Question

Young's double slit experiment is performed with monochromatic light of wavelength \( 6000 \text{Ã…} \). If the intensity of light at a point on the screen where the path difference is \( 2000 \text{Ã…} \) is \( I_1 \) and the intensity of light at a point where path difference is \( 1000 \text{Ã…} \) is \( I_2 \), then the ratio \( I_1: I_2 \) is:

• A. \( 1:3 \)
• B. \( 2:1 \)
• C. \( 1:1 \)
• D. \( 4:5 \)

Hint:

In YDSE, intensity variation depends on the cosine square of the phase difference, affecting the bright and dark fringes.

Answer 1

The Correct Option is A

To determine the ratio \( I_1 : I_2 \) in Young's double slit experiment, we need to analyze the interference pattern and how the path difference affects the intensity of light at different points on the screen. 1. Given Data: - Wavelength of light, \( \lambda = 6000 \text{Ã…} \) - Path difference at point 1, \( \Delta_1 = 2000 \text{Ã…} \) - Path difference at point 2, \( \Delta_2 = 1000 \text{Ã…} \) - Intensity at point 1, \( I_1 \) - Intensity at point 2, \( I_2 \) 2. Calculate the Phase Difference: - The phase difference \( \phi \) is related to the path difference \( \Delta \) by: \[ \phi = \frac{2\pi \Delta}{\lambda} \] - For point 1: \[ \phi_1 = \frac{2\pi \times 2000 \text{Ã…}}{6000 \text{Ã…}} = \frac{2\pi}{3} \] - For point 2: \[ \phi_2 = \frac{2\pi \times 1000 \text{Ã…}}{6000 \text{Ã…}} = \frac{\pi}{3} \] 3. Calculate the Intensity: - The intensity \( I \) at a point on the screen is given by: \[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \] where \( I_0 \) is the intensity of light from a single slit. - For point 1: \[ I_1 = 4I_0 \cos^2\left(\frac{\phi_1}{2}\right) = 4I_0 \cos^2\left(\frac{\pi}{3}\right) = 4I_0 \left(\frac{1}{2}\right)^2 = I_0 \] - For point 2: \[ I_2 = 4I_0 \cos^2\left(\frac{\phi_2}{2}\right) = 4I_0 \cos^2\left(\frac{\pi}{6}\right) = 4I_0 \left(\frac{\sqrt{3}}{2}\right)^2 = 3I_0 \] 4. Determine the Ratio \( I_1 : I_2 \): - The ratio of the intensities is: \[ I_1 : I_2 = I_0 : 3I_0 = 1 : 3 \] 5. Final Answer: - The ratio \( I_1 : I_2 \) is: \[ \boxed{1:3} \] This corresponds to option (1).