The photoelectric threshold wavelength of silver is $3250 \times 10^{-10}m$. The velocity of the electron ejected from a silver surface by ultraviolet light of wavelength $2536 \times 10^{-10} m $ is (Given $ h = 4.14 \times 10^{-15} eVs$ and $c = 3 \times 10^8 \, ms^{-1}$)

Question

A photon and an electron have the same energy $ E $. If $ \lambda_p $ is the wavelength of the photon and $ \lambda_e $ is the de Broglie wavelength of the electron, then the ratio $ \frac{\lambda_p}{\lambda_e} $ is:

Answer 1

To solve the problem, we will use the photoelectric equation:

E_{k} = h \nu - h \nu_0

where:

E_{k} is the kinetic energy of the ejected electron.
\nu is the frequency of the incident light.
\nu_0 is the threshold frequency.
h is Planck's constant.

First, we need to calculate the threshold frequency \nu_0 and the frequency \nu of the incident light using the formula:

\nu = \frac{c}{\lambda}

Given:

Threshold wavelength \lambda_0 = 3250 \times 10^{-10} m
Incident wavelength \lambda = 2536 \times 10^{-10} m
Speed of light c = 3 \times 10^8 ms^{-1}
Planck's constant h = 4.14 \times 10^{-15} eVs

Step 1: Calculate \nu_0

\nu_0 = \frac{c}{\lambda_0} = \frac{3 \times 10^8}{3250 \times 10^{-10}} = \frac{3}{32.5} \times 10^{15} \approx 0.0923 \times 10^{15} Hz

Step 2: Calculate \nu

\nu = \frac{c}{\lambda} = \frac{3 \times 10^8}{2536 \times 10^{-10}} = \frac{3}{25.36} \times 10^{15} \approx 0.1183 \times 10^{15} Hz

Step 3: Calculate the kinetic energy E_{k}

E_{k} = h(\nu - \nu_0) = 4.14 \times 10^{-15} \times (0.1183 - 0.0923) \times 10^{15} eV

E_{k} = 4.14 \times 10^{-15} \times 0.026 \times 10^{15} = 0.10764 eV

Step 4: Convert the kinetic energy into velocity:

Using the relation E_{k} = \frac{1}{2} m v^2, where m = 9.11 \times 10^{-31} kg (mass of electron), we get:

0.10764 \, eV = \frac{1}{2} \times 9.11 \times 10^{-31} \times v^2

Convert E_{k} from eV to Joules:

0.10764 \, eV = 0.10764 \times 1.6 \times 10^{-19} \, J

v^2 = \frac{2 \times 0.10764 \times 1.6 \times 10^{-19}}{9.11 \times 10^{-31}}

Calculate v:

v = \sqrt{\frac{2 \times 0.10764 \times 1.6 \times 10^{-19}}{9.11 \times 10^{-31}}} = \sqrt{3.778 \times 10^{11}} \approx 6.14 \times 10^5 \, ms^{-1}

The closest option is \approx 6 \times 10^5 \, ms^{-1}.

Answer 2