# Photon Energy Calculator

Created by Davide Borchia
Last updated: Feb 18, 2023

In 1905 Einstein, starting from the groundbreaking work by Planck on the black-body radiation, confirmed the quantization of the energy of a photon: our photon energy calculator will teach you that fundamental relationship.

In our photon energy calculator you will learn:

• What is a photon, and why are they so important;
• The equation for the energy of a photon: how to calculate the energy from the frequency;
• How to calculate the energy from the wavelength; and
• How to calculate the number of photons in a light beam.

## What is a photon?

A photon is the building block of light. Both a particle (since they carry ) and a wave (we talked of this characteristic in our De Broglie wavelength calculator), a photon is the quanta of light: you can't find something smaller (less energetic) than a photon and still call it light.

It's hard to talk of photons without terribly understating their importance. They are everywhere, and since they are electromagnetic radiation, they make up everything, from radio waves to visible light, from microwaves to gamma rays.

A detailed description of photons takes much more time than we have. So, let's be short!

## The quantization of light

At the end of the XIX century, a catastrophe happened in the world of physics. Not a literal one, but the ultraviolet catastrophe, an unexplained behavior of the radiation emitted by a black body. With a lot of intelligence and a bit of desperation, Max Planck introduced an arbitrary quantization, a discretization of the behavior, that fitted both the experimental data and the first modelization given by Wien's law.

A few years later, Einstein, studying the emission of electrons by a material hit by light (the photoelectric effect), proved the ideas of Planck right, showing that light arrived on the material in the form of discrete energy packets, quanta. Einstein didn't stop here and derived the Planck-Einstein relation (or Planck's equation), which tells us how to calculate the energy of a photon in a few symbols.

## Calculate the energy from the frequency of a photon

Planck's equation first appeared in the calculations of the from the frequency. It simply states:

$E = h\cdot\nu$

Where:

• $E$ is the energy of a photon;
• $h$ is the Planck's constant; and
• $\nu$ is the photon's frequency.

🙋 The Planck's constant has value $h = 6.62607015\cdot 10^{-34}\ \text{J}/\text{{Hz}}$

Let's try the formula for the energy of a photon: let's take a photon with frequency $\nu = 729.422\ \text{THz}$. Apply the Planck-Einstein relation:

\begin{align*} E &= h\cdot \nu\\ &=6.626\!\cdot \!10^{-34}\ \!\frac{\text{J}}{\text{{Hz}}}\! \cdot \!7.29\!\cdot \!10^{14}\ \!\text{Hz}\\ &= 4.83\cdot10^{-19}\ \text{J} \end{align*}

This energy is extremely small, and luckily so: this photon is in the visible part of the spectrum, and it would be problematic if it was dangerous to us.

## How to calculate the energy of a photon: energy from wavelength calculations

Frequency and wavelength are intimately related by the velocity of the wave in the medium. When talking about photons, the velocity is $c$, a fundamental constant in physics. Knowing that $\nu = \tfrac{c}{\lambda}$, where $\lambda$ is the wavelength, we can derive the equation of the energy of a photon from the wavelength:

$E = \frac{h\cdot c}{\lambda}$

Now that you know how to calculate the energy from the wavelength and the frequency let's scale up from a single photon to multiple ones.

## Calculations for more than a photon

Since light is quantized, we can calculate the number of photons in a light beam of given energy and frequency by simply dividing the energy of the beam by the result of the formula for the energy of a photon:

$n_{\text{ph}}=\frac{E_{\text{beam}}}{h\cdot \nu}$
Davide Borchia
Wavelength
mm
Frequency
GHz
Energy
meV
Energy of the beam
meV
Number of photons
x10³⁰
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