Einstein’s Equation of Photoelectric Effect: Explanation and Formula

Einstein’s Equation of Photoelectric Effect: We know that solar energy is one of the best sources of renewable energy. But how do solar panels convert light energy to electric energy? These days there is a lot of craze for digital cameras. How do digital cameras detect lights coming from objects? The answer to both of the questions is the photoelectric effect. This explains the particle nature of light. This phenomenon played a major role in the development of modern physics because of the puzzling questions it raised about the particle nature of light versus wave nature of the light behaviour, which was finally resolved by Albert Einstein. This phenomenon explains that light has a particle nature and is made up of energy packets called photons. In this article, we will learn more about the Photoelectric effect and its applications.

Photoelectric Effect

The photoelectric effect is the phenomenon of emission of electrons when electromagnetic radiation, such as light, hits a metal surface. The photoelectric effect was first discovered by Heinrich Rudolf Hertz, and the experimental verification was done by Wilhelm Hallwachs and Philipp Lenard. Later on, Albert Einstein gave the mathematical description of how the photoelectric effect was caused by the absorption of quanta of light (photon). They observed that when the wavelength of the light is less than a certain value (threshold value) then, the radiation is absorbed, and the electrons are emitted. For the ejection of an electron from metal, we need the minimum amount of energy above a certain value; that frequency is known as the threshold frequency. For a single electron, the minimum amount of energy required to come out of the metal surface is known as the work function of the metal.

Learn about Photoelectric Effect here

Experimental Study of Photoelectric Effect

To study the photoelectric effect, an experimental setup was made, as shown in the figure below. Two metal plates – emitter and detector are sealed in the vacuum chamber. Light of short wavelength passes through a transparent window in the wall of the chamber and falls on the emitter plate. The emitter plate (cathode) is connected to the negative terminal of high voltage DC. The detector plate (anode) is connected to the positive terminal of the DC supply. We can change the potential difference between the cathode and the anode can be changed with the help of rheostat, batteries and the commutator. By changing the terminal of the DC supply, the anode potential can be made positive or negative with respect to the cathode. When the light fall on the emitter, it emits electrons. These electrons move towards the collector and are observed by it, then it flows through the ammeter and batteries and is back to the emitter (cathode), and hence electric current is established in the circuit. Such a current is called photocurrent.

Einstein’s Theoretical Explanation of Photoelectric Effect

The photoelectric effect is the process in which the ejection of electrons from the surface of metal takes place when the light of a required or lower wavelength falls on them. The photoelectric effect is an important concept that helps us to clearly understand the quantum nature of light and electrons.
Einstein postulated that light consists of a large number of identical particles called photons. The energy carried by each photon of light is dependent on its frequency. When a photon hits a metal surface, all the energy carried by the photon is transferred to the electron. This energy is consumed by the electron to remove itself from the metal surface, and the rest of the energy remained after emission gets transformed as kinetic energy. The minimum energy to be required to just come out of the surface of the metal by an electron is called the Work function. The corresponding frequency is called threshold frequency.
According to Einstein, the energy of each photon is given by,
\({E_i} = hv\)
Where,
\({E_i} =\) Energy of incident photon in joule.
\(h =\) planks constant \(\left( {6.626\; \times {\text{ }}{{10}^{ – 34}}\,{\text{J}}\,{\text{s}}} \right)\).
\(v =\) frequency of incident photon in \(\rm{Hz}\).
Thus the Kinetic energy of photon coming out during the photoelectric effect can be written as,
\(KE =\) Energy of incident photon \(-\) Work function
\( = {E_i} – h{v_0}\)
\( = hv – h{v_0}\)
Where \(v_0\) is the threshold frequency, \(KE\) is the maximum kinetic energy of the ejected electron.

Factors Affecting Photocurrent

The current produced due to the photoelectric effect depends on the following factors.
1. The intensity of incident radiation.
2. A potential difference between metal plate and collector.
3. Frequency of incident radiation.

Effects of Intensity of Incident Radiation on Photocurrent

For a constant value of frequency and the potential difference between the metal plate and collector, the photoelectric current is observed to increase with the intensity of incident radiation. The photoelectric current gives an idea about the rate of production of photoelectrons. As the intensity increase, the number of photons hitting the electron increase, and more photoelectrons will be produced.

Effects of Potential Difference Between Metal Plate and Collector

Keeping the intensity and frequency of light constant, the photoelectric current increases with an increase in the forward potential difference between the metal plate and collector up to a characteristic value. When the potential difference is increased beyond the characteristic value, there will not be much effect on photocurrent. The photocurrent (maximum) at this forward potential difference is called saturation current.
But when the polarity of the plates are changed, the photocurrent starts decreasing with an increase in potential difference (Reversed polarity). The reversed potential difference at which the photocurrent will become zero is called stopping potential.

Effect of Frequency on Photocurrent

Keeping the intensity and potential difference between plates constant, initially, the photocurrent increase with the increase in the frequency of incident light. With the increase in frequency, more electrons will have enough energy (kinetic energy) to reach the collector. But with further increase frequency, it will reach saturation current, as most of the electrons will reach the collector, and no new photoelectrons are produced.

Applications of Photoelectric Effect

1. Used to generate electricity in Solar Panels. For accepting a wide range of wavelengths, these are made with combinations of materials.
2. Motion and Position Sensors: In this case, a photoelectric material is placed in front of a UV or IR LED. The electronic circuit registers a change in potential difference when an object comes between the photosensor and Light-emitting diode (LED).
3. Lighting sensors that are used in smartphones enable automatic adjustment of screen brightness according to the lighting. This happens because the amount of current generated via the photoelectric effect is dependent on the intensity of light falling on the sensor.
4. Photoelectric sensors are used in Digital cameras.
5. \(X-\)Ray Photoelectron Spectroscopy (XPS): XPS is used to study the important aspects of the chemistry of a surface, such as elemental composition, chemical composition, the empirical formula of compounds and chemical state. XPS uses \(x-\)rays to irradiate a surface and measure the kinetic energies of the emitted electrons. We can know the elemental composition, chemical composition, the empirical formula of compounds and chemical state.

Summary

Some materials emit electrons when light with a suitable wavelength falls on them. This phenomenon is called the photoelectric effect. The work function \((\varphi )\) of a material is the minimum energy needed per photon to extract an electron from its surface. Electrons ejected during the photoelectric effect are called photoelectrons. The photoelectric effect will occur when the frequency of the incident light is more than the threshold frequency \((v_0)\). Different materials have different threshold frequencies. The rate of production of photoelectrons is proportional to the intensity of the incident light at a constant frequency. [1] The kinetic energy of the photoelectrons increases with the frequency of the incident light, and it does not depend on the intensity. Einstein gave the maximum kinetic energy of the ejected electron is \({K_{max}} = E – \varphi = h\left( {v – {v_0}} \right)\).

Solved Example on the Einstein’s Equation of Photoelectric Effect

Q.1. A monochromatic source of light operating at \(200\;\rm{W}\) emits \(4 \times 10^{20}\) photons per second. The photons fall on a metal having a threshold wavelength of \(800\;\rm{nm}\). Then find the kinetic energy of photoelectrons.
Ans: The energy of each photon \(\frac{{200\,\;{\text{J}}\,{{\text{s}}^{ – 1}}}}{{4 \times {{10}^{20}}\,{{\text{s}}^{ – 1}}}} = 5 \times {10^{ – 19}}\,{\text{J}}\)
The energy required by an electron to just come out of the metal will be,
\(\phi = h{v_0} = \frac{{hc}}{{{\lambda _0}}}\)
\( = \frac{{\left( {6.63 \times {{10}^{ – 34}}\,{\text{J}}\,{\text{s}}} \right) \times \left( {3 \times {{10}^8}\,{\text{m}}\,{{\text{s}}^{ – 1}}} \right)}}{{800 \times {{10}^{ – 9}}\,{\text{m}}}}\)
\( = 2.5 \times {10^{ – 19}}\,{\text{J}}\)
Then the kinetic energy of the electron will be,
\(KE = hv – h{v_0}\)
\(5 \times {10^{ – 19}}\,{\text{J}} – 2.5 \times {10^{ – 19}}\,{\text{J}}\)
\(2.5 \times {10^{ – 19}}\,{\text{J}}\).

FAQs on the Einstein’s Equation of Photoelectric Effect

Q.1. What is the photoelectric effect?
Ans: Some materials emit electrons when light with a suitable wavelength falls on them. This phenomenon is called the photoelectric effect.

Q.2. What is the formula of the photoelectric effect?
Ans: \({E_k} = hv – \phi \), where \(E_k\) is the maximum kinetic energy of the ejected electron, \(k\) is the plank constant, \(\phi \) is the work function, \(v\) is the frequency.

Q.3. Who gave the photoelectric effect?
Ans: The photoelectric effect was first introduced by Heinrich Rudolf Hertz, and the experimental verification was done by Wilhelm Hallwachs and Philipp Lenard. The mathematical theory to analyse the photoelectric effect was given by Elbert Einstein.

Q.4. How to Explain the Work Function of Photoelectric Effect?
Ans: Work function is defined as the minimum energy required to eject the electrons from the metal surface.

Q.5. What is the stopping potential?
Ans: The smallest magnitude of the negative anode potential with respect to the cathode to just stop the photocurrent, is called the stopping potential.

Practice question on Einstein’s Equation of Photoelectric Effect here

We hope you find this article on ‘Einstein’s Equation of Photoelectric Effect‘ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.