Electromagnetics

Map of content for the fundamentals of electromagnetics — the unified theory of electric and magnetic fields, their sources, their interactions, and their propagation as electromagnetic waves. The path: vector calculus prerequisites → electrostatics → magnetostatics → time-varying fields → Maxwell’s equations → electromagnetic waves → transmission lines → transients on transmission lines → impedance matching and Smith chart.

Vector calculus prerequisites

The math machinery beneath everything. Vector Calculus and Complex Analysis covers it in detail (see Mathematical methods); Electromagnetics reviews and applies it to fields.

• Position vector — the vector from origin to a point in space; the natural variable for fields.
• Distance vector — the source-to-observation vector that drives Coulomb’s law and Biot-Savart.
• Cartesian coordinates, Cylindrical coordinates, Spherical coordinates — three coordinate systems; pick by the symmetry of the problem.
• Gradient, Divergence, Curl — the three first-order differential operators on fields.
• Right-hand rule — the orientation convention behind cross products, curl, and signed flux throughout EM.
• Laplacian — second-order operator ${\nabla }^2=\nabla \cdot \nabla$, central to Poisson and wave equations.
• Curl of curl identity — $\nabla \times (\nabla \times \mathbf{F})=\nabla (\nabla \cdot \mathbf{F})-{\nabla }^2\mathbf{F}$. The key identity for deriving the EM wave equation.
• Line integral, Flux integral — integrating fields along curves and through surfaces.
• Stokes’ theorem, Divergence theorem — the integral theorems linking volume/surface to surface/boundary; the mathematical backbone of Maxwell’s equations in their integral form.

Electrostatics

Fields due to stationary charges, governed by $\nabla \cdot \mathbf{D}={\rho }_v$ and $\nabla \times \mathbf{E}=0$.

• Coulomb’s law — force between point charges; the starting axiom.
• Electric field — force per unit test charge; the vector field that mediates electric interaction.
• Electric flux density — $\mathbf{D}=ϵ\mathbf{E}$; the “free-source” field whose divergence is the free charge density.
• Charge density — volume, surface, line distributions of charge.
• Gauss’s law — $\nabla \cdot \mathbf{D}={\rho }_v$; the most useful electrostatic identity for problems with symmetry.
• Electric potential — scalar function $V$ with $\mathbf{E}=-\nabla V$; the line-integral of the field.
• Equipotential surface — surface of constant $V$; $\mathbf{E}$ is always perpendicular to it.
• Poisson’s equation — ${\nabla }^{2}V=-{\rho }_v/ϵ$; reduces electrostatics to a boundary-value problem.
• Laplace’s equation — ${\nabla }^{2}V=0$ in charge-free regions; the equilibrium PDE.
• Conduction current and Ohm’s law — $\mathbf{J}=\sigma \mathbf{E}$; how currents flow in conductors under applied fields.
• Permittivity — material constant $ϵ={ϵ}_r{ϵ}_0$ relating $\mathbf{D}$ and $\mathbf{E}$.
• Dielectric — insulating material that polarizes under applied field.
• Polarization (dielectric) — the $\mathbf{P}$ field; dipole moment per unit volume in a polarized medium.
• Electric boundary conditions — how $\mathbf{E}$ and $\mathbf{D}$ jump at material interfaces.
• Capacitance — $C=Q/V$; geometry-only property of conductor pairs.
• Electrostatic energy — $\frac{1}{2}ϵE^2$ per unit volume; energy stored in the electric field.

Magnetostatics

Fields due to steady currents, governed by $\nabla \cdot \mathbf{B}=0$ and $\nabla \times \mathbf{H}=\mathbf{J}$.

• Magnetic field — $\mathbf{B}$ (flux density) and $\mathbf{H}$ (intensity); paired fields with constitutive relation $\mathbf{B}=\mu \mathbf{H}$.
• Permeability — material constant $\mu ={\mu }_r{\mu }_0$ relating $\mathbf{B}$ and $\mathbf{H}$.
• Lorentz force — $\mathbf{F}=q(\mathbf{E}+\mathbf{u}\times \mathbf{B})$; the force law that defines the fields.
• Biot-Savart law — direct integration formula for $\mathbf{H}$ from any current distribution.
• Ampère’s law — integral and differential forms; the magnetostatic shortcut for symmetric currents.
• Magnetic flux — $\Phi =\int \mathbf{B}\cdot d\mathbf{s}$; the conserved quantity through closed surfaces.
• Magnetic boundary conditions — how $\mathbf{B}$ and $\mathbf{H}$ jump at material interfaces.
• Vector magnetic potential — $\mathbf{A}$ with $\mathbf{B}=\nabla \times \mathbf{A}$; the magnetic analog of $V$.
• Flux linkage — $\Lambda =N\Phi$; the total flux coupled to an $N$-turn coil, used to define inductance.
• Inductance — $L=\Lambda /I$ for self-inductance; mutual $M$ for inductively coupled circuits.
• Magnetic energy — $\frac{1}{2}\mu H^2$ per unit volume; energy stored in the magnetic field.

Time-varying fields

When fields vary with time, $\mathbf{E}$ and $\mathbf{B}$ couple. New phenomena emerge: induced EMF, displacement current, electromagnetic waves.

• Faraday’s law — $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$; a changing magnetic flux induces an EMF.
• Lenz’s law — the sign rule for induced currents: they oppose the change in flux.
• Displacement current — Maxwell’s addition to Ampère’s law: ${\partial }_t\mathbf{D}$ acts as an effective current.
• Capacitor displacement current example — the canonical charging-capacitor problem showing why the displacement-current term is mandatory.
• Charge continuity equation — $\nabla \cdot \mathbf{J}=-{\partial }_t{\rho }_v$; charge conservation.
• Charge relaxation — exponential decay of free volume charge inside a conductor with time constant $\tau =ϵ/\sigma$.

Maxwell’s equations

The four PDEs that unify all of classical electromagnetism.

• Maxwell’s equations — the four equations (Gauss, Gauss for magnetism, Faraday, Ampère-Maxwell) and their integral counterparts.

Electromagnetic waves

The waves that emerge when displacement current closes the feedback loop between $\mathbf{E}$ and $\mathbf{B}$.

• Plane wave — the simplest EM wave: planar phase fronts, TEM polarization, propagating at $c/\sqrt{{ϵ}_r{\mu }_r}$.
• TEM mode — transverse-electromagnetic field structure: both $\mathbf{E}$ and $\mathbf{H}$ perpendicular to the propagation direction.
• Poynting vector — $\mathbf{S}=\mathbf{E}\times \mathbf{H}$; energy flux of the EM field.

Transmission lines

One-dimensional guided EM waves. The bridge between fields and circuits.

• Transmission line — physical structure (coax, microstrip, two-wire), lumped-element model, wavelength criterion.
• Air line — the lossless air-dielectric reference case ($R^{\prime }=G^{\prime }=0$, $u_p=c$); the cleanest baseline TL.
• Telegrapher’s equations — the PDEs of voltage and current on a TL; reduce to wave equation in the lossless case.
• Characteristic impedance — $Z_0=\sqrt{L^{\prime }/C^{\prime }}$ (lossless); the intrinsic impedance of a propagating wave.
• Reflection coefficient — $\Gamma =(Z_L-Z_0)/(Z_L+Z_0)$; the load mismatch ratio.
• Standing wave ratio — $S=(1+|\Gamma |)/(1-|\Gamma |)$; engineering metric for matching quality.
• Wave impedance and input impedance — $Z(d)$ along the line; $Z_{\text{in}}$ as seen from the generator.
• Transmission line power flow — average incident, reflected, and delivered power.

Transients on transmission lines

Time-domain response to step, pulse, and switching inputs.

• Bounce diagram — graphical method for tracking wavefronts bouncing between source and load.
• Step response on transmission line — DC turn-on transient; rings before settling to lumped-circuit DC value.

Impedance matching and the Smith chart

Engineering practice for eliminating reflections.

• Impedance matching — matching network design: quarter-wave transformer, single-stub, lumped-element.
• Matched segment reflection containment — the subtle property that matching networks confine (rather than eliminate) reflections to the load-side segment.
• Smith chart — graphical calculator for impedance, admittance, $\Gamma$, SWR, and matching network design. The applied workflow built on the Möbius transformation of $z\mapsto (z-1)/(z+1)$.

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The math underwriting this material is in Mathematical methods — vector calculus and complex analysis. Transmission-line analysis uses transform methods from Signals and systems — phasors are $e^{j\omega t}$ from Euler’s formula in steady state, and the time-domain analysis of bounce diagrams parallels Laplace-transform analysis of circuits with delays. The Smith chart specifically is the most-cited engineering application of Möbius transformations and conformal mapping from complex analysis.

For sinusoidal steady-state analysis common throughout EM (especially TLs and antennas), the Phasor convention $e^{j\omega t}$ is used — consistent with Phasor relationships for circuit elements from Continuous-Time Signals and Systems. Complex impedance is $Z=R+jX$ (engineering convention).