Study notes

Qwestrum Engineering360 · Electrical & Electronics · Electromagnetic Theory

Electromagnetic Waves

A uniform plane wave carries E and H perpendicular to each other and to the direction of travel, related by the intrinsic impedance η = √(μ/ε); in conductors the field decays over one skin depth δ.

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.

TEM, TE, TM modes in waveguides
$Reflection coefficient Γ = \frac{(\eta_{2} - \eta_{1})}{(\eta_{2} + \eta_{1})}$
Standing wave ratio SWR = (1 + |Γ|)/(1 − |Γ|)

Topic details

Introduction

In free space η₀ = √(μ₀/ε₀) ≈ 377 Ω relates the electric and magnetic field amplitudes (E = ηH). In a lossy medium the propagation constant γ = α + jβ has an attenuation part α (Np/m) and a phase part β (rad/m).

Scope in B.Tech and GATE syllabus

Skin depth δ = 1/√(πfμσ) is the depth at which the field falls to 1/e; it shrinks with frequency, which is why high-frequency currents crowd to conductor surfaces and why hollow conductors work as well as solid ones at RF.

Key relations & formulas

Intrinsic impedance \eta = \sqrt{\frac{\mu}{\varepsilon}} \approx 377 \Omega

(free space)

Formulas (Indian textbook notation)

$Propagation constant \gamma = \alpha + j\beta; v_{p} = \frac{\omega}{\beta}$

Skin depth \delta = \frac{1}{\sqrt}{\pi f \mu \sigma}

(good conductor)

Notation and sign conventions

Relation 1 —

Intrinsic impedance \eta = √

Intrinsic impedance \eta = \sqrt{\frac{\mu}{\varepsilon}} \approx 377 \Omega

(free space)

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 —

Propagation constant \gamma = \alpha + j\beta; v_{p} = \frac{\omega}{\beta}

Formulas (Indian textbook notation)

$Propagation constant \gamma = \alpha + j\beta; v_{p} = \frac{\omega}{\beta}$

Relation 3 —

Skin depth \delta = 1/√

Skin depth \delta = \frac{1}{\sqrt}{\pi f \mu \sigma}

(good conductor)

Fundamentals and definitions

At an interface between media of impedance η₁ and η₂, the reflection coefficient is Γ = (η₂ − η₁)/(η₂ + η₁) and the transmission coefficient τ = 1 + Γ. A matched interface (η₂ = η₁) gives Γ = 0 and no reflection.

Governing relations in practice

The standing-wave ratio SWR = (1+|Γ|)/(1−|Γ|) measures the mismatch; SWR = 1 is perfect match, ∞ is total reflection. Phase velocity v_p = ω/β.

Design and analysis considerations

In a good conductor α = β = 1/δ, so the wave both attenuates and rotates in phase rapidly; almost all power is reflected, which is why metals shield fields.

Assumptions and validity limits

State assumptions explicitly before using any relation for electromagnetic waves — 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 electromagnetic waves.
4. Use equation 1:

Intrinsic impedance \eta = √

.
5. Use equation 2:

Propagation constant \gamma = \alpha + j\beta; v_{p} = \frac{\omega}{\beta}

.
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

Electromagnetic Waves 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 electromagnetic waves with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use electromagnetic waves?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Using 377 Ω inside a dielectric (divide by √ε_r)
• Confusing attenuation constant α (Np/m) with phase constant β (rad/m)
• Forgetting skin depth decreases as √f
• Sign error in Γ by swapping which medium the wave enters

Quick revision checklist

Before attempting electromagnetic waves problems, confirm you can:

1. TEM, TE, TM modes in waveguides
2.

Reflection coefficient Γ = \frac{(\eta_{2} - \eta_{1})}{(\eta_{2} + \eta_{1})}

3. Standing wave ratio SWR = (1 + |Γ|)/(1 − |Γ|)

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.

Skin depth in copper

Problem

Find the skin depth in copper (σ = 5.8×10⁷ S/m, μ = μ₀) at 1 MHz.

Solution

δ = 1/√(π f μ σ).
π f μ σ = π × 10⁶ × 4π×10⁻⁷ × 5.8×10⁷.
= π × 10⁶ × 1.2566×10⁻⁶ × 5.8×10⁷ ≈ 2.29×10⁸.
δ = 1/√(2.29×10⁸) = 1/1.513×10⁴ = 6.6×10⁻⁵ m = 66 µm.

Conceptual check — Electromagnetic Waves

Problem

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

Exams & GATE

Sadiku — plane wave in lossy medium attenuation.

📖 Standard books (India)

Elements of Electromagnetics — Matthew Sadiku
Read: Syllabus unit
Fields, Maxwell equations, and waves

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📚Key formulas & points