Youngs Double Slit Experiment

To solve the given problem on the Young's double slit experiment, we need to derive the expression for the intensity of light at a point on the screen based on the intensity through each slit and the interference pattern created. Let's break down the steps involved:

Conceptual Background:

In Young's double slit experiment, two coherent light sources interfere, producing an interference pattern of bright and dark fringes. The intensity at any point on the screen depends on the path difference between the light waves originating from the two slits and reaching that point.

Interference Formula:

The intensity of light at a point on the screen can be given as:

Where:

• and are the intensities from each slit (here both are ).
• is the phase difference between the two waves.

Phase Difference:

The phase difference is given by:

Where:

• is the wavelength of light.
• is the path difference between the waves from the two slits.

For a point located at a distance from the central maximum on the screen:

where is the distance between the slits and is the distance from the slits to the screen.

Intensity Calculation:

Substitute the values in the intensity formula:

Simplify to:

Conclusion:

Thus, the correct expression for the intensity at point is , which matches the option:

This derivation is in line with the principle of superposition and the conditions given (i.e., ).

Given:

• Wavelength of light: λ
• Intensity through each slit: I₀
• Slit separation: d (with d << D)
• Screen distance: D
• Observation point distance from center: x

Step 1: Calculate Path Difference (Δ)

For small angles (d << D), the path difference between light from the two slits reaching point P is:

Step 2: Determine Phase Difference (φ)

The phase difference corresponding to this path difference is:

Step 3: Calculate Resultant Intensity

For two coherent sources with intensity I₀ each, the resultant intensity is given by:

Since I₁ = I₂ = I₀:

Using the trigonometric identity :

Substituting φ from Step 2:

Given Information:
Intensity of each slit,
Distance of the point from the central maximum, , where is the fringe width.

Step-by-Step Explanation:

Step 1: Formula for Intensity in Young’s double-slit experiment:

The resultant intensity ( ) at a point on the screen is given by:

where is the intensity from each slit, and is the phase difference between waves from the two slits.

Step 2: Calculate the phase difference ( ) at the given point clearly:

The distance between central maximum and the given point P is . The phase difference corresponding to one fringe width ( ) is . Thus, clearly:

Step 3: Substitute into intensity formula clearly:

Since :

Step 4: Clearly state the final intensity value:

Final Conclusion:
The total intensity at point P (at ) is clearly .