Electromagnetic Waves— Notes, Formulas & Revision

Maxwell noticed Ampère's law (∮B·dl = μ₀I_c) breaks down when applied to a charging capacitor — no real current crosses the gap, yet B clearly exists around it.

He postulated a NEW current, the displacement current I_d = ε₀(dΦ_E/dt), produced by a changing electric flux — not by moving charges.

The complete Ampère–Maxwell law: ∮B·dl = μ₀(I_c + I_d). With this, the magnetic field is continuous everywhere — inside the wire AND inside the capacitor gap.

I_d has the SAME units (ampere) and produces the SAME magnetic field as a real conduction current of equal magnitude.

Inside a charging parallel-plate capacitor, dE/dt is uniform between the plates, so I_d = ε₀·A·(dE/dt) = C·(dV/dt) = I_c. The two currents are exactly equal.

The conception of displacement current closed Maxwell's equations and predicted that ANY changing E creates B and vice-versa — the seed of electromagnetic waves.

In a conductor, I_c dominates; in vacuum or a perfect dielectric, only I_d exists.

An electromagnetic wave is a self-propagating, transverse oscillation of coupled electric (E) and magnetic (B) fields.

E ⊥ B ⊥ direction of propagation. E × B always points along the propagation direction (Poynting vector).

All EM waves travel at the SAME speed in vacuum: c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s — independent of wavelength, frequency, or amplitude.

E and B are in phase: when E peaks, B peaks at the same point and time.

Ratio of field amplitudes: E₀/B₀ = c. So B is much smaller numerically than E (in SI units), but they carry equal energy density.

In a medium of refractive index n: v = c/n. Frequency f stays fixed (set by the source); wavelength shrinks to λ_medium = λ_vac/n.

EM waves DO NOT need a medium — Maxwell's prediction was confirmed by Hertz's 1887 experiment using spark-gap oscillators.

EM waves carry both energy (Poynting vector) and momentum (radiation pressure).

The electromagnetic spectrum spans many decades of frequency (≈ 10³ to 10²² Hz) — all the same kind of wave, only λ and source differ.

Radio waves (≥ 10⁻¹ m): generated by oscillating LC circuits; used in AM/FM, TV, mobile, radar.

Microwaves (1 mm – 30 cm): produced by klystrons / magnetrons; used in radar, satellite communication, microwave ovens (resonate water at 2.45 GHz).

Infrared (700 nm – 1 mm): from hot bodies / molecular vibrations; used in thermal imaging, IR remote controls, night vision.

Visible light (400–700 nm): the narrow band the human eye detects; produced by electron transitions in atoms.

Ultraviolet (10–400 nm): from very hot sources / electron transitions; causes sunburn, fluorescence, sterilisation. Mostly absorbed by the ozone layer.

X-rays (0.01–10 nm): from sudden deceleration of high-energy electrons (bremsstrahlung) or inner-shell transitions; used in medical imaging and crystallography.

γ-rays (< 0.01 nm): from nuclear transitions and cosmic events — highest frequency, shortest λ, most penetrating, most ionising.

All EM waves travel at c in vacuum. They differ in λ and frequency — and therefore in HOW they interact with matter.

The Poynting vector S = (1/μ₀)·(E × B) gives the instantaneous power per unit area carried by an EM wave, in the direction of propagation.

Units: W/m². It is the rate at which EM energy flows through a unit area perpendicular to S.

For a plane wave with E and B in phase: S = (1/μ₀)·E·B and S oscillates as sin²(ωt), always non-negative.

Average intensity ⟨S⟩ = I = ½ε₀cE₀² = E_rms B_rms/μ₀ = c·u_avg, where u_avg is the time-averaged energy density.

Energy density splits equally between E-field and B-field: u_E = u_B = ½ε₀E²(t) at every instant.

Intensity of a point source falls as 1/r² because the same power spreads over a sphere of area 4πr².

S is intimately tied to radiation pressure: P = I/c (absorber) or 2I/c (perfect reflector).

Electromagnetic waves carry momentum p = E/c (per photon, p = h/λ). When they hit a surface, they exert a pressure called radiation pressure.

Perfect absorber: momentum E/c is fully transferred → pressure P_abs = I/c.

Perfect reflector: photon reverses direction, momentum change is 2E/c → pressure P_ref = 2I/c.

Partial absorber (reflectance R, 0 ≤ R ≤ 1): P = (1+R)·I/c.

Effect is tiny in everyday life — solar radiation gives ~5 μPa on absorbing surfaces and ~10 μPa on mirrors — but real and measurable.

Applications: solar sails (NEA Scout, IKAROS spacecraft) propel themselves without fuel; optical tweezers manipulate cells using laser radiation pressure; comet tails are pushed away from the Sun by both radiation pressure and solar wind.

Astrophysical importance: in massive stars, radiation pressure can dominate gravity (Eddington limit) — sets the maximum stable luminosity of a star.

For any wave, the basic relation v = fλ ties speed, frequency, and wavelength together.

For EM waves in vacuum, v is fixed at c ≈ 3 × 10⁸ m/s. So f and λ are INVERSELY proportional.

Frequency f is the number of full cycles passing a point per second (Hz = s⁻¹). It is set by the source.

Wavelength λ is the spatial period — the distance between two consecutive crests of the wave.

Period T = 1/f is the time for one full cycle; angular frequency ω = 2πf.

When an EM wave enters a medium of index n: f stays the same, v drops to c/n, so λ shrinks to λ_vac/n. Wave-number k = 2π/λ therefore GROWS in the medium.

Photon energy is fixed by f, not λ: E = hf. So in a medium, photon energy is unchanged — only wavelength is.