Foundation lesson
Fields and waves
A field gives a value at every place and every time. In a light wave the field swings back and forth hundreds of trillions of times a second, too fast for any detector to follow, so what is measured is the energy delivered on average.
Written for this edition, not translated from Einstein.
What is a wave in a field, and what does a light detector measure?
A field assigns a value to every place at every moment. The temperature in a room is a field of single numbers. The electric field is a field of arrows, each with a size and a direction.
In a wave, the field at each place rises and falls in a repeating cycle, and the pattern travels. For the simplest wave the field goes as cos(kx − ωt). The wavelength λ is the distance from one crest to the next; the frequency ν is the number of crests passing a point each second; the crests travel at c = λν. The quantities in the cosine are k = 2π/λ and ω = 2πν.
The field at x and t is E nought times the cosine of k x minus omega t, and the speed c equals the wavelength times the frequency.
Visible light oscillates about 6 × 10¹⁴ times a second, far too fast for any detector to follow. A detector responds to the energy delivered, which goes as the square of the field, averaged over many cycles. Over a whole cycle the average of cos² is exactly one half:
The average of cosine squared of omega t, over a whole cycle, is one half.
A small lamp sending power P out equally in all directions spreads it over spheres of area 4πr². The intensity, the power arriving on each square metre, at distance r is therefore P/(4πr²): at twice the distance, a quarter of the intensity.
The Doppler shift of sound is a useful analogy for a change of frequency that comes with motion, but it has a limit. Sound travels in air, so a moving source and a moving listener are different situations, told apart by motion relative to the air. §7 of the relativity paper derives the Doppler shift of light from the relative velocity of source and observer alone; no medium enters the calculation.
Worked example: A 600-terahertz wave and a one-watt lamp
- Wavelength of light at 600 terahertz, 6.0 × 10¹⁴ Hz: λ = c/ν = 299 792 458 ÷ 6.0 × 10¹⁴ = 4.997 × 10⁻⁷ m, which is 499.7 nanometres, blue-green light.
- A lamp giving out 1 W equally in all directions. At 1 m the sphere has area 4π × 1² = 12.57 m², so the intensity is 1 ÷ 12.57 = 0.0796 W/m².
- At 2 m the sphere has area 4π × 2² = 50.27 m², so the intensity is 0.0199 W/m², a quarter of the value at 1 m.
The intensity equals the power divided by four pi r squared.
Where this lesson stops
A field is a value assigned to every place and time. A light detector records the energy delivered averaged over many cycles, not the field at one instant.
If you came here from a passage, Back returns you to the exact place you left.