Electromagnetic Waves

Maxwell's key addition: a changing electric flux produces a magnetic field exactly as a real current does. ΦE is the electric flux through a surface. Id has the same unit as real current (ampere) and produces the same magnetic effect. Resolves the inconsistency in Ampere's law for a capacitor circuit.

Generalised Ampere's law: magnetic circulation is due to both conduction current I and displacement current Id. In a capacitor gap where I = 0, the displacement current bridges the discontinuity. This equation, with Faraday's law, predicts self-sustaining EM waves.

Electric field lines originate from positive charges and terminate on negative charges. Net electric flux through a closed surface equals the enclosed charge divided by ε₀. Electric monopoles (charges) exist — a fundamental asymmetry with magnetism.

Net magnetic flux through any closed surface is zero — magnetic monopoles do not exist. Every magnetic field line forms a closed loop. Magnetic field lines that enter a surface must exit it. This is the magnetic analogue of Gauss's law but with zero source term.

A changing magnetic field induces a circulating electric field. This is Faraday's law in integral form. The negative sign (Lenz's law) means the induced E opposes the change in B. This is what makes EM waves self-sustaining: changing B → induced E → changing E → induced B → ...

A changing electric field or a real current produces a circulating magnetic field. Maxwell's equations are symmetric between E and B (except for source terms — no magnetic monopoles or magnetic currents in nature). Together, all four equations are the foundation of classical electrodynamics.

Wave equations for E and B fields, derivable directly from Maxwell's equations. Both E and B satisfy identical wave equations, confirming they propagate at the same speed. Wave speed v = 1/√(μ₀ε₀) = c. This predicted value matched the known speed of light, confirming light as an EM wave.

Speed of light in vacuum, derivable purely from electrical constants μ₀ and ε₀. Maxwell's greatest prediction: c = 1/√(μ₀ε₀) ≈ 2.998×10⁸ m/s, matching the measured speed of light — proving light is an electromagnetic wave. In SI units, c is now defined exactly as 299,792,458 m/s.

EM waves travel slower in a medium. n is the refractive index, μr the relative permeability, εr the relative permittivity (dielectric constant). For most optical materials, μr ≈ 1, so n ≈ √εr. Water: εr ≈ 80, but n ≈ 1.33 because εr is frequency-dependent.

A plane EM wave propagating in the +x direction. E and B oscillate in phase (reach maxima simultaneously), perpendicular to each other and to the direction of propagation. Wave is transverse. k = 2π/λ is the wave number, ω = 2πf. E, B, and k̂ form a right-handed triad.

Wave number k (rad m⁻¹) relates wavelength to angular frequency via the dispersion relation ω = ck. Standard relations: c = fλ. For a wave in a medium: v = ω/k = fλ. These relations hold for all EM waves regardless of frequency — all travel at c in vacuum.

In an EM wave, the electric and magnetic field amplitudes are related by E₀ = cB₀ at all instants. Although E and B oscillate together in phase, E is much larger numerically (since c is large): for B₀ = 1 μT, E₀ = 300 V/m. In a medium: E/B = v = c/n.

Energy is shared equally between E and B fields: uE = uB at every instant. Instantaneous total energy density u = ε₀E² = B²/μ₀. Time-averaged density uses the factor ½ (since ⟨sin²⟩ = ½). This equal energy sharing is a deep consequence of E = cB and c = 1/√(μ₀ε₀).

The Poynting vector S gives the instantaneous power flow per unit area (energy flux) in the direction of wave propagation (E×B direction). SI unit: W m⁻². Magnitude = energy density × wave speed. The Poynting vector is the electromagnetic analogue of the intensity vector.

Intensity (average power per unit area) is the time average of the Poynting vector magnitude. All forms are equivalent; use the one with known quantities. SI unit: W m⁻². For a point source of power P: I = P/(4πr²) — intensity falls as 1/r². This is the inverse-square law.

EM waves carry momentum and exert pressure on surfaces they strike. For partial reflection with reflectivity r: P = I(1+r)/c. Radiation pressure is tiny under ordinary conditions but important in stellar interiors (where it balances gravity), laser cooling, and solar sails.

An EM wave carrying energy U also carries momentum p = U/c. Rate of momentum transfer (force) to a perfectly absorbing surface of area A: F = IA/c. This is the basis of radiation pressure. The photon relation E = pc (for massless photons) is the quantum version of the same result.