Question

For a metal of work function 6.6 eV, which of the following wavelengths of incident radiation does not give rise to the photoelectric effect? (Take Planck’s constant as 6.6 $\times$ 10⁻³⁴ Js)

• A. 50 nm
• B. 100 nm
• C. 150 nm
• D. 200 nm

Hint:

Remember the inverse relationship: Shorter wavelength = Higher energy. If the energy is too low at 200 nm, it will definitely be too low for any wavelength longer than that.

Answer 1

The Correct Option is D

Step 1: Understanding the Topic:
This problem is part of "Modern Physics" and the "Photoelectric Effect." For an electron to be ejected from a metal surface, the incident light must provide at least a minimum amount of energy, known as the "Work Function" ($\phi$). If the energy of the incoming photon is less than the work function, no emission occurs, regardless of the light's intensity.
Step 2: Key Formulas and Approach:

Energy of a photon: $E = h \nu = hc / \lambda$.
Condition for emission: $E \geq \phi$.
Useful shortcut: Energy in eV $\approx 1240 / \lambda$ (with $\lambda$ in nm).

Step 3: Detailed Explanation:

Identify the requirement: We need to find which wavelength results in a photon energy $E$ that is less than $6.6 \text{ eV}$.
Calculate energy for the longest given wavelength (200 nm): Since $E$ and $\lambda$ are inversely proportional, the longest wavelength will have the lowest energy. \[ E = \frac{hc}{\lambda} = \frac{6.6 \times 10^{-34} \times 3 \times 10^8}{200 \times 10^{-9}} \] \[ E = \frac{19.8 \times 10^{-26}}{2 \times 10^{-7}} = 9.9 \times 10^{-19} \text{ Joules} \]
Convert to eV: Divide by the charge of an electron ($1.6 \times 10^{-19} \text{ C}$). \[ E = \frac{9.9 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 6.18 \text{ eV} \]
Compare with Work Function: $6.18 \text{ eV}$ is less than the required $6.6 \text{ eV}$. Therefore, $200 \text{ nm}$ light does not have enough energy to trigger the effect.
Since $200 \text{ nm}$ fails, all other options (50, 100, 150 nm) being shorter wavelengths will have higher energies and {will} cause the effect.
Step 4: Final Answer:
Radiation with a wavelength of 200 nm will not give rise to the photoelectric effect.