Electromagnetic Waves

A changing magnetic field makes an electric field. Maxwell realised the reverse must also hold, and adding that one term to Ampère’s law turned electricity and magnetism into a theory of light.

Where Ampère’s law fails#

Ampère’s law, ∮B⃗·dl⃗ = μ₀i, holds only while the electric field through the loop is steady. Charge a capacitor with circular plates and put a compass between the plates: it deflects, so there is a magnetic field there. Take a loop γ lying between the plates, parallel to them. Its circulation of B is not zero, yet no charge crosses the flat surface it bounds: i = 0. And the field due to the currents in the wires alone, worked out from Biot–Savart, would give zero circulation. Something else between the plates makes magnetic field: the changing electric field.

The displacement current#

Just as Faraday’s law says ∮E⃗·dl⃗ = −dΦ_B/dt, a changing electric flux makes a magnetic circulation:

With both a current and a changing electric field present, Maxwell’s generalised Ampère’s law is:

Current is always continuous#

If charge piles up inside a closed surface, conduction current i₁ in exceeds i₂ out: d(q_inside)/dt = i₁ − i₂. By Gauss’s law Φ_E = q_inside/ε₀, so ε₀ dΦ_E/dt = d(q_inside)/dt, i.e. i_d = i₁ − i₂.

Counting displacement current, as much current flows out of any closed surface as flows in.

Four equations sum up all of electricity and magnetism. In empty space they have a remarkable solution: a wave of electric and magnetic fields that travels at exactly the speed of light.

Maxwell’s equations#

In vacuum there are no charges or currents, leaving

A plane wave#

Try a wave travelling along x, with E along y and B along z:

Faraday’s law on a rectangle in the x–y plane (sides at x₁ and x₂, height l): the circulation of E is E₀l[sin ω(t − x₂/c) − sin ω(t − x₁/c)], and −dΦ_B/dt works out to cB₀l times the same bracket. They agree if

Ampère–Maxwell law on a rectangle in the x–z plane does the same job for B and gives B₀ = μ₀ε₀cE₀. Together with E₀ = cB₀:

With μ₀ and ε₀ measured in purely electrical and magnetic experiments, this gave c = 2.99793 × 10⁸ m/s: the measured speed of light. That was the proof that light is an electromagnetic wave. (Today c and μ₀ are fixed by definition and ε₀ is defined through this equation.)

Sunlight warms your face and pushes on solar sails. An electromagnetic wave carries both energy and momentum, shared equally between its electric and magnetic fields.

Energy density#

In a small volume of the wave the energy is ½ε₀E² dV (electric) plus B²dV/2μ₀ (magnetic). With E = E₀ sin ω(t − x/c) and B = B₀ sin ω(t − x/c), and sin² averaging to ½:

Using E₀ = cB₀ and μ₀ε₀ = 1/c², the two terms are equal: on average the electric and magnetic fields carry the same energy.

Intensity#

Intensity is the energy crossing unit area per unit time, perpendicular to the direction of travel. The energy in a cylinder of area A and length cΔt (u_avcΔtA) crosses A in time Δt, so

Momentum and radiation pressure#

A portion of the wave carrying energy U also carries momentum

• Fully absorbed: the surface receives momentum U/c; the pressure is I/c.
• Fully reflected: the momentum reverses (p → −p), the surface receives 2U/c; the pressure is 2I/c.

Maxwell’s equations don’t pick a wavelength. Radio waves, microwaves, light and gamma rays are the same phenomenon over an enormous range of scales.

The electromagnetic spectrum#

Visible light spans only about 380–780 nm. Known electromagnetic waves stretch from about 30 fm (30 × 10⁻¹⁵ m) to 30 km. All travel at c in vacuum, so frequency and wavelength are tied by ν = c/λ. The boundaries between named regions are not sharp; gamma rays and X-rays overlap a good deal, and it is only on average that gamma rays are shorter.

Sources and uses

Every electromagnetic wave comes ultimately from accelerated charges, whose changing fields make up the wave.

• Radio waves: charges oscillating in LC circuits; radio and TV broadcasting.
• Microwaves: oscillating currents in electric circuits; radar, cooking, communication.
• Infrared: emitted by atoms and molecules of hot bodies; used in physiotherapy.
• Visible light: emitted by atoms, linked to the acceleration of their electrons.
• Ultraviolet: produced the same way; the Sun emits a lot, harmful in large doses.
• X-rays: most commonly from fast electrons slowing down in a metal target; medical diagnosis, harmful to tissue.
• Gamma rays: emitted by atomic nuclei; the shortest wavelengths we usually meet.

Radiation and the atmosphere#

The atmosphere reaches about 300 km up, in rough layers without sharp boundaries:

• Troposphere, up to about 12 km: almost all clouds, water vapour and ice. Air at its top is about a tenth as dense as at the ground.
• Stratosphere, 12–50 km, with the ozone layer in its upper part; air at its top is about 10⁻³ times as dense as at the ground.
• Mesosphere, about 50–80 km; above it, the ionosphere.

The Sun is the main source of radiation reaching us, and much of it is absorbed on the way down. Visible light is only weakly absorbed. Much of the infrared is absorbed and heats the air. The ozone layer absorbs most of the ultraviolet and shorter wavelengths, converting it to infrared that heats the atmosphere and ground, and so protects living things.