# Photoelectric Effect Calculator

Created by Davide Borchia
Last updated: Sep 08, 2022

Follow Einstein's steps in your discovery of quantum mechanics: the photoelectric effect calculator will guide you through the intricacies of the atomic world: you will learn the fundamental relationship between photons and quantum mechanics, and much more! Keep reading our short article to learn:

• What is the photoelectric effect;
• How to calculate the photoelectric effect: the equation with the threshold frequency and with the work function;
• How to use the threshold frequency in the equation for the electron's kinetic energy in a practical example

## What is photoelectric effect? Einstein's Nobel winning work

Einstein is known for his work on relativity and the infamous $E = mc²$ equation. However, in the physics environment, we like to remember him for his Nobel-worthy work on the photoelectric effect: the discovery of the relationship between light energy and the emission of electrons by a metal proved, once and for all, the value of the quantum theory.

But what is the photoelectric effect? The photoelectric effect is the emission of electrons by a material as a response to a specific wavelength of incident light. At the end of the XIX century, observing the behavior of metal when the light shined on them collided with the physics theories of the time. In particular, the energy of emitted electrons depended on the frequency of the incident light rather than on the intensity. The power only affected the number of emitted electrons. With his theory of quantization of light, Planck gave Einstein the basis to understand the physics behind the photoelectric effect.

In the new perspective, light is composed of photons with the energy associated with their frequency, thanks to Planck's relationship. We discussed it in our energy of a photon calculator.

Einstein polished the theory, finally sealing the link between photon and electron energy.

## The formula of photoelectric effect: how to calculate the work function

To calculate the photoelectric effect formula, we must define the main quantities involved. First, what do we calculate in the photoelectric effect equation? The kinetic energy of the emitted electrons. To do so, we compute the difference between two quantities:

• The energy of the incident photon; and
• The energy required to extract an electron from a given material**: we call this quantity work function.

Here is the formula for the photoelectric function:

$K_{\text{e}} = h\cdot\nu - W$

Where:

• $K_{\text{e}}$ — The maximum kinetic energy of the electron;
• $h$Planck's constant;
• $\nu$ — The frequency of the incident photon; and
• $W$ — The work function.

We can rewrite the equation for the photoelectric effect to highlight another fundamental quantity, the threshold frequency, the lowest frequency for which a photon can extract an electron. We use the threshold frequency to calculate the photoelectric effect's electron kinetic energy with the following formula:

$K_{\text{e}} = h\left(\nu-\nu_0\right)$

$\nu_0$ is the threshold frequency.

## Calculate the photoelectric effect's emission of an electron in a few selected metals

Now you know how to calculate the photoelectric effect: its formula is straightforward. It's time to apply it to a real-world situation!

There's no formula telling us how to calculate the work function: the value of this quantity or the related threshold frequency is measured by experimental means.

For titanium the work function in electronvolts is $W_{\text{Ti}} = 4.33\ \text{eV}$. We calculate the threshold frequency with the equation $E=h\cdot \nu$:

$\nu_0 = \frac{W}{h} = 1,047\ \text{THz}$

🙋 If you are wondering what are electronvolts, head to our volt to electronvolt calculator for a quick explanation!

This frequency falls in the ultraviolet region of the spectrum. Since ultraviolet is more energetic than visible, we can't measure photoelectric emission under visible light. However, we can try to shine a light in the extreme ultraviolet region, let's say, a light with frequency $\nu = 12,000\ \text{THz}$.

🙋 Use our wavelength calculator to convert the frequency of the incident light to the corresponding wavelength: this way, it's much easier to place it correctly on the spectrum! You can also try our energy to wavelength calculator to skip a step!

Let's try the photoelectric effect equation (the threshold frequency one):

\begin{align*} K_{\text{e}} &= h\!\cdot\!\left(\nu-\nu_0\right)\\ K_{\text{e}} &=\! h\!\cdot\!\left(12\!-\!1.047\right)\!\times\!10^{15}\ \text{Hz}\\ &=45.3\ \text{eV} \end{align*}

Simple enough!

Davide Borchia
Incident light
Frequency
THz
Work function
Threshold frequency
THz
Ejected electron
Maximum kinetic energy
eV
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