When light of a certain wavelength is incident on a metal surface, it can cause the emission of photoelectrons. The energy of these photoelectrons depends on the frequency (or wavelength) of the incident light. The maximum kinetic energy of the emitted photoelectrons is given by the equation:

\[E_{\text{{max}}} = hf - \phi\]

where \(E_{\text{{max}}}\) is the maximum kinetic energy of the photoelectrons, \(h\) is Planck's constant, \(f\) is the frequency of the incident light, and \(\phi\) is the work function of the metal (the minimum energy required to remove an electron from the metal surface).



The stopping potential is the minimum potential difference that needs to be applied across the metal surface to stop the photoelectrons from reaching the other electrode. This stopping potential is a measure of the maximum kinetic energy of the emitted photoelectrons.

In the given problem, the stopping potential for the photoelectrons is 3.2 V when the light of a certain wavelength is incident on the metal surface. This means that the maximum kinetic energy of the photoelectrons is 3.2 eV.



The stopping potential depends on the maximum kinetic energy of the photoelectrons, which in turn depends on the frequency (or wavelength) of the incident light. As the frequency (or wavelength) increases, the maximum kinetic energy of the photoelectrons also increases.

In the second scenario, when the wavelength of the incident light is doubled, the stopping potential increases to 8.8 V. This means that the maximum kinetic energy of the photoelectrons is now 8.8 eV.



The energy of a photon is given by the equation:

\[E = hf = \frac{hc}{\lambda}\]

where \(E\) is the energy of the photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the light.

The maximum kinetic energy of the emitted photoelectrons can be calculated by subtracting the work function of the metal from the energy of the incident photons:

\[E_{\text{{max}}} = E - \phi\]

Since the stopping potential is a measure of the maximum kinetic energy of the photoelectrons, we can equate the stopping potential to the maximum kinetic energy:

\[V_s = \frac{E_{\text{{max}}}}{e}\]

where \(V_s\) is the stopping potential and \(e\) is the charge of an electron.

By substituting the equations for \(E\) and \(E_{\text{{max}}}\) into the equation for \(V_s\), we can derive an expression for the stopping potential in terms of the wavelength:

\[V_s = \frac{hc}{\lambda e} - \frac{\phi}{e}\]

Using this equation, we can find the wavelength of the second light source by equating the stopping potential in the two scenarios:

\[\frac{hc}{\lambda_1 e} - \frac{\phi}{e} = \frac{hc}{\lambda_2 e} - \frac{\phi}{e}\