The cutoff wavelength for the emission of 7. photoelectrons from a particular surface is \(500 \mathrm{nm}\). Find the maximum kinetic energy of the ejected photoelectrons when the surface is illuminated with light of wavelength \(600 \mathrm{nm}\).

First, we need to find the work function of the material using the given cutoff wavelength. We will use the relationship between wavelength, frequency, and the speed of light: \(c = \lambda f\) Where: - \(c\) is the speed of light (given by \(3 \times 10^8 \mathrm{m/s}\)) - \(\lambda\) is the wavelength of the light - \(f\) is the frequency of the light Using the equation above, we can find the frequency at the cutoff wavelength: \(f_c = \frac{c}{\lambda_c}\) We are given the cutoff wavelength \(\lambda_c = 500 \times 10^{-9} \mathrm{m}\). Substitute the values to calculate \(f_c\): \(f_c = \frac{3 \times 10^8 \mathrm{m/s}}{500 \times 10^{-9} \mathrm{m}} = 6 \times 10^{14} \mathrm{Hz}\) Now, we can find the work function: \(W = h f_c\) Substitute Planck's constant and frequency \(f_c\) to find the work function: \(W = (6.626 \times 10^{-34} \mathrm{Js})(6 \times 10^{14} \mathrm{Hz}) = 3.98 \times 10^{-19} \mathrm{J}\)

Now, we will calculate the energy of the incident photons at the given wavelength (\(\lambda = 600 \mathrm{nm}\)). First, we will find the frequency of the light using the relationship between wavelength and frequency: \(f = \frac{c}{\lambda}\) Substitute the values to find \(f\): \(f = \frac{3 \times 10^8 \mathrm{m/s}}{600 \times 10^{-9} \mathrm{m}} = 5 \times 10^{14} \mathrm{Hz}\) Now, we can calculate the incident photon energy using Planck's formula: \(E = h f\) Substitute Planck's constant and frequency \(f\) to find the incident photon energy: \(E = (6.626 \times 10^{-34} \mathrm{Js})(5 \times 10^{14} \mathrm{Hz}) = 3.31 \times 10^{-19} \mathrm{J}\)

To find the maximum kinetic energy of the ejected photoelectrons, we will use the energy conservation equation: \(K = E - W\) Substitute the values for the incident photon energy (\(E\)) and work function (\(W\)): \(K = 3.31 \times 10^{-19} \mathrm{J} - 3.98 \times 10^{-19} \mathrm{J} = -0.67 \times 10^{-19} \mathrm{J}\) Since the kinetic energy is negative, it means that the incident photons' energy at a wavelength of \(600 \mathrm{nm}\) is not sufficient enough to overcome the work function and eject photoelectrons from the surface. Therefore, there will be no ejected photoelectrons in this case.