Question

Among the following the correct statement(s) is/are (1) Increase in the frequency of the incident radiation increases the kinetic energy of photoelectrons. (2) Threshold wavelength depends upon work function. (3) The study of photoclectric effect is useful in understanding quantisation of energy. (4) To cross the threshold energy intensity of the light must be increased. (1) I, II and III (2) II, III and IV (3) I, III and IV (4) I, II, III and IV

Short answer

Option (1) I, II and III

Step-by-step solution

Understand the statements

Go through each of the given statements and understand the concepts they refer to: (1) Kinetic energy of photoelectrons and frequency, (2) Threshold wavelength and work function, (3) Photoclectric effect and energy quantisation, (4) Threshold energy and light intensity.

Evaluate Statement 1

According to the photoelectric effect, the kinetic energy of photoelectrons increases with an increase in the frequency of the incident radiation: Thus, statement (1) is correct.

Evaluate Statement 2

Threshold wavelength \(\theta_t \) is related to the work function \(\theta_0 \) by the equation \(\theta_t =\frac{hu}{\theta_0}\). So, statement (2) is correct.

Evaluate Statement 3

The photoelectric effect demonstrates that energy is quantized, being transferred in discrete quanta (photons). Therefore, statement (3) is correct.

Evaluate Statement 4

Increasing the intensity of light affects the number of photoelectrons but does not impact the threshold energy which depends on frequency, not intensity. Thus, statement (4) is incorrect.

Conclusion

Based on the analysis, the correct statements are (1), (2), and (3), matching with option (1).

Key concepts

Kinetic Energy of Photoelectrons

When light hits a material, it can eject electrons from that material in a process called the photoelectric effect. These ejected electrons are known as photoelectrons. Their kinetic energy, which is the energy due to their motion, is influenced by the frequency of the incident light.
This relationship is described by the equation:
\( KE = hf - \theta \)
Here:

• \(KE\) is the kinetic energy of the photoelectron
• \(h\) is Planck's constant
• \(f\) is the frequency of the incident light
• \(\theta\) is the work function of the material

As the frequency of the light increases, the kinetic energy of the ejected photoelectrons also increases, hence statement (1) in the exercise is correct.

Threshold Wavelength

The threshold wavelength is the maximum wavelength of light that can eject electrons from a material. When light of this wavelength or shorter hits the material, photoelectrons are emitted. The threshold wavelength is related to the work function of the material and the speed of light.
The mathematical relationship is given by:
\( \lambda_t = \frac{hc}{\theta} \)
Here:

• \(\lambda_t\) is the threshold wavelength
• \(h\) is Planck's constant
• \(c\) is the speed of light
• \(\theta\) is the work function of the material

This shows that the threshold wavelength depends on the work function. The work function is the minimum energy required to eject an electron from the material. So, statement (2) in the exercise is correct.

Work Function

The work function represents the minimum energy needed to eject an electron from a material's surface. It is a property intrinsic to each material, depending on how tightly electrons are bound within the atomic structure.
Mathematically, it can be represented by:
\( \theta = hf_0 \)
Here:

• \( \theta \) is the work function
• \(h\) is Planck's constant
• \(f_0\) is the threshold frequency

The work function must be overcome by the energy of the incident photons to release electrons. If the photon's energy \( hf \) is less than the work function, no electrons will be emitted, regardless of light intensity. This explains why the intensity does not affect the threshold energy.

Quantisation of Energy

The photoelectric effect provided significant evidence for the quantisation of energy, a core concept in quantum mechanics. This means that energy is transferred in discrete units called quanta (singular: quantum). In the context of the photoelectric effect, light energy is transferred in packets called photons. Each photon carries an energy of:
\( E = hf \)
Here:

• \(E\) is the energy of the photon
• \(h\) is Planck's constant
• \(f\) is the frequency of the photon

This became a pivotal understanding in physics, exemplified in Einstein's explanation of the photoelectric effect, for which he received the Nobel Prize. So, statement (3) correctly points out that the study of the photoelectric effect aids in understanding energy quantisation.