Photons are little packets of energy, which exhibit interesting wave-like and particle-like behavior. Photons are both electromagnetic waves, such as visible light, or x-rays, but also are quantized in energy like particles. The energy of a photon is therefore a multiple of a fundamental constant, called Planck's constant, *h* = 6.62607015 × 10^{-34} J s_._

## Calculate the Energy of a Photon

We can calculate the energy of a photon in two ways. If you already know the frequency, *f*, of the photon in Hz, then use **E****=** * hf*. This equation was first suggested by Max Planck, who theorized that photon energy is quantized. Therefore, sometimes this energy equation is referred to as Planck's equation.

Another form of Planck's equation uses the simple relationship that ** c = λ f**, where

*λ*is the wavelength of the photon, and

*c*is the speed of light, which is a constant and is 2.998 × 10

^{8}m/s. If you know the frequency of the photon, you can easily calculate the wavelength by the following formula:

**.**

*λ*=*c*/*f*Now we can calculate the energy of a photon by either version of Planck's equation: ** E = hf** or

*E*=

*. Often we use the units of eV, or electron volts, as the units for photon energy, instead of joules. You can use*

**hc / λ***h*= 4.1357 × 10

^{-15}eV s, which results in a more reasonable energy scale for photons.

## Which Photons are More Energetic?

The formula makes it very easy to see how the energy depends on the frequency and wavelength of a photon. Let's look at each of the formulas shown above, and see what they imply about the physics of photons.

First, because the wavelength and the frequency always multiply to equal a constant, if photon A has a frequency that is two times that of photon B, the wavelength of photon A must be 1/2 of the wavelength of photon B.

Second, you can learn a lot about how the frequency of a photon can provide a relative idea of it's energy. For example, since photon A has a higher frequency than photon B, we know it is twice as energetic. In general, we can see that energy scales directly with frequency. Similarly, because the energy of a photon is inversely related to its wavelength, if photon A has a shorter wavelength than photon B, it is again, more energetic.

## Simple Photon Energy Calculator

It may be useful to quickly estimate photon energy. Because the relationship between photon wavelength and frequency is so simple, and the speed of light is roughly 3 × 10^{8} m/s, then if you know the order of magnitude of either the frequency or wavelength of the photon, you can easily calculate the other quantity.

The wavelength of visible light is approximately 10 ^{−8} meters, so *f* = 3 × (10^{8} / 10 ^{−7}) = 3 × 10^{15} Hz. You can even forget the 3 if you are just trying to get a quick order of magnitude estimate. Next, *E* = *hf*, so if *h* is about 4 × 10 ^{−15} eV, then a quick estimate for the energy of a visible light photon is *E* = 4 × 10 ^{−15}× 3 × 10^{15}, or around 12 eV.

That's a good number to remember in case you want to quickly figure out if a photon is above or below the visible range, but this whole procedure is a good way to make a quick estimate of photon energy. The quick and easy procedure could even be considered a simple photon energy calculator!

#### References

- Young, H. D., Freedman, R. A., (2012) University Physics. San Francisco, CA: Addison-Wesley.
- National Institute of Standards and Technology: Planck's Constant

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