Problem 22 - Master Resource Book in JEE Main Physics

Question

In a photoelectric experiment, the wavelength of the incident light is decreased from \(6000 \AA\) to \(4000 \AA\), while the intensity of radiation remains the same (a) the cut-off potential will increase (b) the cut-off potential will decrease (c) the KE of the emitted photoelectron will increase (d) the photoelectric current will increase

Step-by-Step Solution

Answer

(a) The cut-off potential will increase, and (c) the KE of the emitted photoelectron will increase.

Step 1: Understanding the Photoelectric Effect

The photoelectric effect is a phenomenon where electrons are emitted from a material when it is exposed to light. The energy of these electrons is determined by the frequency (or equivalently, the wavelength) of the incident light according to the equation, \( E = hf \), where \( h \) is Planck’s constant and \( f \) is the frequency of light. The kinetic energy (KE) of the emitted photoelectrons is given by \( KE = hf - \phi \), where \( \phi \) is the work function of the material.

Step 2: Analyzing the Impact of Wavelength Decrease

When the wavelength of incident light decreases from \(6000 \ \text{Å} \) to \(4000 \ \text{Å} \), the frequency of the light increases because wavelength and frequency are inversely proportional (\( f = \frac{c}{\lambda} \), where \( c \) is the speed of light and \( \lambda \) is the wavelength). An increase in frequency leads to an increase in the energy of the photons \( (E = hf) \). Hence, the kinetic energy of the emitted electrons also increases because \( KE = hf - \phi \).

Step 3: Determining Changes in Cut-off Potential

The cut-off potential is the minimum potential that stops the most energetic emitted photoelectrons. As the kinetic energy of the emitted photoelectrons increases, more energy is required to stop them, indicating an increase in the cut-off potential. Hence, when the frequency increases due to a decrease in wavelength, the cut-off potential increases.

Step 4: Evaluating the Impact on Photoelectric Current

The photoelectric current is proportional to the number of photoelectrons emitted. Since the intensity of the incident light remains unchanged, the number of photons (and hence the number of emitted photoelectrons) in the light remains the same. Thus, the photoelectric current does not change when the wavelength decreases, provided the intensity is constant.

Key Concepts

Wavelength and Frequency

In the context of the photoelectric effect, the behavior of light is often characterized by its wavelength and frequency. Both of these properties are intrinsically linked by the formula \( f = \frac{c}{\lambda} \),where \( f \) is the frequency, \( c \) is the speed of light, and \( \lambda \) is the wavelength. This mathematical relationship shows that as the wavelength of the incident light decreases, the frequency increases, and vice versa.
The energy of a photon is directly proportional to its frequency and can be calculated using Planck’s equation:\( E = hf \).Here, \( h \) represents Planck’s constant, a fundamental constant in quantum mechanics.

• Shorter wavelengths mean higher frequencies, which in turn, means more energetic photons.
• This increase in photon energy will affect subsequent interactions with materials, such as the kinetic energy of emitted photoelectrons.

Kinetic Energy of Photoelectrons

The kinetic energy of photoelectrons released in the photoelectric effect is crucial for understanding how energy is exchanged at the atomic level. When light of a certain frequency strikes a material, electrons are emitted as a result. The kinetic energy (KE) of these photoelectrons can be described by the equation:\( KE = hf - \phi \).In this equation, \( hf \) represents the energy of the incoming photons, while \( \phi \) is the work function, or the minimum energy needed to eject an electron from the material.

• Higher frequency light increases \( hf \), meaning more energy is available to the photoelectrons after overcoming the work function \( \phi \).
• Thus, when wavelength decreases (and frequency increases), the kinetic energy of the photoelectrons rises.

This increased kinetic energy plays a critical role in determining other aspects of the photoelectric effect, such as the cut-off potential.

Cut-off Potential

In a photoelectric experiment, the cut-off potential is defined as the minimum voltage required to stop the most energetic photoelectrons from reaching the anode. This is tied to the energy exchange in the photoelectric effect and is crucial for experimental measurements.
When the kinetic energy of photoelectrons increases due to higher photon energy (from reduced wavelength), the cut-off potential also increases. This increase is a direct consequence of needing more energy to halt the electrons:

• An increase in frequency (caused by a decrease in wavelength) demands a higher cut-off potential.
• Therefore, this potential is a measure of the energy carried by the most energetic photoelectrons.

Understanding cut-off potential helps in applications like determining Planck’s constant or examining unknown materials' work functions.

Photoelectric Current

The photoelectric current is a measure of the flow of photoelectrons emitted when light strikes a material. Conceptually, it is proportional to the number of electrons ejected, which is dependent on the intensity of the incident light.
In scenarios where the intensity remains constant, alterations in wavelength (or frequency) do not influence the photoelectric current, provided sufficient energy is present to overcome the work function:

• Intensity, in this context, refers to the number of incoming photons per unit area per unit time.
• If the intensity doesn't change, the count of emitted photoelectrons remains the same.

Therefore, when wavelength decreases but intensity is constant, the photoelectric current remains unaffected.