Electromagnetic waves, radiation, and relativistic electromagnetism

Ask what a wave carries

An electromagnetic wave is a space-time pattern of electric and magnetic fields that transports energy and momentum. It is not a material substance sliding through an invisible support. The fields have local values: E tells how a charge at rest would be pushed, while B contributes to the force on a moving charge. A travelling pattern can exist in a vacuum even where the charge density and current density vanish.

Three questions organize this lesson. Which field patterns are allowed by Maxwell’s equations? How does a localized source launch a disturbance, and which part survives far away? How do observers in relative motion divide the same electromagnetic field into electric and magnetic components? Answering one question does not automatically answer the others. A useful plane-wave solution is not a complete antenna, and a local field invariant is not a test that proves a source is radiating.

The first reading follows the wave, energy, dipole and frame examples. The advanced return follows retarded boundary conditions, the distinction between source time and arrival time, tensor conventions, and causal material response. The quantitative examples use chosen analytic inputs. They are not measurements of an antenna, a particular dielectric, or a charged particle in a laboratory.

State the vacuum problem before simplifying it

In an inertial frame, the vacuum equations are ∇·E=ρ/ε₀, ∇·B=0, ∇×E=−∂B/∂t, and ∇×B=μ₀J+(1/c²)∂E/∂t, with ε₀μ₀c²=1. In a source-free region set ρ=0 and J=0 there. Sources can still be elsewhere. Their past behavior and the boundary or initial data select the particular solution in the region.

Vacuum means more than replacing a material constant by a number. Here there is no polarization, magnetization or conduction response to solve. The speed c is the vacuum causal speed. In modern SI its numerical value is exact, while this lesson keeps μ₀ and ε₀ symbolic rather than treating the historical numerical value of μ₀ as an exact measured input.

A field problem also needs admissible initial data. At an initial time the divergence constraints must hold, and the curl equations determine the subsequent evolution. A boundary may impose incoming waves or reflections. One cannot choose arbitrary E and B functions independently and then declare that any pattern resembling a sinusoid is a source-free electromagnetic wave.

Derive the wave equations without losing their assumptions

Take the curl of Faraday’s law and use ∇×(∇×E)=∇(∇·E)−∇²E. In the source-free region ∇·E=0, while the time derivative of the Ampère–Maxwell equation supplies ∂(∇×B)/∂t=(1/c²)∂²E/∂t². The result is ∇²E−(1/c²)∂²E/∂t²=0. Repeating the calculation for B gives the same wave operator on B.

The derivation explains why the displacement-current term matters. Dropping it before taking the curl removes the term that allows a vacuum electromagnetic wave. It also explains the scope of the result: derivatives commute in the smooth region being considered, and the divergence equations were used. At ideal sheets or point sources, distributional source terms and jump conditions must be retained.

With sources present, the corresponding equations are ∇²E−c⁻²∂²E/∂t²=∇ρ/ε₀+μ₀∂J/∂t and ∇²B−c⁻²∂²B/∂t²=−μ₀∇×J. They organize the source problem but do not make E and B independent. Charge conservation and the remaining Maxwell equations still connect the source data and the fields.

Keep the first-order Maxwell constraints

A second-order wave equation is necessary for a source-free field component, but it is not sufficient for a complete electromagnetic solution. For example, E=E₀ x̂ cos(kx−ckt), B=0 satisfies the scalar wave equation component by component. Nevertheless ∇·E is generally nonzero and the Ampère–Maxwell equation is not satisfied. It is a longitudinal scalar-wave construction, not the proposed vacuum electromagnetic wave.

For a nonzero-frequency plane-wave mode with wavevector k, the divergence equations require k·Ẽ=0 and k·B̃=0. Faraday’s law requires k×Ẽ=ωB̃, and the other curl equation then requires ω²=c²|k|². Thus the two transverse electric components specify the mode; the magnetic components are constrained by them.

Uniform static background fields can be superposed on a wave. They should not be mistaken for an oscillating longitudinal polarization. Nor should the plane-wave transverse statement be transferred to arbitrary superpositions, waveguides, evanescent fields or near-source configurations. Those problems can have longitudinal components relative to a selected direction while still satisfying Maxwell’s equations.

Fix a directed plane wave and a physical scale

Choose the unit propagation vector n=(2/3,−1/3,2/3), and the transverse unit vectors e₁=(2/3,2/3,−1/3) and e₂=(−1/3,2/3,2/3). Their dot products vanish pairwise, their lengths are one, and n×e₁=e₂. These three directions specify a right-handed geometry without silently identifying the propagation direction with a coordinate axis.

Let E_w>0 be an electric-field scale in V/m and L_w>0 an independent length in metres. Set k_w=1/L_w and ω_w=c/L_w. The phase is φ=n·r/L_w−ct/L_w. The wavelength is 2πL_w and the period is 2πL_w/c. Length and time cannot both be chosen independently once vacuum propagation is imposed.

Our family is E/E_w=e₁ cosφ+ηe₂ sinφ and cB/E_w=e₂ cosφ−ηe₁ sinφ. Directly taking n×E gives cB, with the required sign. The three selected values η=0, 2/5 and 1 describe a line, an ellipse and a circle. A closed set of phase samples represents these periodic fields; equal values at φ=0 and 2π are the same phase, not an additional physical state.

Separate polarization from propagation

Polarization describes how the electric vector changes in the transverse plane. It does not describe motion of a parcel of vacuum. For η=2/5 the transverse components satisfy (E·e₁/E_w)²+(E·e₂/(2E_w/5))²=1. The curve is an ellipse in field-component space. Its axes carry electric-field units, whereas the axis along n in a spatial wave picture carries length.

Orientation conventions matter. Our plotted phase increases from e₁ toward +e₂ near φ=0, but at a fixed position increasing physical time decreases φ. At that event ∂E/∂t points toward −e₂ when η>0. A label such as right-circular must also specify whether the observer looks along or against propagation and which handedness convention is used. We instead state the vectors and the time direction explicitly.

A coherent single-frequency ellipse does not describe unpolarized light. Unpolarized radiation requires an ensemble or time-dependent statistical description of polarization, not the claim that a single instantaneous field lacks a direction. A real optical beam may also have finite bandwidth, a spatially varying polarization and a longitudinal correction associated with its finite transverse extent.

Recognize the plane-wave idealization

The monochromatic plane wave extends over an infinite transverse plane and over all time. Its local energy density is well defined, but its total energy integrated over all space is not a finite pulse energy. Multiplying its field by an arbitrary sharp spatial window generally breaks Maxwell’s constraints at the window. A finite beam or pulse must be constructed with appropriate sources, boundaries or a superposition of allowed modes.

For a finite-bandwidth packet, each vacuum Fourier component satisfies ω=c|k|. Diffraction and interference determine how their combination evolves. A narrow beam can be approximately transverse and approximately planar locally, without being an exact plane wave globally. This distinction matters whenever one interprets a diagram as a spatial snapshot or infers a total energy from an infinite-wave density.

The source-free equations are not a story in which an electric field creates a magnetic field at one place, which later creates a separate electric field farther away in a sequence of little machines. They are simultaneous local evolution and constraint equations. A disturbance propagates through their coupled solution, with its initial or source data determining the allowed field pattern.

Derive the local energy balance

Dot the Ampère–Maxwell equation with E and Faraday’s law with B/μ₀. The vector identity ∇·(E×B)=B·(∇×E)−E·(∇×B) combines the two results into ∂u/∂t+∇·S=−J·E, where u=ε₀E²/2+B²/(2μ₀) and S=E×B/μ₀. In vacuum these are the usual field energy density and energy flux.

Integrating over a fixed volume gives d∫u dV/dt=−∮S·n_out dA−∫J·E dV. The surface integral is outward power, not a field magnitude. Positive J·E transfers energy from the field to charges; a driven source can have the opposite sign. The volume and its outward normal must be specified before assigning a sign to an energy ledger.

This local balance explains why a static or near field can have a nonzero Poynting vector without being an outgoing radiation field. S describes the chosen local energy-flow convention, while radiation is a statement about source-dependent solutions and their far-zone transport. Sources, stresses and stored energy must be included in a complete apparatus account.

Average real fields, not complex amplitudes squared

For the chosen travelling wave, define q(φ)=cos²φ+η²sin²φ. The electric and magnetic energy densities are each ε₀E_w²q/2. Their sum is ε₀E_w²q, and S=ε₀cE_w²q n. The equality of the two instantaneous energy contributions follows from this one travelling-wave solution; it is not a separate rule for every electromagnetic field.

The cycle average is ⟨q⟩=(1+η²)/2. The linear case reaches zero energy density at its quadrature phases, whereas the circular case has constant energy density even though its vector direction changes. At a zero-field instant, dividing S by its magnitude is undefined. The wave’s prescribed propagation vector n is still part of the solution, but it is not obtained by normalizing a zero vector.

Complex phasors are a convenient linear representation. If physical fields are the real parts of Ẽe⁻ⁱωᵗ and H̃e⁻ⁱωᵗ, their time-averaged flux is ⟨S⟩=Re(Ẽ×H̃*)/2. One must not insert an unconjugated complex field into the real quadratic expression and interpret the result as instantaneous power. The amplitude convention is responsible for the factor one-half.

Transport momentum and distinguish stress conventions

In vacuum the field momentum density is g=S/c². To make the momentum-flux sign explicit, define Π_ij=uδ_ij−ε₀E_iE_j−B_iB_j/μ₀. The local balance is ∂g_i/∂t+∂_jΠ_ij=−(ρE+J×B)_i. Momentum gained by matter is lost by the field once the surface flux is included.

Some treatments instead call σ_ij=−Π_ij the Maxwell stress and write the matter force in terms of ∇·σ−∂g/∂t. Other authors call Π itself the stress tensor. The sign convention, the surface normal and the force being calculated matter more than the name. Here Π always means transported field momentum flux; σ means the opposite traction convention.

For a single travelling vacuum wave, Π=u nn. An ideal stationary absorber at normal incidence receives mean momentum I/c per unit area and time, where I=|⟨S⟩|. A perfect specular mirror reverses the normal momentum and receives 2I/c. These are apparatus statements with specified transmission, absorption and motion, not universal formulas for light in arbitrary material.

The magnetic part of the force on an individual point charge is perpendicular to its velocity and does no instantaneous work. This does not prevent a wave from carrying momentum. The electric field does work on charges, while the combined Lorentz forces, material response and field momentum account for recoil and pressure. Energy and momentum conservation must be followed together.

Test the travelling-wave shortcuts with a standing wave

Superpose two equal linearly polarized waves travelling along +n and −n. Write ξ=n·r/L_w and τ=ct/L_w. Their fields are E/E_w=2e₁cosξ cosτ and cB/E_w=2e₂sinξ sinτ. The minus sign in the backward wave’s magnetic field is essential. It is what changes the magnetic sum into a product of sines.

The total normalized energy density is 2(cos²ξ cos²τ+sin²ξ sin²τ), while the normalized directed flux along n is sin(2ξ)sin(2τ). The cycle-averaged flux vanishes. Instantaneously energy can flow in either direction, and the electric and magnetic energy contributions need not be equal. A point with vanishing E can have nonzero B, or conversely.

The local balance is especially instructive. In units where time is τ and distance along n is ξ, ∂[u/(ε₀E_w²)]/∂τ=−2cos(2ξ)sin(2τ), while ∂[S·n/(ε₀cE_w²)]/∂ξ has the opposite value. Zero flux at one point does not imply zero flux divergence there. Energy density can change at that point because neighboring fluxes differ.

The chosen space-time grid spans complete 2π cycles in both ξ and τ, with explicitly repeated closure endpoints. It illustrates a periodic ideal standing field, not a finite cavity with unspecified walls. A real resonator requires boundary conditions, modes, losses and sources; its allowed frequencies and energy exchange cannot be inferred from this unbounded superposition alone.