Answer is option 1.
I want to know here that in that the phase difference given in the question is for centre of screen but the y value given is for 1st maxima, so both will have different phase difference then how their path difference could be same?
In Young's double-slit experiment, the phase difference given in the question (π/2) is specified at the central point on the screen.
This phase difference is not the same for the first maxima (which is located at some distance away from the central point).
To answer your specific question about phase and path difference:
Phase Difference at the Central Point (π/2):
At the central point (y = 0), the waves from the two slits arrive with a phase difference of . This phase difference is due to some factor like a small path difference, a variation in optical path length, or phase shift introduced by the slits themselves.
For the First Maxima (y ≠ 0):
The first maxima occurs when the path difference is exactly one wavelength (), leading to constructive interference.
The phase difference at the first maxima would typically be 2π (full cycle), which is different from the at the center.
Thus, while the phase differences are different at the center and at the first maxima, the path difference can still lead to constructive interference at both points if we consider that the phase shift at the center is caused by something additional (like an initial phase offset or different path lengths through the slits), whereas at the maxima, it is the standard for constructive interference.
In summary, the path difference remains consistent based on the geometry of the setup (like , , and ), but the phase differences vary due to additional shifts introduced by the slits or experimental conditions.
Hence, the path difference could be the same across different points of constructive interference, but the phase difference varies depending on position and initial conditions.