Calculating the Energy of a Particle of Electromagnetic Radiation Using Wavelength

Radiation from the electromagnetic spectrum in order of decreasing wavelength begins at the low end with radio waves and microwaves. Next is light radiation from the infrared region, the visible and the ultraviolet. Highest in frequency are the x-rays and cosmic radiation. The wavelength range is from approximately 100,000 centimeters to one ten-billionth of a centimeter!

The Nature of Light

Light is not fully understood by scientists. So far it has best been described as possessing two natures. Sometimes it acts as a wave, and sometimes it acts as a particle. Thus light has a wavelength. When it acts as a particle, that particle is called a photon.

Wavelength of Light or Radiation

Light or radiation travels in waves, like those made in the water of a still pond when a tiny stone is dropped in it. Waves have high "crests" and low "troughs." The distance between either two crests or two troughs is equal to the wavelength. What is the formula to calculate energy using wavelength?

E = hc / w

The above equation reads, energy (E) equals Planck's constant (h) times the speed of light (c) divided by the wavelength (w). This energy is in terms of a single unit of light called a photon.

Equation Simplification

Since h is a constant and c is a constant, they can be combined to make a single constant, if the units chosen are consistent throughout. If wavelength is measured in centimeters, then Planck's constant is 6.6 times ten to the minus twenty-seventh power grams-centimeters squared per second and the speed of light is 3.0 times ten to the tenth power centimeters per second. Multiplying these two gives approximately 2 times 10 to the minus sixteenth power grams-centimeters cubed per second squared. The overall equation is thus,

E = 2.0 x 10^-16 / w

The energy has the units grams-centimeters squared per second squared. That was officially simplified to "ergs."

Examples

Calculation One:

What is the energy of one photon of yellow light? Yellow light has a wavelength of approximately 570 nanometers, or after conversion, 0.57 x 10^-4 centimeters. The equation becomes,

E = 2.0 x 10^-16 / 0.57 x 10^-4 = 3.5 x 10^-12 ergs

Calculation Two:

What is the energy of an x-ray photon? With a wavelength of 1.0 x 10^-10 meters and converting to centimeters, we get,

E = 2.0 x 10^-16 / 1.0 x 10^-8 = 2 x 10^-8 ergs

Thus an x-ray photon has an energy nearly ten thousand times the energy of a photon of yellow light. This should come as no surprise.

References and Resources

University of Colorado at Boulder: Planck's Constant and the Energy of a Photon