Photoelectric Effect and Stopping Potential with Work Function and Derivation

What is the Photoelectric Effect?

In 1905, Albert Einstein gave his Nobel prize-winning paper on the discrete quantisation of energy in electromagnetic radiation. The core premise of this work was based on the data obtained from the photoelectric effect experiments done by Aleksandr Stoletov. To explain the photoelectric effect, they found a correlation between the intensity of light and the number of electrons emitted. According to the photoelectric effect, when an electromagnetic wave of a specific frequency (energy) is bombarded on a metal, the electrons inside the metal are ejected and become known as photoelectrons through a process known as photoemission.

Work Function

The work function draws a barrier between emission and absorption. If the incident electromagnetic wave does not have energy equal to the work function of the material, electrons will not be ejected from the atoms of the metal. Quantised wave packets can activate emission if they have energies greater than the work function of the metal. 

In solid-state physics, the work function is the minimum thermodynamic work needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Work function is a property of a material, which is defined as the minimum quantity of energy required to remove an electron to infinity from the surface of a given solid.

What is Stopping Potential?

Stopping potential, or cut-off potential, is defined as the required potential for stopping the removal of an electron from a metal surface when the incident light energy is greater than the work potential of the metal on which the incident light is focused. The work function and stopping potential tend to mean the same thing in terms of the photoelectric effect.

Derivation of Maximum Kinetic Energy:

The concept of the derivation of Maximum Kinetic Energy using Einstein’s explanation of the photoelectric effect in simpler terms:

Einstein figured out that light energy comes in small packets called photons. Each photon has a certain amount of energy, and when it hits a metal surface, an electron can absorb that energy. If the absorbed energy is enough to overcome the “stickiness” of the metal (called work function), the electron gains extra energy and gets kicked out from the metal.

Energy of the photon (hf) ≥ Work function (Φ) + Maximum Kinetic Energy (KE_max)

By rearranging the formula, we find the maximum kinetic energy:

K.E_max = hf – Φ

Two important things to note:

1. Threshold Frequency: If the frequency of light is below a certain value (ν₀ = Φ/h), the absorbed energy is not enough to free the electron, so nothing happens.
2. Frequency Dependence: If the light frequency is above the threshold, the more the frequency increases, the more extra energy the electron gets. This is different from what classical physics would predict, where brighter light should just give more energy. In the photoelectric effect, it’s the frequency that matters, not the brightness.

Derivation of Photoelectric Effect and Stopping Potential Formula

To derive Einstein’s photoelectric equation formula for stopping potential, we need to identify the required variables and parameters first. As we know   

c=νλ

where c is the speed of light, 𝛎 is the frequency of the electromagnetic wave, and λ is the measure of the wavelength of the electromagnetic wave.

ν=c/λ

⇒E=hv

⇒E=hν……..(1)

Here, E is the energy of the electromagnetic wave.

Therefore, we can use the known parameters to calculate the kinetic energy of the photoelectrons.

KEmax=hc/λ−ϕKEmax=hc/λ−ϕ……..(2)

The stopping potential ϕ can be given as

ϕ=hc/λ−KE_max

Now the kinetic energy can be written as

KE_max=1/2mv²_max

Also, the energy of a photo equal to the work function can be written as

ϕ=hc/λ₀

ϕ=hν₀

So, the equation for the stopping potential can be written as

hν₀=hc/λ−1/2mv²_max

hν₀=hν−1/2mv²_max

We can also write

1/2mv²_max=hν−hv₀

1/2mv²_max=h(ν−ν₀)

The photoelectric effect can be derived from the concept of Planck’s quantum theory. Photons travel in discrete energy packets, where the energy of one photon is given by the equation (1). νν is the frequency of the incident photon. ‘h’ is the Planck’s constant whose value is h=6.6260715×10−34 J.sh=6.6260715×10−34 J.s the units are Joule-Second.

To summarise the variables used in the derivation of the photoelectric equation, we have

• h, which is the Planck constant.
• ν, which is the frequency of the radiation in question.
• λ, which is the wavelength of the radiation.
• c, which is the speed of light in a vacuum.
• E, which is the energy of the photon.

There is a single interaction that sums up the photoelectric effect, the incidence of a photon on the metal surface, and its interaction with the electron bound to the surface of the metal within the interatomic interactions. 

Dependence of Photoelectric Effect on Different Parameters

1. For a light of any given frequency, the photoelectric current is directly proportional to the intensity of light.
2. If the frequency of light is no greater than the threshold frequency, the emission of photoelectrons will be stopped, no matter how intense the light rays are.
3. The maximum kinetic energy of the photoelectrons emitted is independent of the intensity of light. As the frequency of light increases over the threshold frequency, the kinetic energy of the electrons also increases.
4. The process of photoemission is instantaneous, as it depends on the quanta of energy the radiation brings with it.

Applications of the Photoelectric Effect:

• Solar Panels: They use special metals to turn light into electricity from various wavelengths.
• Motion and Position Sensors: These sensors use a material that reacts to light (UV or IR) along with LEDs. When something blocks the light, a change in electricity is noted.
• Lighting Sensors (e.g., in smartphones): Adjust screen brightness automatically based on the light intensity, thanks to the photoelectric effect generating current.
• Digital Cameras: Detect and capture light by using sensors that respond to different colors.
• X-Ray Photoelectron Spectroscopy (XPS): Uses X-rays to learn about surfaces, like elemental and chemical composition.
• Burglar Alarms: Photoelectric cells help in detecting intruders.
• Photomultipliers: Used to find very low levels of light.
• Video Camera Tubes (early TV): Photoelectric effect was key in early television cameras.
• Night Vision Devices: They work because of the photoelectric effect.
• Chemical Analysis: Photoelectric effect aids in studying nuclear processes and analyzing materials by examining the energy of emitted electrons.

Conclusion

The phenomenon of the photoelectric effect was discovered by the works of Heinrich Hertz, Albert Einstein, Max Planck, and others during the late 19th to early 20th century. The photoelectric effect can be defined as a phenomenon during which electrons are emitted from the surface of the metal when the light of a suitable wavelength is incident on it. These electrons are termed photoelectrons. 

This emission occurs due to the electrons absorbing energy from the electromagnetic radiation and using this agitation to break free from the intra-atomic forces of attraction in the metal. This spontaneous emission of electrons is called photoemission. The current generated due to this emission is referred to as photocurrent. This phenomenon converts light energy into electrical energy.

Factors Affecting Stopping Potential

Several factors influence the value of stopping potential. Understanding these factors is crucial for studying various phenomena and practical applications. Let’s take a closer look at them:

1. Frequency of Incident Light

The frequency of the incident light has a direct impact on the energy carried by photons. Higher frequencies result in more energetic photons, which, in turn, possess greater kinetic energy. Consequently, higher-frequency light requires a higher stopping potential to halt the emitted electrons.

2. Work Function of the Metal

The work function (ϕ) of the metal refers to the minimum energy required to remove an electron from its surface. Metals with higher work functions demand greater stopping potentials to overcome the stronger binding forces holding the electrons within the material.

3. Charge of the Particle

The charge of the particle experiencing the electric field also influences the stopping potential. Heavier particles, such as ions, require different calculations than electrons due to their differing masses and charges.

4. Intensity of Incident Light

The intensity of the incident light affects the number of photons striking the metal surface. However, it does not directly impact the stopping potential, as it only alters the rate of electron emission, not the kinetic energy of individual electrons.

5. Nature of the Electric Field

The type of electric field used to stop the charged particles is a critical factor. Uniform electric fields are often utilized, but non-uniform fields can also play a role in certain experiments.

6. Temperature

At higher temperatures, the kinetic energy of the electrons within the metal also increases. As a result, the stopping potential must be adjusted accordingly.