Threshold frequency and wavelength of electrons in the photoelectric effect

[Ezequiel]

Homework Statement

Electrons are emitted from a metal as a consequence of their absorption of energy from a light beam. Find the threshold frequency of the metal and the wavelength of emitted electrons.

Wavelength of incident light λ = 500 nm
Work function of the metal $\phi$ = 2.1 eV

Homework Equations

Threshold frequency:

f₀ = $\frac{\phi}{h}$

The Attempt at a Solution

Threshold frequency:

f₀ = $\frac{2.1 eV}{4.136 \times 10^{-15} eV·s }$ = 5.08 × 10¹⁴ Hz

Is this correct?

How can I find the wavelength of emitted electrons?

 

[mfb]

Confirmation of f₀

Try to find the electron energy first.

 

[Ezequiel]

Thanks for the confirmation.

As I understand the photoelectric effect, one photon transfers all of its energy to an electron, so the energy absorbed by any electron must be the same (for a monochromatic beam), in this case hc/(500 nm) = 2.48 eV. Electrons need at least 2.1 eV to escape this metal, so they must have a maximum kinetic energy of 0.38 eV. Since not all of them have the same kinetic energy (due to losses) I assume they must have different wavelengths as well, how can I find the wavelength of emitted electrons?

 

[mfb]

You can assume that they all have 0.38 eV. If you like, use "<=" in the calculations, but that won't change much.

 

[Ezequiel]

Ok, so it would be λ = $\frac{hc}{\sqrt{2mc^2K}} \approx$ 2 nm, right?