Electromagnetism

Classical electromagnetism describes electric and magnetic fields, their sources, their forces on matter, and their propagation. Circuit models approximate those fields when component dimensions are small compared with the relevant wavelengths. High-frequency interconnects require a field or transmission-line model because voltage and current no longer change everywhere at once.

Charge, Current, Fields, and Force #

Electric charge is conserved. Charge density ρ describes charge per volume, while current density J describes charge flow per area. Their local continuity relation is:

∂ρ/∂t + ∇ · J = 0

An electric field E exerts force on charge whether the charge moves or not. A magnetic field B contributes a velocity-dependent force. Together they form the Lorentz force:

F = q(E + v × B)

The magnetic term is perpendicular to velocity, so it changes direction but does no mechanical work by itself. The electric term can transfer energy to or from a charge. In a conductor, an electric field drives charge carriers and thus produces current; material properties and geometry determine the resulting macroscopic resistance.

Maxwell’s Equations in Differential and Integral Form #

Maxwell’s equations in vacuum with sources, using SI units, are:

∇ · E = ρ/ε₀                 ∮ E · dA = Qenclosed/ε₀
∇ · B = 0                    ∮ B · dA = 0
∇ × E = -∂B/∂t              ∮ E · dl = -dΦB/dt
∇ × B = μ₀J + μ₀ε₀∂E/∂t    ∮ B · dl = μ₀I + μ₀ε₀dΦE/dt

The differential forms describe field behavior at a point. The integral forms relate fields on a closed surface or loop to enclosed charge, current, or changing flux. Gauss’s electric law associates electric divergence with charge. Gauss’s magnetic law states that magnetic flux has no source or sink in classical theory. Faraday’s law associates a circulating electric field with changing magnetic flux. Ampère’s law with Maxwell’s displacement-current term associates a circulating magnetic field with conduction current and changing electric flux.

The integral equations follow from the differential equations under the regularity conditions required by the divergence and Stokes theorems. Interface boundary conditions follow by shrinking a Gaussian surface or loop across a material boundary; surface charge and surface current can then create discontinuities in selected field components.

Induction and Electromagnetic Waves #

Faraday induction explains generators, transformers, and inductors. A changing magnetic flux through a circuit produces electromotive force:

emf = -dΦB/dt

The minus sign expresses Lenz’s law: the induced response opposes the flux change that produced it. Mutual induction couples circuits through shared flux, while self-induction relates a circuit’s changing current to its own induced voltage.

In a source-free, homogeneous vacuum, taking curls of Maxwell’s equations gives:

∇²E = μ₀ε₀ ∂²E/∂t²
∇²B = μ₀ε₀ ∂²B/∂t²

Those wave equations propagate at c = 1/√(μ₀ε₀). A plane wave in vacuum has mutually perpendicular E, B, and propagation directions. Material permittivity, permeability, and conductivity change phase velocity, attenuation, reflection, and dispersion.

Energy, Poynting Flow, and Impedance #

Vacuum electromagnetic energy density is:

u = (ε₀|E|² + |B|²/μ₀) / 2

The Poynting vector gives power flow per area:

S = E × H

Poynting’s theorem balances decreasing field energy with outward energy flux and work performed on charge. The model is useful for circuits because power reaches a load through fields surrounding the conductors rather than by treating electrons as packets that carry energy along the wire at signal velocity.

Wave impedance relates electric and magnetic field magnitudes. In vacuum, Z₀ = √(μ₀/ε₀), approximately 377 ohms. A transmission line instead has a characteristic impedance determined by conductor geometry and material properties. A mismatch causes reflections; matching source, line, and load impedances controls those reflections.

Radiation and Model Limits #

Accelerating charges and time-varying current distributions can radiate. Antenna geometry determines how near fields couple to stored electric and magnetic energy and how far fields carry energy away. A charge moving at constant velocity in an inertial frame does not radiate merely because it moves.

Maxwell’s equations are classical. Special relativity relates electric and magnetic fields between inertial frames, while quantum electrodynamics describes their quantum interaction with matter.