Problem 14 - Master Resource Book in JEE Main Physics

Question

Given that a photon of light of wavelength \(10,000 \AA\) has an energy equal to \(1.23 \mathrm{eV}\). When light of wavelength \(5000 \AA\) and intensity \(I_{0}\) falls on a photoelectric cell, the surface current is \(0.40 \times 10^{-6} \mathrm{~A}\) and the stopping potential is \(1.36 \mathrm{~V}\), then the work function is (a) \(0.43 \mathrm{eV}\) (b) \(0.55 \mathrm{eV}\) (c) \(1.10 \mathrm{eV}\) (d) \(1.53 \mathrm{eV}\)

Step-by-Step Solution

Answer

The work function is 1.10 eV, option (c).

Step 1: Understand Energy-Wavelength Relationship

The energy of photons is related to their wavelength through a constant. For a photon of wavelength \(10,000 \text{ Å}\), the energy is given as \(1.23 \text{ eV}\). This allows us to calculate the energy for other wavelengths.

Step 2: Calculate Energy of 5000 Å Photon

Apply the energy-wavelength relationship to find the energy of a photon with wavelength \(5000 \text{ Å}\). Since energy and wavelength are inversely proportional, we use the ratio: \[ E_{1} \times \lambda_{1} = E_{2} \times \lambda_{2} \] Given \(E_{1} = 1.23\ \text{eV}\) and \(\lambda_{1} = 10,000 \text{ Å}\), we solve for \(E_{2}\): \[ E_{2} = \frac{E_{1} \times \lambda_{1}}{\lambda_{2}} = \frac{1.23 \times 10,000}{5000} = 2.46 \text{ eV} \]

Step 3: Use Photoelectric Equation to Find Work Function

The photoelectric equation relates the maximum kinetic energy of photoelectrons to the photon energy and the work function \( \phi \). The equation is: \[ E_{\text{photon}} = \phi + E_{\text{kinetic}} \] where \( E_{\text{kinetic}} = eV_{0} \) (\(e\): charge of electron, \(V_{0}\): stopping potential).Given \(E_{\text{photon}} = 2.46 \text{ eV}\) and stopping potential \(V_{0} = 1.36 \text{ V}\), then: \[ 2.46 = \phi + 1.36 \] Solve for \( \phi \): \[ \phi = 2.46 - 1.36 = 1.10 \text{ eV} \]

Step 4: Choose the Correct Option

Compare the calculated work function \(1.10 \text{ eV}\) with the given options: (a) \(0.43 \text{ eV}\), (b) \(0.55 \text{ eV}\), (c) \(1.10 \text{ eV}\), (d) \(1.53 \text{ eV}\). The correct answer is option (c) \(1.10 \text{ eV}\).

Key Concepts

Wavelength-Energy Relationship

In the intriguing world of photons and their behavior, the relationship between wavelength and energy plays a pivotal role, especially in the context of the photoelectric effect. According to the fundamental principles of physics, the energy of a photon is inversely proportional to its wavelength. This means that as the wavelength of a photon decreases, its energy increases. Conversely, as the wavelength increases, the energy decreases. This relationship is mathematically expressed by the formula:\[ E = \frac{hc}{\lambda} \]where:

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J s})\),
• \( c \) is the speed of light in vacuum \((3.0 \times 10^8 \text{ m/s})\),
• \( \lambda \) is the wavelength of the photon.

So, understanding this relationship helps us calculate the energy of photons at different wavelengths, which is crucial when studying the photoelectric effect. For instance, a light photon with a wavelength of \(10,000 \text{ Å}\) has an energy of \(1.23 \text{ eV}\). We can then apply the same principles to find the energy for any other given wavelength. This foundational concept sets the stage for exploring the photoelectric effect.

Work Function

The concept of a work function is a cornerstone of the photoelectric effect. It represents the minimum energy required to eject an electron from the surface of a material when exposed to light. Each material has a unique work function, influenced by the nature of its atoms and the forces holding its electrons.The work function \( \phi \) is typically measured in electron volts (eV). It plays a crucial role in determining whether or not electrons will be emitted when the material is struck by light of a particular energy.In mathematical terms, the work function is part of the photoelectric equation:\[ E_{\text{photon}} = \phi + E_{\text{kinetic}} \]where:

• \( E_{\text{photon}} \) is the energy of the incoming photon,
• \( \phi \) is the work function of the material,
• \( E_{\text{kinetic}} \) is the maximum kinetic energy of the emitted electrons.

In practical applications, knowing the work function of a material helps predict if photoelectric emission will occur when it is exposed to light of a given energy.

Stopping Potential

The concept of stopping potential is integral to understanding how the energy of photoelectrons is measured in the photoelectric effect. When light shines on a material and electrons are emitted, not all electrons have the same energy. The stopping potential is the voltage required to bring the most energetic electrons to a stop, preventing them from reaching the opposite electrode.If \( V_{0} \) is used as a stopping potential, the maximum kinetic energy of the electrons \( E_{\text{kinetic}} \) can be expressed as:\[ E_{\text{kinetic}} = eV_{0} \]where:

• \( e \) is the charge of the electron, \((1.602 \times 10^{-19} \text{ C})\),
• \( V_{0} \) is the stopping potential in volts.

The concept of stopping potential allows scientists to measure and calculate the kinetic energy of the electrons emitted, thus linking the stopping potential directly with the work function and helping to better understand the complexities of electron motion influenced by light.

Kinetic Energy of Electrons

Kinetic energy is the energy an object possesses due to its motion. In the realm of the photoelectric effect, it refers to the energy carried by the electrons that are emitted when light hits a material surface. Understanding this aspect is fundamental because it connects the properties of the incident light through the photoelectric equation.The expression for the kinetic energy of photoelectrons is given by:\[ E_{\text{kinetic}} = E_{\text{photon}} - \phi \]This equation highlights that the kinetic energy of emitted electrons depends on:

• \( E_{\text{photon}} \) - energy of the incident photon,
• \( \phi \) - the work function of the material.

These electrons will only have enough kinetic energy to overcome the threshold energy defined by the work function to be emitted. The kinetic energy can then be affected by factors such as stopping potential, which provides a direct measurement of this energy in experiments. Through the measurement of kinetic energy, one can gather insights into the interaction of photons with matter, impacting fields ranging from materials science to quantum physics.