electricity and magnetism; the displacement-current term ∂D/∂t that Maxwell added predicts electromagnetic waves travelling at v = 1/√(με).
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Displacement current completes continuity in capacitors
  • PoyntingvectorS=E×HPoynting vector S = E \times H — power flow density
  • ConstitutiverelationsD=εE,B=μH,J=σEConstitutive relations D = \varepsilon E, B = \mu H, J = \sigma E

Topic details

Introduction

The four equations state: electric flux originates on charge (∇·D = ρ), no magnetic charge exists (∇·B = 0), a changing magnetic field induces an electric field (Faraday), and both current and a changing electric field produce a magnetic field (Ampere–Maxwell).

Scope in B.Tech and GATE syllabus

The crucial addition is the displacement current ∂D/∂t, which keeps current continuous through a capacitor gap and, combined with Faraday’s law, gives the wave equation. Deriving v = 1/√(με) is a standard exam derivation.

Key relations & formulas

∇×E=−∂B/∂t∇ \times E = -∂B/∂t
(Faraday)
∇×H=J+∂D/∂t∇ \times H = J + ∂D/∂t
(Ampere-Maxwell)

Formulas (Indian textbook notation)

  • ∇⋅D=ρv;∇⋅B=0∇ \cdot D = \rho_{v}; ∇ \cdot B = 0

Formulas (Indian textbook notation)

  • Waveequation:∇2E=με∂2E/∂t2Wave equation: ∇^{2}E = \mu \varepsilon ∂^{2}E/∂t^{2}

Notation and sign conventions

Relation 1 —
∇×E=−∂B/∂t∇ \times E = -∂B/∂t
∇×E=−∂B/∂t∇ \times E = -∂B/∂t
(Faraday)
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
∇×H=J+∂D/∂t∇ \times H = J + ∂D/∂t
∇×H=J+∂D/∂t∇ \times H = J + ∂D/∂t
(Ampere-Maxwell)
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
∇⋅D=ρv;∇⋅B=0∇ \cdot D = \rho_{v}; ∇ \cdot B = 0

Formulas (Indian textbook notation)

  • ∇⋅D=ρv;∇⋅B=0∇ \cdot D = \rho_{v}; ∇ \cdot B = 0
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
Waveequation:∇2E=με∂2E/∂t2Wave equation: ∇^{2}E = \mu \varepsilon ∂^{2}E/∂t^{2}

Formulas (Indian textbook notation)

  • Waveequation:∇2E=με∂2E/∂t2Wave equation: ∇^{2}E = \mu \varepsilon ∂^{2}E/∂t^{2}
Write this relation with symbols exactly as in Elements of Electromagnetics — Matthew Sadiku before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

Taking the curl of Faraday’s law and substituting Ampere–Maxwell (in a source-free region) yields ∇²E = με ∂²E/∂t², a wave equation whose speed is 1/√(με). In free space this is c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s.

Governing relations in practice

The Poynting vector S = E×H gives the instantaneous power flow per unit area; its time-average measures real transmitted power. For a plane wave the average is ½E₀²/η.

Design and analysis considerations

The constitutive relations D = εE, B = μH, J = σE connect the fields in a medium and must be applied with the medium’s ε, μ and σ.

Assumptions and validity limits

State assumptions explicitly before using any relation for maxwell equations — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Electromagnetic Theory viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Electromagnetic Theory papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to maxwell equations.
4. Use equation 1:
∇×E=−∂B/∂t∇ \times E = -∂B/∂t
.
5. Use equation 2:
∇×H=J+∂D/∂t∇ \times H = J + ∂D/∂t
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Maxwell Equations appears in RF, power apparatus, and communications. In Indian electrical curricula this topic is tested because it connects theory to fields, Maxwell equations, and waves.
GATE and semester exams often combine maxwell equations with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use maxwell equations?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Omitting the displacement-current term ∂D/∂t (leaving Ampere’s law incomplete)
• Using free-space c inside a dielectric instead of 1/√(με)
• Confusing instantaneous and time-averaged Poynting power
• Forgetting ∇·B = 0 has no source term (unlike ∇·D = ρ)

Quick revision checklist

Before attempting maxwell equations problems, confirm you can:
1. Displacement current completes continuity in capacitors
2.
PoyntingvectorS=E×HPoynting vector S = E \times H
— power flow density
3.
ConstitutiverelationsD=εE,B=μH,J=σEConstitutive relations D = \varepsilon E, B = \mu H, J = \sigma E
Revise the solved examples in Elements of Electromagnetics — Matthew Sadiku and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Wave speed in a dielectric

Problem

A lossless dielectric has ε_r = 4 and μ_r = 1. Find the speed of an electromagnetic wave and the wavelength at 300 MHz.

Solution

v = c/√(ε_r μ_r) = (3×10⁸)/√(4×1) = 3×10⁸/2 = 1.5×10⁸ m/s.
λ = v/f = 1.5×10⁸ / 300×10⁶.
λ = 1.5×10⁸ / 3×10⁸ = 0.5 m.

Conceptual check — Maxwell Equations

Problem

In a Electromagnetic Theory semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of maxwell equations." What should a complete answer include?

Exams & GATE

Sadiku — derive wave equation and speed v = 1/√(με).

📖 Standard books (India)

  • Elements of Electromagnetics — Matthew Sadiku

    Read: Syllabus unit

    Fields, Maxwell equations, and waves

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