Electromagnetic waves
How electromagnetic waves are born
A charge at rest has only an electric field around it. A charge moving at a steady speed is a steady current, and it makes a steady magnetic field. Neither of these sends anything outward.
But an accelerated charge does. Maxwell’s theory shows that accelerated charges radiate electromagnetic waves.
Picture a charge oscillating up and down at a frequency ν. The electric field near it oscillates at the same frequency.
An oscillating electric field is a changing electric flux, a displacement current, so it produces an oscillating magnetic field. That changing magnetic field, by Faraday’s law, produces an oscillating electric field. And so on.
The two fields keep regenerating each other and spread outward through space. This is an electromagnetic wave, and its frequency equals the frequency of the oscillating charge. The energy it carries comes from the source.
For example, charges oscillating in an aerial at 100 megahertz send out waves of 100 megahertz. Their wavelength is c over ν: 3 × 10⁸ over 10⁸, which is 3 metres.
In 1887 Heinrich Hertz produced and detected such waves in his laboratory, confirming Maxwell’s prediction. A few years later Jagadish Chandra Bose, working in Kolkata, made much shorter waves, from 25 mm down to 5 mm.
At around the same time, Guglielmo Marconi in Italy sent electromagnetic waves over distances of many kilometres. This was the start of radio communication.
Could we make visible light with a circuit? Light needs about 10¹⁴ hertz, but even modern electronic circuits reach only about 10¹¹ hertz. Light comes from charges inside atoms instead.
Nature of electromagnetic waves
Take a wave travelling along z. Its electric field points along x, and its magnetic field along y. The two fields are perpendicular to each other and to the direction of travel. Electromagnetic waves are transverse.
At any point, E_x equals E₀ sin(kz − ωt), and B_y equals B₀ sin(kz − ωt). Watch them: they rise, fall and pass through zero together. E and B are in phase.
Here k equals 2π over λ, the wave number, and ω equals 2πν. The pattern moves along z with speed ω over k, which is νλ.
Maxwell found this speed from his equations: c equals 1 over √(μ₀ε₀). Put in μ₀ = 4π × 10⁻⁷ T m A⁻¹ and ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻².
μ₀ε₀ comes to 1.113 × 10⁻¹⁷. Its square root is 3.336 × 10⁻⁹, and one over that is 3.00 × 10⁸ metres per second. Exactly the measured speed of light.
So light is an electromagnetic wave. Electricity, magnetism and optics became one subject. In a material medium the speed is v equals 1 over √(με), which is less than c.
The amplitudes are linked: B₀ equals E₀ over c. A wave with E₀ = 30 volt per metre has B₀ = 30 over 3 × 10⁸, which is 10⁻⁷ tesla. The magnetic field looks tiny in SI units.
The direction of travel is the direction of E × B. With E along x and B along y, x̂ × ŷ is ẑ: the wave moves along plus z. Unlike sound, it needs no medium; it crosses empty space.
An example. A 25 megahertz wave travels along x. At one point E is 6.3 volt per metre along y. Then B is 6.3 over 3 × 10⁸, which is 2.1 × 10⁻⁸ tesla.
Its direction: E × B must point along x, and ĵ × k̂ is î. So B points along z. And the wavelength is c over ν: 3 × 10⁸ over 25 × 10⁶, which is 12 metres.
Energy, momentum and pressure
Stand in the sunshine and your skin warms up. That energy has travelled 150 million kilometres through empty space. Electromagnetic waves carry energy.
Wherever there are electric and magnetic fields, there is energy. As the wave moves, the energy stored in its fields moves along with it.
Electromagnetic waves also carry momentum. If a surface completely absorbs a total energy U from the wave, the momentum delivered to it is p equals U over c.
If the surface is a perfect mirror, the wave bounces back and its momentum reverses. Then the momentum delivered is twice as much: 2U over c.
Momentum delivered every second is a force, and force per unit area is a pressure. So light pushes on whatever it falls on. This is radiation pressure.
An example. Light with an energy flux of 18 watt per square centimetre falls normally on a non-reflecting surface of area 20 square centimetres. Find the average force over 30 minutes.
Energy arriving per second: 18 × 20, which is 360 watt. In 30 minutes, 1800 seconds, U equals 360 × 1800, which is 6.48 × 10⁵ joule.
Momentum delivered: p equals U over c, 6.48 × 10⁵ over 3 × 10⁸, which is 2.16 × 10⁻³ kilogram metre per second. Average force: p over t, 2.16 × 10⁻³ over 1800, which is 1.2 × 10⁻⁶ newton.
If the surface were a perfect reflector, the force would double to 2.4 × 10⁻⁶ newton.
Sunlight at the ground brings about 1000 watt per square metre. Its pressure is about 1000 over 3 × 10⁸, roughly 3 × 10⁻⁶ pascal. Far too small to feel, though it helps push the tails of comets away from the Sun.
Try it: a travelling EM wave
- Drag to turn the view. E (amber) points along x, B (blue) along y, and the wave travels along z.
- Pause and look at the probe: E and B are zero together and largest together. They are in phase.
- Raise the frequency: λ = c/ν gets shorter, but ω/k stays 3 × 10⁸ m/s for every frequency.
- Change E₀ and watch B₀ = E₀/c. Then reverse the wave: B flips, so E × B still points along the motion.
Find the magnetic field
A plane electromagnetic wave of frequency 25 MHz travels in free space along the x-direction. At a particular point, E = 6.3 ĵ V/m. What is B at that point?
- 2.1 × 10⁻⁸ k̂ T
- 2.1 × 10⁻⁸ ĵ T
- −2.1 × 10⁻⁸ k̂ T
- 1.9 × 10⁹ k̂ T
Answer
A. Size: B = E/c = 6.3/(3 × 10⁸) = 2.1 × 10⁻⁸ T. Direction: E × B must point along +x. Since ĵ × k̂ = î, B is along +k̂. B is never parallel to E, and −k̂ would send the wave along −x. 1.9 × 10⁹ T comes from multiplying by c instead of dividing.
Read the wave equation
The magnetic field of a plane EM wave is B_y = 2 × 10⁻⁷ sin(0.5 × 10³ x + 1.5 × 10¹¹ t) T. What are its wavelength and the amplitude of its electric field?
- λ = 1.26 cm, E₀ = 60 V/m
- λ = 2 mm, E₀ = 60 V/m
- λ = 1.26 cm, E₀ = 6.7 × 10⁻¹⁶ V/m
- λ = 12.6 m, E₀ = 60 V/m
Answer
A. Compare with sin(kx + ωt): k = 0.5 × 10³ m⁻¹, so λ = 2π/k = 6.28/500 = 1.26 × 10⁻² m. E₀ = cB₀ = 3 × 10⁸ × 2 × 10⁻⁷ = 60 V/m. Check: ω/k = 1.5 × 10¹¹/500 = 3 × 10⁸ m/s. (1/k = 2 mm forgets the 2π; dividing B₀ by c gives 6.7 × 10⁻¹⁶.) The + sign means the wave travels along −x.
Summary
- Plane wave along z (B_y in phase):
E_x = E₀ sin(kz − ωt) - Speed of light, ω/k = νλ:
c = 1/√(μ₀ε₀) - Amplitudes:
B₀ = E₀/c - Momentum, full absorption:
p = U/c
- Accelerated charges radiate EM waves; an oscillating charge gives a wave of the same frequency.
- E and B are perpendicular to each other and to the direction of travel (transverse), in phase, and the wave moves along E × B.
- EM waves need no medium. In a medium their speed is v = 1/√(με).
- EM waves carry energy and momentum, so they exert radiation pressure; a perfect reflector receives 2U/c.
- Hertz (1887) first produced them; J.C. Bose made 25–5 mm waves; Marconi sent them over many kilometres.