Griffiths Electrodynamics Problem 5.27 – A Complete Guide

David J. Griffiths’ “Introduction to Electrodynamics” is widely regarded as a foundational textbook in electromagnetism. Among the many problems in the book, Problem 5.27 from Chapter 5 stands out as a deep dive into the applications of electromagnetic wave theory. This article explores Problem 5.27 in Griffiths Electrodynamics, providing a detailed explanation, mathematical derivation, physical insights, and helpful tips to understand and solve it.

What is Griffiths Problem 5.27 About?

Problem 5.27 is part of Chapter 5: Electromagnetic Waves, which deals with the behavior and properties of electromagnetic radiation in various media. This problem specifically tests your understanding of the boundary conditions for electromagnetic fields and the reflection and transmission of EM waves at an interface between two media.

Typically, this problem presents a plane electromagnetic wave incident at a boundary between two dielectric media and asks you to:

• Determine reflected and transmitted fields,

• Apply Maxwell’s boundary conditions,

• Use the Fresnel equations, and

• Analyze the physical interpretation of reflected and transmitted intensities.

This exercise is essential for students aiming to understand wave optics, fiber optics, and electromagnetic theory more deeply.

Understanding the Physical Setup

Incident, Reflected, and Transmitted Waves

The problem describes a plane wave incident from one dielectric medium onto another. Key features typically include:

• The wave is linearly polarized, often in the y-direction.

• It hits a boundary at z = 0, going from medium 1 (z < 0) to medium 2 (z > 0).

• The electric field of the wave is given in complex exponential form.

Medium Properties

Each medium is characterized by its own:

• Permittivity: ε₁ and ε₂,

• Permeability: μ₁ and μ₂,

• Index of refraction: n=μϵn = \sqrt{\mu \epsilon}n=μϵ​,

• Possibly different wave speeds v=1/μϵv = 1/\sqrt{\mu \epsilon}v=1/μϵ​.

This setup simulates real-world scenarios like light hitting glass or radio waves crossing from air to a metal.

Solving Griffiths Problem 5.27 Step-by-Step

1. Write the General Form of the Fields

Start by expressing all the fields (incident, reflected, and transmitted) as phasors:

• Incident: E⃗i=E0ei(k1z−ωt)y^\vec{E}_i = E_0 e^{i(k_1z – \omega t)} \hat{y}Ei​=E0​ei(k1​z−ωt)y^​

• Reflected: E⃗r=rE0ei(−k1z−ωt)y^\vec{E}_r = rE_0 e^{i(-k_1z – \omega t)} \hat{y}Er​=rE0​ei(−k1​z−ωt)y^​

• Transmitted: E⃗t=tE0ei(k2z−ωt)y^\vec{E}_t = tE_0 e^{i(k_2z – \omega t)} \hat{y}Et​=tE0​ei(k2​z−ωt)y^​

Where $r$ and $t$ are the reflection and transmission coefficients.

2. Apply Boundary Conditions at the Interface

The four Maxwell’s boundary conditions at $z=0$ are:

• E1t=E2tE_{1t} = E_{2t}E1t​=E2t​: Tangential E-fields must be continuous.

• H1t=H2tH_{1t} = H_{2t}H1t​=H2t​: Tangential H-fields must be continuous.

For normal incidence and plane waves in linear media, this boils down to:

• Ei+Er=EtE_{i} + E_{r} = E_{t}Ei​+Er​=Et​

• Ei−ErZ1=EtZ2\frac{E_{i} – E_{r}}{Z_1} = \frac{E_{t}}{Z_2}Z1​Ei​−Er​​=Z2​Et​​

Where Z1=μ1/ϵ1Z_1 = \sqrt{\mu_1 / \epsilon_1}Z1​=μ1​/ϵ1​​ and Z2=μ2/ϵ2Z_2 = \sqrt{\mu_2 / \epsilon_2}Z2​=μ2​/ϵ2​​ are the wave impedances.

3. Solve for Reflection and Transmission Coefficients

Solving the boundary equations gives:

• r=Z2−Z1Z2+Z1r = \frac{Z_2 – Z_1}{Z_2 + Z_1}r=Z2​+Z1​Z2​−Z1​​

• t=2Z2Z2+Z1t = \frac{2Z_2}{Z_2 + Z_1}t=Z2​+Z1​2Z2​​

These are the Fresnel equations for normal incidence, essential in wave optics and telecommunications.

Physical Interpretation and Applications

H3: Reflection and Transmission Power

The power carried by an electromagnetic wave is given by the Poynting vector:

• S⃗=1μE⃗×B⃗\vec{S} = \frac{1}{\mu} \vec{E} \times \vec{B}S=μ1​E×B

The average power is proportional to the square of the electric field divided by impedance:

• ⟨S⟩∝∣E∣2Z\langle S \rangle \propto \frac{|E|^2}{Z}⟨S⟩∝Z∣E∣2​

So, reflected and transmitted powers can be derived from:

• R=∣r∣2R = |r|^2R=∣r∣2 → Reflection coefficient (power)

• T=Z2Z1∣t∣2T = \frac{Z_2}{Z_1}|t|^2T=Z1​Z2​​∣t∣2 → Transmission coefficient (power)

And conservation of energy ensures that:

• $R+T=1$

H3: Special Cases

Let’s consider some special physical scenarios:

1. Identical Media

• If Z1=Z2Z_1 = Z_2Z1​=Z2​: then $r=0,t=1$. No reflection, complete transmission.

2. Perfect Conductor

• If Z2→0Z_2 \to 0Z2​→0: then $r=-1$, total reflection with a phase flip.

3. Air to Glass Transition

• Realistic scenario: partial reflection and partial transmission.

These cases are important for designing antennas, coatings, and fiber optics.

Real-World Relevance and Applications

H3: Anti-Reflective Coatings

Understanding wave impedance mismatch allows engineers to design anti-reflection coatings that minimize reflection and maximize transmission.

H3: Fiber Optic Communications

In fiber optics, light transitions between media with different refractive indices, leading to total internal reflection – a principle closely tied to this problem.

H3: Radar and Radio Waves

Radar engineers analyze reflected EM waves from different surfaces. This problem mirrors such setups, helping to calculate return signals and reflection losses.

Tips for Solving Similar Problems in Electrodynamics

H3: Stick to the Phasor Approach

Using phasor notation simplifies dealing with sinusoidal functions and allows easier application of boundary conditions.

H3: Master Maxwell’s Equations

Every boundary condition, field relationship, and wave interaction boils down to Maxwell’s equations. Revisit them frequently.

H3: Use Symmetry

If a problem has normal incidence, symmetry simplifies the math. For oblique incidence, expect more complex vector decomposition.

H3: Check Units and Physical Meaning

Always verify:

• Are units consistent?

• Do the coefficients add up to 1?

• Does the phase of reflection make sense?

Conclusion

Griffiths Electrodynamics Problem 5.27 is not just another textbook exercise—it’s a gateway into the real physics of electromagnetic waves at interfaces. By understanding wave reflection, transmission, impedance, and boundary conditions, you’re mastering concepts that are crucial in optics, communications, radar, and materials science.

Whether you’re a student preparing for exams or a professional brushing up on theory, revisiting problems like 5.27 offers both deep insight and practical utility. Use this guide to sharpen your problem-solving skills and deepen your physical intuition.