Plane Waves & Intrinsic Impedance

The ratio that connects electric and magnetic fields in a traveling wave

ElectromagneticsElectrical Engineering Year 2Free preview
⏱️ About 18 min

In Module 6 you derived the wave equation and found that electromagnetic disturbances travel at c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s. But what is the relationship between the electric and magnetic fields that make up that wave? The answer is a single number — and it behaves just like an impedance.

💡
The big idea: In a plane wave, the electric and magnetic field amplitudes are locked in a fixed ratio called the intrinsic impedance η = E/H, a property of the medium itself — 377 Ω in free space.
🎯 By the end, you'll be able to
  • Describe the plane-wave solution to the wave equation and the transverse, perpendicular relationship between E and H
  • Define intrinsic impedance η = E/H and derive η = √(μ/ε) for a lossless medium
  • Verify the free-space intrinsic impedance η₀ ≈ 377 Ω numerically
  • Explain qualitatively how lossy media make η complex and cause E-H phase shift and attenuation

Picking Up from the Wave Equation

In Module 6 you combined Faraday's law and Ampere's law (with displacement current) to derive the electromagnetic wave equation. The solution revealed that disturbances in E and H travel together at speed v=1/√(με), which in free space gives c=1/√(μ₀ε₀)≈3×10⁸ m/s. Now we ask: for a given electric field amplitude, how large is the magnetic field that accompanies it? Maxwell's equations enforce a fixed ratio between them, depending only on the material properties of the medium.

This ratio, called the intrinsic impedance of the medium, plays the same conceptual role that resistance or characteristic impedance plays in circuit theory. When we reach transmission lines later in this module, you will see that the characteristic impedance Z₀ of a line is the direct circuit-theory analog of the intrinsic impedance η of a medium.

The Plane-Wave Solution

For a uniform plane wave traveling in the +z direction, the electric and magnetic fields take the form of sinusoidal traveling waves. The electric field oscillates along one transverse axis (say, x̂) and the magnetic field oscillates along the other transverse axis (ŷ). Neither field has a component in the direction of propagation — this is the transverse property of plane waves.

\[ \vec{E}(z,t) = E_0 \, \cos(\omega t - \beta z) \, \hat{x} \]
Electric field of a plane wave traveling in +z, polarized along x.
\[ \vec{H}(z,t) = H_0 \, \cos(\omega t - \beta z) \, \hat{y} \]
Magnetic field of the same wave, polarized along y — perpendicular to both E and the propagation direction.

E, H, and the Direction of Propagation

Notice the geometry: E points along x̂, H points along ŷ, and the wave travels along ẑ. These three directions form a right-handed coordinate system, which follows directly from Maxwell's equations. The cross product E×H always points in the direction of wave propagation, a fact we will exploit when we study the Poynting vector in the next lesson.

The two fields are in phase in a lossless medium — they reach their maxima and minima at the same instant and position. Their amplitudes E₀ and H₀ are not independent, however. Substituting the plane-wave solutions back into Maxwell's curl equations yields a direct proportionality:

\[ \frac{E_0}{H_0} = \sqrt{\frac{\mu}{\varepsilon}} \equiv \eta \]
Definition of intrinsic impedance: the ratio of electric to magnetic field amplitude in a plane wave.
🔑 Intrinsic Impedance Is a Material Property

η = √(μ/ε) depends only on μ and ε, the permeability and permittivity of the medium — not on frequency, wave amplitude, or the source. Every lossless medium has a fixed intrinsic impedance, just as every resistor has a fixed resistance. In free space, this value turns out to be approximately 377 Ω — one of the most quoted numbers in all of electromagnetics.

Verifying η₀ ≈ 377 Ω

The free-space intrinsic impedance is computed from the fundamental constants μ₀ and ε₀. We use μ₀=4π×10⁻⁷ H/m and ε₀=8.854×10⁻¹² F/m. First, we form the ratio μ₀/ε₀:

\[ \frac{\mu_0}{\varepsilon_0} = \frac{4\pi \times 10^{-7}}{8.854 \times 10^{-12}} = \frac{1.2566 \times 10^{-6}}{8.854 \times 10^{-12}} = 1.41947 \times 10^{5} \]
Step 1: form the ratio of permeability to permittivity in free space.
\[ \eta_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \sqrt{1.41947 \times 10^{5}} = 376.7 \; \Omega \approx 377 \; \Omega \]
Step 2: take the square root to obtain the free-space intrinsic impedance.
📝 Worked example: Verify that the free-space intrinsic impedance η₀ = √(μ₀/ε₀) is approximately 377 Ω, given μ₀ = 4π×10⁻⁷ H/m and ε₀ = 8.854×10⁻¹² F/m.
  1. Compute μ₀ = 4π×10⁻⁷ = 1.2566×10⁻⁶ H/m.
  2. Form the ratio μ₀/ε₀ = (1.2566×10⁻⁶)/(8.854×10⁻¹²) = 1.41947×10⁵.
  3. Take the square root: √(1.41947×10⁵) = 376.7 Ω.
  4. Round to three significant figures: η₀ ≈ 377 Ω.
✓ η₀ ≈ 377 Ω

Intrinsic Impedance in Dielectrics

For a lossless dielectric with relative permittivity εᵣ and relative permeability μᵣ (most dielectrics are nonmagnetic, so μᵣ=1), the intrinsic impedance simplifies. Since μ=μᵣμ₀ and ε=εᵣε₀:

\[ \eta = \sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_r \mu_0}{\varepsilon_r \varepsilon_0}} = \eta_0 \sqrt{\frac{\mu_r}{\varepsilon_r}} \]
Intrinsic impedance of a general lossless medium in terms of relative material properties.

Lossy Media: A Conceptual Note

When a medium has nonzero conductivity σ, the wave equation picks up a damping term, and the intrinsic impedance η becomes a complex number rather than a real scalar. Physically: the wave amplitude decays exponentially as it travels (attenuation), and E and H are no longer perfectly in phase — H lags E by a small angle. These effects matter in seawater, conductors, and lossy dielectrics, and are treated in advanced electromagnetic theory. For this module, we restrict ourselves to lossless or low-loss media where η is real.

✨ Why 377 Ω Matters

The value 377 Ω appears everywhere in electromagnetics. It sets the ratio of E to H for every wave in free space, determines how much power a given field carries, and surfaces again as a reference impedance in antenna and transmission-line problems. The fact that it is real (not complex) is what makes free space lossless — every joule of energy in the E-field is matched by a corresponding amount in the H-field, with no phase lag and no dissipation.

✏️ Practice: A lossless nonmagnetic dielectric has relative permittivity εᵣ = 4 (μᵣ = 1). (a) Find its intrinsic impedance η. (b) If the electric field amplitude in this medium is E₀ = 10 V/m, find the magnetic field amplitude H₀ in mA/m.
mA/m
Solution
  1. For a nonmagnetic dielectric (μᵣ=1): η = η₀/√εᵣ = 377/√4 = 377/2 = 188.5 Ω.
  2. The intrinsic impedance relates the field amplitudes: H₀ = E₀/η = 10/188.5 = 0.05305 A/m.
  3. Convert to mA/m: H₀ = 0.05305 × 1000 = 53.05 mA/m.

Check your understanding

1. In a uniform plane wave traveling in the +z direction, the electric field points along x̂. Which direction does the magnetic field H point?
E, H, and the propagation direction form a right-handed set. If E is along x̂ and the wave travels along ẑ, then H must point along ŷ.
2. The intrinsic impedance of a lossless medium depends on:
η = √(μ/ε) is purely a material property — it depends only on the permeability and permittivity of the medium, not on frequency, amplitude, or source.
🔎 Related lessons
Wave Polarization & the Poynting Vector
✅ Key takeaways
  • A uniform plane wave has E and H fields that are transverse to the direction of propagation and perpendicular to each other, forming a right-handed (E, H, k) triplet.
  • In a lossless medium, E and H are in phase and their amplitude ratio is the intrinsic impedance η = √(μ/ε), a material property.
  • In free space, η₀ = √(μ₀/ε₀) ≈ 377 Ω, verified numerically from μ₀ = 4π×10⁻⁷ H/m and ε₀ = 8.854×10⁻¹² F/m.
  • For a nonmagnetic dielectric, η = η₀/√εᵣ; in lossy media, η becomes complex, and the wave attenuates as it propagates.
➡️ Now that we know the fixed relationship between E and H in a plane wave, we can ask: which way does the E-field point as the wave travels, and how much power does the wave actually carry? The next lesson introduces wave polarization and the Poynting vector.
Want to test yourself on this? Try the Electrical Aptitude test →