A radiation of wavelength 'λ' is incident on metal surface having 620 nm threshold wavelength. The stopping potential of electron emitted is 4.9 V. What is the value of λ (in nm

To determine the wavelength 'λ' of the incident radiation, we can use the photoelectric effect principles, which relate the energy of the incoming photons to the work function of the metal and the kinetic energy of the emitted electrons. Let's break this down step by step.

Understanding the Photoelectric Effect

The photoelectric effect occurs when light (or radiation) hits a metal surface and causes the emission of electrons. The key points to consider are:

• The energy of the incident photon is given by the equation: E = h * f, where h is Planck's constant and f is the frequency of the radiation.
• The work function (φ) of the metal is the minimum energy required to remove an electron from the surface. This is related to the threshold wavelength (λ₀) by the equation: φ = h * c / λ₀, where c is the speed of light.
• The stopping potential (Vₛ) is the potential needed to stop the emitted electrons, which relates to their kinetic energy: K.E. = e * Vₛ, where e is the charge of an electron.

Given Values

From the problem, we have:

• Threshold wavelength (λ₀) = 620 nm
• Stopping potential (Vₛ) = 4.9 V

Calculating the Work Function

First, we need to calculate the work function (φ) of the metal using the threshold wavelength:

Convert the threshold wavelength from nanometers to meters:

λ₀ = 620 nm = 620 x 10^-9 m

Now, using the equation for work function:

φ = h * c / λ₀

Substituting the values:

• Planck's constant (h) = 6.626 x 10^-34 J·s
• Speed of light (c) = 3.00 x 10⁸ m/s

Calculating φ:

φ = (6.626 x 10^-34 J·s * 3.00 x 10⁸ m/s) / (620 x 10^-9 m)

φ ≈ 3.21 x 10^-19 J

Finding the Energy of the Emitted Electrons

The kinetic energy of the emitted electrons can be calculated using the stopping potential:

K.E. = e * Vₛ

Where e (the charge of an electron) is approximately 1.602 x 10^-19 C.

Calculating K.E.:

K.E. = (1.602 x 10^-19 C) * (4.9 V) ≈ 7.85 x 10^-19 J

Total Energy of the Incident Photon

The total energy of the incident photon must equal the work function plus the kinetic energy of the emitted electron:

E = φ + K.E.

Substituting the values we calculated:

E = (3.21 x 10^-19 J) + (7.85 x 10^-19 J) ≈ 1.10 x 10^-18 J

Relating Energy to Wavelength

Now, we can find the wavelength 'λ' of the incident radiation using the energy equation:

E = h * c / λ

Rearranging gives us:

λ = h * c / E

Substituting the known values:

λ = (6.626 x 10^-34 J·s * 3.00 x 10⁸ m/s) / (1.10 x 10^-18 J)

Calculating λ:

λ ≈ 1.80 x 10^-6 m = 1800 nm

Final Result

Therefore, the wavelength 'λ' of the incident radiation is approximately 1800 nm.