Electromagnetic Waves

30-second intuitionA time-varying electric field and magnetic field can sustain each other and propagate energy through space. In a uniform far-field plane wave, $\mathbf E$, $\mathbf H$, and the propagation direction are mutually perpendicular.

Wave equation

For a homogeneous source-free medium,

\[\nabla^2\mathbf E-\mu\epsilon\frac{\partial^2\mathbf E}{\partial t^2}=0,\]

with an analogous equation for $\mathbf H$.

The phase velocity in a lossless medium is

\[v_p=\frac{1}{\sqrt{\mu\epsilon}}.\]

Frequency and wavelength

\[f=\frac{\omega}{2\pi},\qquad \lambda=\frac{v_p}{f}.\]

In free space,

\[\lambda_0=\frac{c}{f}.\]

A useful mental rule is

\[\boxed{\lambda_0(\text{cm})\approx\frac{30}{f(\text{GHz})}.}\]

Propagation constant

\[\gamma=\alpha+j\beta.\]

A wave varying along $z$ can be written

\[E(z)=E_0e^{-\alpha z}e^{-j\beta z}.\]

Here $\alpha$ is attenuation constant and $\beta$ is phase constant. In a lossless medium,

\[\beta=\frac{2\pi}{\lambda}.\]

Intrinsic impedance

In a general conducting medium,

\[\eta=\sqrt{\frac{j\omega\mu}{\sigma+j\omega\epsilon}}.\]

For a lossless dielectric,

\[\eta=\sqrt{\frac{\mu}{\epsilon}}.\]

In free space, $\eta_0\approx376.73\ \Omega$.

Plane-wave field relation

For propagation along $+z$ with $\mathbf E$ along $x$,

\[\mathbf H=\frac{1}{\eta}\hat{\mathbf z}\times\mathbf E.\]

The time-average power density is

\[\langle\mathbf S\rangle=\frac12\operatorname{Re}\{\mathbf E\times\mathbf H^*\}.\]

Lossy media

When $\sigma\neq0$, propagation attenuates. In a good conductor, fields penetrate only a skin depth

\[\delta\approx\sqrt{\frac{2}{\omega\mu\sigma}}.\]

Thus the same wave equations connect free-space radiation to conductor shielding and skin effect.

Worked example — 100 MHz versus 10 GHz

At 100 MHz,

\[\lambda\approx3\ \text{m}.\]

At 10 GHz,

\[\lambda\approx3\ \text{cm}.\]

A 10-cm object is electrically tiny at 100 MHz but several wavelengths across at 10 GHz. The same physical object can therefore move from lumped-circuit behavior to strong scattering/resonance simply by changing frequency.

Measurement

Wave quantities are inferred from:

phase/amplitude with VNA measurements;
field probes;
antenna measurements;
time-domain reflectometry;
material-property extraction from transmission/reflection;
optical interferometry at much higher frequencies.

Engineering reality

Real propagationMultipath, dispersion, anisotropy, absorption, roughness, diffraction, finite apertures and near-field structure can invalidate the simple uniform plane-wave picture.

References

R. E. Collin, Foundations for Microwave Engineering.
D. M. Pozar, Microwave Engineering.
D. J. Griffiths, Introduction to Electrodynamics.