2 coherent narrow slits emitting light of wavelength lamda in the same phase are placed parallel to each other at a small separation of 2lamda.The light is collected on a screen S which is placed at a distance D from slit S1 as shown in figure. The finite distance x such that intensity at P is equal to intensity at O is

In this scenario, we’re dealing with a classic interference pattern produced by two coherent narrow slits emitting light waves of the same wavelength, λ. To find the distance x from the center of the interference pattern where the intensity at point P equals the intensity at point O, we need to delve into some fundamental principles of wave interference.

Understanding the Setup

We have two slits, S1 and S2, separated by a distance of 2λ. When light passes through these slits, it creates an interference pattern on a screen placed at a distance D from the slits. The key points to consider are:

• The light waves from both slits are coherent, meaning they maintain a constant phase difference.
• The wavelength of the light is λ.

Intensity in Interference Patterns

The intensity of light at any point on the screen can be described using the formula:

Where:

• I₀ is the maximum intensity.
• Δφ is the phase difference between the waves from the two slits at that point.

Calculating Phase Difference

The phase difference Δφ can be expressed in terms of the path difference between the two waves reaching a point P on the screen:

For our setup, the path difference (d) can be calculated using the geometry of the situation. If we define the horizontal distance from the center line to point P as x, the path difference can be approximated as:

Thus, the phase difference becomes:

Finding the Condition for Equal Intensities

We need to find the point P where the intensity at P equals the intensity at O. At point O (the center), the phase difference is zero, hence:

For the intensity at point P to equal that at point O:

This means:

From this, it follows that:

This occurs at:

Solving for the Position x

Substituting Δφ into the equation:

From this, we can solve for x:

This means that the distance x from the center line to the point P where the intensity is equal to that at O is λ/4. If you want to consider the situation in the context of the screen distance D, keep in mind that this distance remains valid as long as the small angle approximation holds true.

Conclusion

The distance x where the intensity at point P equals the intensity at point O is given by the formula x = λ/4. This result illustrates the principles of interference and the relationship between path difference, phase difference, and intensity in wave phenomena. If you have any further questions or need clarification, feel free to ask!