Photoelectric Effect

The plot of frequency vs stopping potential for a photoemissive material is a straight line. The equation governing the relationship is given by the photoelectric equation: where is the kinetic energy of the emitted photoelectron, is Planck's constant, is the frequency of the incident radiation, and is the work function of the material. The stopping potential is related to the kinetic energy of the emitted electron: where is the charge of the electron. By equating the two expressions for , we get: This equation represents a straight line of the form . Thus, the plot of vs is a straight line with slope and intercept , which represents the work function of the material. The intercept on the -axis gives the value of the work function , and the slope is related to . 

Einstein’s Photoelectric Equation and Calculation of Cut-off Potential

According to Einstein’s photoelectric equation:

Where is the maximum kinetic energy of the photoelectrons.

The kinetic energy of the photoelectrons is related to the cut-off potential by:

Therefore, we can write:

Substituting the given values:

Substitute , , , and :

Thus, the cut-off potential for the photoelectrons is .

Photoelectric Effect – Reasoning Based Questions

(a) Why do all the photoelectrons not eject with the same kinetic energy when monochromatic light is incident on a metal surface?

Even though monochromatic light (single frequency) has photons of the same energy, photoelectrons are emitted from different depths of the metal surface. Those emitted from deeper layers lose some energy in overcoming internal resistance and collisions with atoms before reaching the surface.

Therefore, the kinetic energy of photoelectrons is given by:

Hence, photoelectrons do not have identical kinetic energies.

(b) Why is the saturation current different for different intensities of light?

Saturation current corresponds to the maximum photocurrent when all emitted electrons are collected. As the intensity of light increases, more photons are incident on the metal per unit time, resulting in the emission of more photoelectrons.

Since current is directly proportional to the number of electrons, saturation current increases with light intensity.

(c) Why does the emission of photoelectrons stop at a certain wavelength even when the intensity of light is kept constant?

The energy of a photon is given by:

As the wavelength increases, the energy of each photon decreases. When becomes large enough that (work function of the metal), photons no longer have enough energy to eject electrons.

Hence, photoemission stops beyond a certain cut-off wavelength , where:

In Robert A. Millikan's experiment on the photoelectric effect, the slope of the cut-off voltage versus frequency plot was found to be . We are tasked with calculating the value of Planck's constant ( ) from this information.

1. Understanding the Photoelectric Effect Equation:

The photoelectric effect is described by the equation:

where:

• is the elementary charge ( ),
• is the cut-off voltage,
• is Planck’s constant,
• is the frequency of the incident light, and
• is the work function of the material.

2. Relating the Slope to Planck’s Constant:

In the experiment, the plot of versus is a straight line. From the photoelectric effect equation, rearranged as:

This equation is in the form , where:

• ,
• ,
• (the slope), and
• (the y-intercept).

Thus, the slope of the plot , so we can calculate Planck's constant using the given slope.

3. Calculating Planck’s Constant:

The slope is given as , and we know the value of . Using the equation:

We can solve for :

4. Performing the Calculation:

5. Final Answer:

The value of Planck's constant is approximately .

The photoelectric effect experiment demonstrated several features that cannot be explained using the wave theory of light. Here are three key features:

1. Discrete Threshold Frequency:

In the photoelectric effect, it was observed that no electrons are emitted from the surface of a material unless the frequency of the incident light exceeds a certain threshold value, regardless of the light's intensity. According to wave theory, the energy of light is spread out over time, so increasing the intensity (i.e., the energy delivered over time) should eventually provide enough energy to release electrons, even at lower frequencies. However, the wave theory fails to explain why a minimum frequency (threshold frequency) is required to release electrons, and this observation can only be explained by the quantum theory, which proposes that light has discrete packets of energy (photons) with energy proportional to frequency .

2. Instantaneous Emission of Electrons:

In the photoelectric effect, electrons are emitted instantly when light of sufficient frequency strikes the surface, with no noticeable delay. According to the wave theory, if light were purely a wave, the energy would accumulate over time to free an electron. Therefore, there should be a delay before electron emission begins. However, experimental results show that electron emission occurs immediately when light of the correct frequency is incident, indicating that light interacts as discrete particles (photons), not as a continuous wave.

3. Effect of Intensity on Electron Kinetic Energy:

Wave theory suggests that the intensity of light should affect the energy of the emitted electrons, as intensity is related to the amplitude of the wave, and higher intensity should impart more energy to the electrons. However, in the photoelectric effect, increasing the intensity of the light (while keeping the frequency constant above the threshold) increases the number of emitted electrons, but does not affect their kinetic energy. This behavior is explained by the particle theory of light, where the energy of each photon is fixed and proportional to the frequency, not the intensity. Intensity simply affects the number of photons striking the surface, not the energy of each photon.

The photoelectric equation is given by:

where is the maximum kinetic energy of emitted electrons, is Planck's constant, is the frequency of the incident light, and is the work function of the metal.

In terms of stopping potential , the maximum kinetic energy can also be expressed as:

where is the charge of the electron.

Equating the two expressions for :

This can be rearranged to:

Comparing with the equation of a straight line , the slope is:

Thus, the charge of an electron is:

Hence, the correct answer is .