Uniform Plane Wave

A uniform plane wave is the simplest solution to the electromagnetic wave equation in free space or a uniform medium. It represents a wave where the electric and magnetic field amplitudes are constant over any plane perpendicular to the direction of propagation. This idealization is the starting point for analyzing antenna radiation, free-space links, and guided wave structures in all practical engineering systems.

Core Concept Explanation

A uniform plane wave has field amplitudes that do not vary in any direction transverse to propagation. If the wave travels in the z direction, then dE/dx = 0 and dE/dy = 0 at every instant. Only the z-dependence and time-dependence remain. This simplification makes the mathematics tractable and, more importantly, provides a very good model for waves far from their source, where the wavefronts become nearly flat.

The defining property of a uniform plane wave is that it is a transverse electromagnetic (TEM) wave. This means the electric field E has no component along the propagation direction (Ez = 0) and the magnetic field H also has no component along that direction (Hz = 0). Both E and H lie entirely in the plane transverse to the direction of travel, and they are mutually perpendicular to each other as well.

From Maxwell's equations applied to a source-free uniform medium, if the wave travels in the z direction and E is polarized in the x direction, then H must be polarized in the y direction. The relationship between them is determined by the intrinsic impedance eta of the medium. The cross product E cross H points in the z direction, confirming that power flows in the direction of propagation. This is the Poynting vector direction.

Mathematical Expression

For a wave traveling in the positive z direction in a lossless medium, the phasor-domain electric field is E = E0 * e^(-j*beta*z) * x_hat. The corresponding magnetic field is H = (E0/eta) * e^(-j*beta*z) * y_hat, where eta = sqrt(mu/epsilon) is the intrinsic impedance. The time-domain expressions are E(z,t) = E0 * cos(omega*t - beta*z) * x_hat and H(z,t) = (E0/eta) * cos(omega*t - beta*z) * y_hat.

Substituting into Gauss's law, div E = dEx/dx = 0, which is automatically satisfied because E does not vary with x. This confirms the consistency of the uniform plane wave assumption. The phase velocity is vp = omega/beta = 1/sqrt(mu*epsilon). In free space, vp = c = 3 x 10^8 m/s.

Practical Understanding

Uniform plane waves are an idealization because they require infinite extent in the transverse plane, which is physically impossible. However, any real wave from a distant source, such as sunlight or a far-field antenna beam, closely approximates a uniform plane wave over a small observation region. Satellite communication link analysis, radar cross-section computation, and optical system design all routinely use the uniform plane wave model.

The TEM nature of the uniform plane wave has a direct consequence: both E and H reach their peaks and zeros at the same position and time. They are in phase spatially and temporally in a lossless medium. In a lossy medium, E and H become phase-shifted relative to each other, and the intrinsic impedance becomes complex. This phase shift reduces real power transfer and results in reactive wave behavior, which is important in lossy dielectric and conductor analysis.

Given:
Free space uniform plane wave
E0 = 10 V/m (peak electric field amplitude)
f = 1 GHz

Why this formula applies:
Uniform plane wave in free space: |H| = |E| / eta0
eta0 = 377 ohms (intrinsic impedance of free space)

Formula:
|H| = E0 / eta0

Substitution:
|H| = 10 / 377

Calculation:
|H| = 0.02653 A/m

Final Answer:
H amplitude = 26.53 mA/m
Note: beta = (2*pi*1e9) / (3e8) = 20.94 rad/m
Wavelength lambda = 2*pi/beta = 0.3 m = 30 cm

Exam Tip: In GATE, if E is given along x and propagation is along z, H must be along y and equals E/eta. If propagation is along -z, H reverses direction: H = -(E/eta)*y_hat. Forgetting this sign reversal is a frequent mistake in problems with reflected waves.

• Uniform plane wave has constant field amplitude over any transverse plane. Fields vary only along the propagation direction.
• It is a TEM wave: Ez = 0 and Hz = 0. Both E and H lie in the transverse plane.
• E, H, and the propagation direction k form a right-handed orthogonal set. E cross H gives the Poynting vector direction.
• In a lossless medium, E and H are in phase and their ratio equals the intrinsic impedance eta = sqrt(mu/epsilon).
• In a lossy medium, E and H are out of phase and the intrinsic impedance is complex.

Quick Revision

• Uniform plane wave: field amplitudes constant over planes perpendicular to propagation.
• TEM mode: Ez = 0, Hz = 0.
• H = (1/eta) * (k_hat cross E), where k_hat is the unit propagation vector.
• Phase velocity: vp = omega/beta = 1/sqrt(mu*epsilon).
• Free space: eta0 = 377 ohms, vp = c = 3 x 10^8 m/s.
• Exam trap: If wave propagates in -z direction, the formula for H becomes H = -(1/eta) * (z_hat cross E). Sign changes matter in reflection problems.
• Real sources approximate uniform plane waves only in the far field. Near field analysis requires more complex wave models.