How Frequency Determines Wavelength and Photon Energy

Electromagnetic radiation can be described by frequency, wavelength, and the energy carried by each photon. Two equations connect those quantities.

Frequency and wavelength

In vacuum,

$$c=f\lambda,$$

where $c$ is the speed of light, $f$ is frequency, and $\lambda$ is wavelength. Because $c$ is fixed, frequency and wavelength are inversely proportional:

$$\lambda=\frac{c}{f}.$$

Increasing frequency therefore means decreasing wavelength. This comparison does not require memorizing every electromagnetic-spectrum boundary.

Frequency and photon energy

The energy of one photon is

$$E=hf=\frac{hc}{\lambda},$$

where $h$ is Planck's constant. A higher-frequency, shorter-wavelength photon carries more energy than a lower-frequency, longer-wavelength photon.

A new example

Compare blue light at $450\,\text{nm}$ with red light at $650\,\text{nm}$. The blue light has the shorter wavelength, so it has the higher frequency and the greater energy per photon.

Using $E=hc/\lambda$,

$$E_{450}\approx4.42\times10^{-19}\,\text{J},$$ $$E_{650}\approx3.06\times10^{-19}\,\text{J}.$$

The ratio can be checked without constants:

$$\frac{E_{450}}{E_{650}}=\frac{650}{450}\approx1.44.$$

Each 450 nm photon carries about 1.44 times the energy of each 650 nm photon.

Per-photon energy is not beam energy

A higher photon energy does not automatically mean that an entire beam carries more energy. Beam energy also depends on the number of photons. Keep energy per photon separate from intensity, power, and total exposure.