Question

If \(\lambda_{0}\) is the threshold wavelength for photoelectric emission from a metal surface, \(\lambda\) is the wavelength of light falling on the surface of metal and \(m\) is the mass of electron, then the maximum speed of ejected electrons is given by (a) \(\left[\frac{2 h}{m}\left(\lambda_{0}-\lambda\right)\right]^{1 / 2}\) (b) \(\left[\frac{2 h c}{m}\left(\lambda_{0}-\lambda\right)\right]^{1 / 2}\) (c) \(\left[\frac{2 h c}{m}\left(\frac{\lambda_{0}-\lambda}{\lambda_{0} \lambda}\right)\right]^{1 / 2}\) (d) \(\left[\frac{2 h}{m}\left(\frac{1}{\lambda_{0}}-\frac{1}{\lambda}\right)\right]^{1 / 2}\)

Short answer

(d) \(\left[\frac{2h}{m}\left(\frac{1}{\lambda_{0}} - \frac{1}{\lambda}\right)\right]^{1/2}\)

Step-by-step solution

Understand the Photoelectric Effect

Photoelectric emission occurs when photons with a certain threshold energy, corresponding to the threshold wavelength \(\lambda_{0}\), strike a metal surface and cause electrons to be ejected. The energy of the incoming photons is given by the equation \(E = \frac{h c}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the incident light.

Apply Conservation of Energy

The maximum kinetic energy \(K_{max}\) of the ejected electrons will be equal to the photon's energy minus the work function \(\phi\) of the metal: \(K_{max} = \frac{h c}{\lambda} - \frac{h c}{\lambda_{0}}\). The work function is the minimum energy required to remove an electron from the metal surface, and it is given by \(\phi = \frac{h c}{\lambda_{0}}\).

Rearrange the Energy Equation

Rearranging the equation for kinetic energy gives us \(K_{max} = h c \left(\frac{1}{\lambda} - \frac{1}{\lambda_{0}}\right)\).

Relate Kinetic Energy to Electron Speed

The maximum kinetic energy can also be expressed in terms of the mass \(m\) and speed \(v\) of the electron: \(K_{max} = \frac{1}{2} m v^2\).

Solve for Electron Speed

Equating the two expressions for \(K_{max}\) and solving for \(v\) yields \(v = \left[\frac{2 K_{max}}{m}\right]^{1/2} = \left[\frac{2h}{m}\left(\frac{1}{\lambda} - \frac{1}{\lambda_{0}}\right)\right]^{1/2}\). So the correct option that represents the maximum speed of ejected electrons is (d).

Key concepts

Threshold Wavelength

The threshold wavelength, denoted by \(\lambda_0\), is a fundamental concept in the study of the photoelectric effect. It represents the maximum wavelength of light that can cause electrons to be ejected from a metal surface. If the incident light has a wavelength longer than the threshold wavelength, its photons do not have enough energy to liberate the electrons from the material.

Understanding the threshold wavelength is crucial because it is directly linked to the work function (\(\phi\)), the minimum energy necessary to remove an electron from the metal. The work function can be calculated using Planck's constant (\(h\)) and the speed of light (\(c\)) as \(\phi = \frac{h c}{\lambda_0}\). This equation demonstrates that as the threshold wavelength increases, the energy needed to eject an electron decreases, which is pivotal for predicting the behavior of a material in response to various wavelengths of light.

Maximum Speed of Ejected Electrons

The maximum speed of the ejected electrons in the photoelectric effect is affected by the energy of the incident photons and the work function of the metal. The kinetic energy imparted to the electron is the difference between the photon's energy and the work function. To find the maximum speed, we equate the kinetic energy (\(K_{max}\)) to \(\frac{1}{2} m v^2\), where \(m\) is the electron's mass and \(v\) is its speed.

The appropriate formula for determining this is option (d) \(v = \left[\frac{2h}{m}\left(\frac{1}{\lambda} - \frac{1}{\lambda_{0}}\right)\right]^{1/2}\), as derived in the solution steps. This signifies that the maximum speed is not only determined by the incident light's wavelength but also by the specific properties of the metal, such as its threshold wavelength and the mass of an electron.

Energy Conservation in the Photoelectric Effect

Energy conservation is a principle that is at the core of the photoelectric effect. When a photon hits a metal surface, its energy is used to perform work to eject an electron. The energy of the photon must be at least equal to the work function for the electron to be emitted.

The total energy of the incoming photon is converted into the work function and the kinetic energy of the ejected electron. Mathematically, this principle is expressed as \(E_{photon} = \phi + K_{max}\), which simplifies to \(\frac{h c}{\lambda} = \frac{h c}{\lambda_{0}} + \frac{1}{2} m v^2\). This conservation law allows us to solve for unknowns, such as the maximum kinetic energy of the electron or the maximum speed of the ejected electrons once the threshold wavelength is known.

Planck's Constant Application

Planck's constant (\(h\)) is a crucial quantity in quantum mechanics that relates the energy of a photon to its frequency. Its application in photoelectric effect problems is vital as it enables the calculation of energies, speeds, and other properties of ejected electrons from the wavelength of incident light.

Through the equation \(E = h f\), where \(f\) is the frequency of the photon, and the relationship between frequency and wavelength (\(f = \frac{c}{\lambda}\)), we can calculate the photon's energy as \(E = \frac{h c}{\lambda}\). This relationship is fundamental when applying energy conservation to determine the kinetic energy and speeds of ejected electrons as seen in the photoelectric equations used earlier in the solution. Planck's constant bridges the gap between the microscopic quantum world and macroscopic observations like the photoelectric effect.