Two coherent sources S1 and S2, each producing an intensity I_0 at th…

The correct answer is Option (1) → \( 4I_0 \)

The phase difference between the two waves at P has two parts. One is the phase difference the sources already have. The other is produced by the path difference. The intensity then follows from \( I=4I_0\cos^{2}(\phi/2) \), with \( \phi \) the total phase difference.

The phase difference due to the path difference \( \lambda/2 \) is:

    \( \begin{aligned} \phi_{\mathrm{path}} &= \frac{2\pi}{\lambda}\times\Delta x \\ &= \frac{2\pi}{\lambda}\times\frac{\lambda}{2} \\ &= \pi \end{aligned} \)

Now add the phase difference \( \pi \) between the sources. The total phase difference at P is then \( 0 \) or \( 2\pi \), depending on which source leads. Taking \( \phi=2\pi \):

    \( \begin{aligned} I &= 4I_0\cos^{2}(\phi/2) \\ &= 4I_0\cos^{2}\pi \\ &= 4I_0 \end{aligned} \)

So the two waves arrive at P in phase. P lies on a bright fringe of intensity \( 4I_0 \), due to constructive interference. If the sources had been in phase, the same point would have been dark.

NCERT (2026-27):
Coherent and Incoherent Addition of Waves – “The amplitude of the resultant displacement is 2a cos (φ/2) and therefore the intensity at that point will be”

    \( I=4I_0\cos^2(\phi/2) \qquad (10.11) \)

“If φ = 0, ± 2π, ± 4π,… […] we will have constructive interference leading to maximum intensity.”