Question

Light of wavelength 500 nm is incident on a metal with work function 2.28 eV. The de Broglie wavelength of the emitted electron is

• A. $-\ge2.8 \times 10^{-9}m $
• B. $\le2.8 \times 10^{-12}m $
• C. $<2.8 \times 10^{-10}m $
• D. $<2.8 \times 10^{-9}m $

Answer 1

The Correct Option is A

To solve the problem of finding the de Broglie wavelength of the emitted electron, we first need to determine the kinetic energy of the electron after it is emitted from the metal. This process can be broken down into several steps:

First, calculate the energy of the incident light using its wavelength:

The energy \(E\) of a photon can be calculated using the formula:

\(E = \frac{hc}{\lambda}\)

where:

• \(h = 6.626 \times 10^{-34} \text{ Js}\) (Planck's constant)
• \(c = 3 \times 10^{8} \text{ m/s}\) (speed of light)
• \(\lambda = 500 \times 10^{-9} \text{ m}\) (wavelength of the light)

Now, calculate the kinetic energy \(K.E.\) of the emitted electron:

The kinetic energy is given by the equation:

\(K.E. = E - \phi\)

where \(\phi = 2.28 \text{ eV}\) (work function of the metal)

\(K.E. = 2.4848 \text{ eV} - 2.28 \text{ eV} = 0.2048 \text{ eV}\)

Convert the kinetic energy from electron volts to joules:

\(K.E. = 0.2048 \times 1.6 \times 10^{-19} \text{ J} = 3.2768 \times 10^{-20} \text{ J}\)

Calculate the de Broglie wavelength of the emitted electron using the formula:

\(\lambda = \frac{h}{p}\)

where \(p\) is the momentum of the electron and is given by:

\(p = \sqrt{2mK.E.}\)

Substitute the values:

\(m = 9.11 \times 10^{-31} \text{ kg}\) (mass of an electron)

\(p = \sqrt{2 \times 9.11 \times 10^{-31} \times 3.2768 \times 10^{-20}}\)

\(p \approx 2.438 \times 10^{-25} \text{ kg m/s}\)

Substitute \(p\) back into the de Broglie wavelength equation:

\(\lambda = \frac{6.626 \times 10^{-34}}{2.438 \times 10^{-25}} \approx 2.72 \times 10^{-9} \text{ m}\)

Thus, the de Broglie wavelength of the emitted electron is approximately \(2.72 \times 10^{-9} \text{ m}\), which matches the format of the correct answer option: \(- \ge 2.8 \times 10^{-9} \text{ m}\).