## What is the Photoelectric Effect?

In an experiment conducted by Heinrich Hertz, an apparatus was made where the incident light was made to fall on the metallic plate, it was discovered that metals emit electrons. The surface electrons are bound to metals with a minimum amount of energy and some of the incident photos enter the surface, they undergo collision with the atoms of the metal, they get absorbed and emit energy to an election, making it photoelectron, where the collision between the photons and electrons ejects the electrons out of the metal and with a negatively charged electron, causes photocurrent and when this current passes it creates an electric field where there is a potential difference at the output due to the anode and cathode of the electrode of the apparatus. This study involves the theory of Quantum physics and electromagnetism involving electromagnetic radiation and electromagnetic wave theory.

## The Photoelectric Equation

The energy of the photon is E=hv which is called the Planck-Einstein equation where h is Planck's constant and ν is the frequency. The work function is denoted as φ and the photoelectric equation is ΔE=hv-φ.

Where ΔE is the difference of the energy or known as the maximum kinetic energy.

The ejected electron's kinetic energy has a relation to its frequency and this frequency has a relation to the wavelength.

$v=f\lambda $ where $f=\frac{v}{\lambda}$ .

Hence$E=\frac{hc}{\lambda}$. Hence the kinetic energy becomes $\Delta E=\left(\frac{hc}{\lambda}\right)-\phi $

## What is Stopping Potential?

In physics, the stopping potential is defined as during the photoelectric effect, voltage difference required to stop the electrons from moving between the plates and creating an electric current between them. This phenomenon is also known as stopping voltage and the electrons involved here are the photoelectron in the photoelectric effect when light falls on the metal and if the frequency of the light is sufficient, the photoelectron is ejected from the metal surface. The term work function is defined as the minimum amount of energy required to eject an electron from the metal surface in the apparatus used to study the photoelectric effect. The photoelectric effect was first observed by the great scientist Heinrich Hertz and later proposed by the great scientist Albert Einstein and stated that the beam of light is not propagating in space like a wave but the light falling on the metal emits a discrete packet of energy called a photon. The stopping voltage is the cutoff voltage that stops the removal of an electron from the metal surface when the light falling on it has its energy greater than the work function of the metal. The maximum kinetic energy of the ejected electron is given by reducing the work function from the energy of one photoelectron. Hence the energy of the electrons reduces. The product of the charge of an electron and the stopping potential exerts the maximum kinetic energy of the electron ejected from the metal plate. It is also known as the negative potential of the metal plate where the photoelectric current equals the null value.

## The Stopping Potential Equation from its Experimental Setup:

In the apparatus setup used to study the phenomenon of photoemission and its photoelectric effect. When the power supply is given to the coated metallic conductors, but due to this the collector plate of the conductors would become negative. If it is negative, then the voltage will also become negative. So at one level of the voltage, which is called the stopping potential or cutoff voltage is when the electrons stop reaching the collector and hence no current flows. In this condition, the collector voltage is independent of the intensity of the light. This power source creates an electric field between the emitter and collector. The electrons which are ejected tend to move through the field but since the collector is negative, it has the maximum potential energy and the electrons' kinetic energy drastically reduces. But to stabilize this experiment, the initial value of the ejected electron's kinetic energy should have a higher value than the value of the potential energy at the collector. So the K.E> P.E at the collector ends and when the collector's voltage becomes equivalent to the stopping potential, then even the ejected electron's kinetic energy equals the potential energy of the collector. So K.E=P.E

Calculating the stopping potential value, the potential energy is equal to the product of the charge of the electron and voltage value.

Since K.E=P.E, that means hv-φ=eV where h is Planck's constant, ν is the frequency, e is the charge of an electron and V is the stopping potential or cutoff voltage

Hence the stopping potential or the cutoff voltage is $V=\frac{\left(hv-\phi \right)}{e}$

For the metal inside the apparatus, the minimum cutoff frequency of the electrons before they are ejected is called the threshold frequency for the photoelectric effect.

Einstein postulated that each photon has specific energy depending on the frequency of the radiation of the incident light. When light or the incident radiation falls on the metallic surface, there is an energy transfer from the incident photon to the electron present in the metal. Now expressing the energy of the photon in terms of kinetic energy will be

$hv=\phi +\frac{1}{2}m{v}^{2}$

The stopping potential's equation can also be rearranged to

$\begin{array}{c}hv-\phi =\frac{1}{2}m{v}^{2}\\ eV=\frac{1}{2}m{v}^{2}\end{array}$

Hence we can rewrite the equation as $hv=\phi +eV$.

Solving for V and dividing both sides of the equation by e. Thus we get,

$V=\frac{hv}{e}-\frac{\phi}{e}$

Therefore, if there is an increase in the frequency of the radiation, then there will be an increase in the stopping potential.

## Practice Problem

1) When radiation of a certain wavelength shines on the cathode of the photoelectric cell, the photocurrent produced can be reduced to zero by applying a stopping potential of 2.63 V. If the work function of the photon emitter is 4 eV, find the wavelength of radiation.

Hint: Use Einstein's photoelectric effect equation

Answer: The condition used according to the photoelectric effect is

${V}_{s}=\frac{hc}{\lambda e}-\frac{\varphi}{e}$Substitute the values of Planck’s constant, stopping potential, work function, speed of light and charge of electron to calculate the value of the wavelength. Note: Since the work function is given in eV it can be used as it is.

$$\begin{array}{l}2.63=\frac{6.6\times {10}^{-34}\text{Js}\times \text{3}\times {10}^{8}\text{m/s}}{\lambda \times 1.6\times {10}^{-19}\text{J}}-4\\ 6.63=\frac{6.6\times {10}^{-34}\text{Js}\times \text{3}\times {10}^{8}\text{m/s}}{\lambda \times 1.6\times {10}^{-19}\text{J}}\\ \lambda =\frac{6.6\times {10}^{-34}\text{Js}\times \text{3}\times {10}^{8}\text{m/s}}{6.63\times 1.6\times {10}^{-19}\text{J}}\\ \lambda =1871\text{nm}\end{array}$$The wavelength of the radiation is 1871 nm.

2) Radiation of wavelength 3000 Ampere measured from the ammeter, falls on a photoelectric surface for which work function is 1.6eV. What is the stopping potential for the emitted electron?

Answer: The condition used according to the photoelectric effect is

${V}_{s}=\frac{hc}{\lambda e}-\frac{\varphi}{e}$Substitute the values of Planck’s constant, stopping potential, work function, speed of light and charge of electron to calculate the value of the stopping potential.

$$\begin{array}{l}{V}_{s}=\frac{6.6\times {10}^{-34}\text{Js}\times \text{3}\times {10}^{8}\text{m/s}}{3000\times {10}^{-10}\text{m}\times 1.6\times {10}^{-19}\text{J}}-1.6\text{\hspace{0.33em}}\text{V}\\ {V}_{s}=4.125\text{\hspace{0.33em}}\text{V-1}\text{.6V}\\ {V}_{s}=2.525\text{V}\end{array}$$The stopping potential is -2.525 volt ( the negative sign indicates it is applied in the opposite direction.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Science (Physics)
- Masters in Science (Physics)

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