Planck-Einstein Relation (E = hf)

In one sentence: E = hf says each packet of light carries energy in proportion to its frequency: higher frequency means more energy per packet, and brightness does not change it.

$E = h f$

Read it out loud: “E equals h times f. E is the energy of one photon, h is Planck's constant, and f is the frequency of the light.”

What each symbol means

Symbol	Meaning
$E$	The energy of one photon, a single packet of light, in joules (J). Atomic physics often uses electron volts (eV) instead: $1\ \text{eV} = 1.602176634 \times 10^{-19}$ J.
$h$	Planck’s constant, $6.62607015 \times 10^{-34}$ J s (joule seconds). Its value is exact by definition in the SI system.
$f$	The frequency of the light, in hertz (Hz), which means waves per second.

What the formula is doing

In 1900 Max Planck found that the radiation from hot objects only fits the measurements if energy is exchanged in fixed steps. In 1905 Albert Einstein went further and proposed that light itself arrives in packets, now called photons. Each photon carries a fixed amount of energy, and the formula $E = hf$ says how much.

The frequency $f$ counts how many wave crests pass a point each second. Double the frequency and each photon carries twice the energy. Nothing else about the light changes that number: the energy of one photon depends only on its frequency.

Brightness is a different quantity. A brighter light has more photons arriving each second, so it carries more energy in total, but every photon is as energetic as before. Brightness answers the question how many, and frequency answers the question how much energy in each one.

This explains a practical difference. Ultraviolet photons carry enough energy each to break chemical bonds, including bonds in the DNA of skin cells, which is why UV light can cause sunburn. Red photons carry much less energy each, too little to do that, so even a very bright red lamp does not cause sunburn.

An everyday picture

Think of a vending machine that accepts only coins. Each coin type has a fixed value. A dollar coin is worth more than a dime, however many dimes you feed in. Frequency is the value of each coin, and brightness is how many coins arrive each second.

Worked example: Green light at 540 THz

Green light has a frequency of $f = 5.4 \times 10^{14}$ Hz, which is 540 THz. Use $h = 6.62607015 \times 10^{-34}$ J s.

Multiply by Planck’s constant: $E = hf = (6.62607015 \times 10^{-34}\ \text{J s})(5.4 \times 10^{14}\ \text{Hz}) \approx 3.578 \times 10^{-19}$ J.
Convert joules to electron volts by dividing by $1.602176634 \times 10^{-19}$ J per eV: $\frac{3.578 \times 10^{-19}}{1.602176634 \times 10^{-19}} \approx 2.233$ eV.
Find the wavelength with $\lambda = c / f = (2.998 \times 10^{8}\ \text{m/s}) / (5.4 \times 10^{14}\ \text{Hz}) \approx 5.55 \times 10^{-7}$ m, or about 555 nm.

Answer: Each photon of this green light carries about $3.58 \times 10^{-19}$ J, or 2.23 eV. A lamp that sends out a million times more photons per second carries a million times more energy each second, but each photon still has the same 2.23 eV.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Thinking brightness changes the energy of each photon. Brightness changes how many photons arrive per second. Frequency sets the energy of each one.
Plugging a wavelength into $E = hf$ . The formula needs the frequency in hertz. If you only have the wavelength, first find $f = c / \lambda$ .
Forgetting the units. $E = hf$ gives joules, not electron volts. Divide by $1.602176634 \times 10^{-19}$ to convert.
Imagining that the energy of a light beam is spread evenly over time and space, like water in a bucket. Each photon is a whole packet that is absorbed in one piece or not at all.

Where you will meet it

Photoelectric effect calculations, the work that earned Einstein the 1921 Nobel Prize in Physics.
LED and laser design, where the color of the light is set by the energy gap of the semiconductor.
Spectroscopy, where the energies of absorbed or emitted light reveal which atoms and molecules are present.
Radiation safety, where the energy of each photon decides whether the light can break chemical bonds.

Questions people ask

Does E = hf mean light is a particle?

Light shows both wave and particle behavior. Spreading out and interfering are wave behaviors. Being absorbed in packets is particle-like. The formula links the two: the frequency belongs to the wave, and the energy belongs to each packet.

What unit is h in?

$h$ is in joule seconds (J s). A frequency is in hertz, which is per second, so $hf$ comes out in joules because the seconds cancel.

What happens if the frequency is too low?

Then each photon carries too little energy for the job. In the photoelectric effect, for example, a photon that is too weak cannot free an electron, however many arrive. See the photoelectric effect page for details.

Photoelectric Effect: Light can knock electrons out of a metal, but only if each packet of light carries enough energy, and the leftover energy becomes the electron’s speed.
De Broglie Wavelength: Every moving object has a wavelength, and the heavier or faster it moves, the shorter that wavelength gets.
Hydrogen Energy Levels: A hydrogen electron can only sit on certain energy steps, and every jump between steps gives off or takes in light of one exact color.