In Bright-Line Spectra each element gave off its own colors. To use those colors as data, we need to describe light with numbers. Light is a wave of electric and magnetic fields, and like the waves in The Wave Equation v = fλ, its speed is its wavelength times its frequency.
Key idea
$$c = \lambda\nu$$ The wavelength $\lambda$ is the distance from one crest to the next (in m or nm, $1\ \text{nm} = 10^{-9}\ \text{m}$). The frequency $\nu$ (Greek "nu") is the number of crests passing a point each second (in hertz, $1\ \text{Hz} = 1\ \text{s}^{-1}$). In a vacuum every kind of light travels at $c = 3.00 \times 10^{8}\ \text{m/s}$, so a longer wavelength means a lower frequency.
Why it works
Stand still and count crests going by. In one second the wave train moves forward $c$ meters, and each crest is $\lambda$ meters behind the one before, so $c/\lambda$ crests pass you: that is $\nu$. Because $c$ is the same for all light, $\lambda$ and $\nu$ are locked together: double one and the other halves. Light differs only in wavelength, which is why radio waves, visible light and X-rays are one family, the electromagnetic spectrum.
| Region | Typical wavelength | Typical frequency |
|---|---|---|
| Radio | 1 m to km | $10^{5}$–$10^{9}$ Hz |
| Microwave | 1 mm to 30 cm | $10^{9}$–$10^{11}$ Hz |
| Infrared | 700 nm to 1 mm | $10^{11}$–$10^{14}$ Hz |
| Visible | 400–700 nm | $4$–$7.5 \times 10^{14}$ Hz |
| Ultraviolet | 10–400 nm | $10^{15}$–$10^{16}$ Hz |
| X-ray and gamma | below 10 nm | above $10^{16}$ Hz |
Worked example: Green light
A green laser pointer emits light of wavelength $525\ \text{nm}$. What is its frequency?
- Change nanometers to meters: $525\ \text{nm} = 525 \times 10^{-9}\ \text{m} = 5.25 \times 10^{-7}\ \text{m}$.
- Solve $c = \lambda\nu$ for $\nu$: $\nu = \dfrac{c}{\lambda} = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{5.25 \times 10^{-7}\ \text{m}}$.
- $\nu = 5.71 \times 10^{14}\ \text{s}^{-1} = 5.71 \times 10^{14}\ \text{Hz}$. The meters cancel, leaving per second.
Answer: $5.71 \times 10^{14}\ \text{Hz}$
Worked example: A radio station
An FM station broadcasts at $98.1\ \text{MHz}$. What is the wavelength of its waves?
- $98.1\ \text{MHz} = 98.1 \times 10^{6}\ \text{Hz} = 9.81 \times 10^{7}\ \text{s}^{-1}$.
- $$\lambda = \dfrac{c}{\nu} = \dfrac{3.00 \times 10^{8}\ \text{m/s}}{9.81 \times 10^{7}\ \text{s}^{-1}} = 3.06\ \text{m}$$
- About the height of a room: radio waves are long, and their frequency is low.
Answer: $3.06\ \text{m}$
Common mistake
Two slips are common. First, thinking a longer wavelength means a higher frequency; it is the reverse, since $\lambda\nu$ is fixed. Red light has the longer wavelength and the lower frequency; violet has the shorter wavelength and the higher frequency. Second, putting nanometers straight into $c = \lambda\nu$, which makes the frequency $10^{9}$ times too small. Check: visible light always has a frequency near $10^{14}$ to $10^{15}\ \text{Hz}$.