Foundations of Electromagnetics

Illustration of Maxwell's equations and electromagnetic field quantities. — Maxwell's equations provide the field-law foundation for the topics collected here. Image source: Wikimedia Commons, Euisudyi, CC BY-SA 4.0.

30-second intuitionElectromagnetics is a theory of fields, sources, propagation and energy transfer. Maxwell's equations tell fields how to diverge and curl; material relations tell the fields how matter responds; boundary conditions connect regions; the Lorentz force tells charged matter how fields push it.

The four Maxwell equations

\[\nabla\cdot\mathbf D=\rho_v\] \[\nabla\cdot\mathbf B=0\] \[\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}\] \[\nabla\times\mathbf H=\mathbf J+\frac{\partial\mathbf D}{\partial t}.\]

Together with

\[\mathbf D=\epsilon\mathbf E,\qquad \mathbf B=\mu\mathbf H,\qquad \mathbf J=\sigma\mathbf E,\]

these generate much of classical EM engineering.

How to navigate the foundations

Maxwell's EquationsMeaning of divergence/curl, displacement current and the unification of electricity, magnetism and light. Electromagnetic WavesWave equation, wavelength, impedance, phase and propagation. Boundary ConditionsHow tangential and normal field components change across interfaces. PolarizationLinear, circular and elliptical field evolution and why it matters for antennas and atoms. Power & Poynting VectorEnergy flow, power density and conservation.

From Maxwell to applications

\[\boxed{ \text{Maxwell} \rightarrow \text{wave equation} \rightarrow \text{propagation / boundaries} \rightarrow \text{guided waves / radiation} \rightarrow \text{devices and systems} }\]

and for matter interaction,

\[\boxed{ \mathbf E,\mathbf B \rightarrow \text{Lorentz force / polarization / magnetization} \rightarrow \text{motion, energy shift or induced current} }\]

Essential scales

Before solving a problem, compare geometry and timescales with:

$L/\lambda$: is the structure electrically small or distributed?
$R/R_{FF}$: near field or far field?
$\delta/t$: does skin effect dominate conductor thickness?
$\omega\epsilon/\sigma$: displacement current or conduction current?
material dispersion/loss across the bandwidth of interest.

See Scaling Laws and Orders of Magnitude.

Measurement bridge

Maxwell’s equations predict fields, but experiments often measure voltages, currents, wave ratios, optical power or detector outputs. The Measurements & Instruments page shows how those observables are connected back to field quantities.

Recommended sequence

Maxwell → waves → boundaries → Poynting → transmission lines → antennas → RF systems → interaction with matter → quantum/atomic sensing.

For compact formulas use Fundamental Equations; for history see History Timeline.

Advanced roadmap — graduate electromagnetic field theory

The next level beyond Maxwell-equation recall is solving boundary-value and inverse problems using dyadic Green functions, surface equivalence, spectral methods, radiation conditions and scattering operators. In a homogeneous isotropic region, an electric field driven by current may be expressed using a dyadic Green tensor:

\[\mathbf E(\mathbf r)=i\omega\mu\int\overline{\overline{\mathbf G}}(\mathbf r,\mathbf r')\mathbf J(\mathbf r')\,d^3r',\]

under the (e^{-i\omega t}) convention and a compatible definition of (\overline{\overline{\mathbf G}}). Boundary and radiation conditions determine the correct Green function.

Advanced sequence: derive Lorentz reciprocity; calculate an angular spectrum with evanescent components; solve a layered-medium reflection problem; formulate an electric-field integral equation; compare FEM/FDTD/MoM convergence on the same finite scatterer. Distinguish mathematical field singularities from poor meshing and erroneous physical models.

Further reading: MIT 6.635 Advanced Electromagnetism; W. C. Chew, Waves and Fields in Inhomogeneous Media.