Resultant Intensity at \(y=\frac{d}{2}, I_{y}=?\) 

The path difference between two waves at \(y=\frac{d}{2}\)

\(\Delta x = d\tan \theta  = \) \(d \times \frac{y}{D} = \) \(\frac{{d \times \frac{d}{2}}}{{10d}} = \) \(\frac{d}{{20}} = \frac{{5\lambda }}{{20}} = \frac{\lambda }{4}\)

Corresponding phase difference, \(\phi=\frac{2 \pi}{\lambda} \Delta x=\frac{\pi}{2}\)

Now, maximum intensity in Young's double slit experiment,

\({I_{\max }} = {I_1} + {I_2} + 2{I_1}{I_2}\)

\({I_0} = 4I\) \((\because \,\,{I_1}\, = \,{I_2}\, = \,I)\)

\(\therefore I=\frac{I_{0}}{4}\)

Required intensity,

\({I_y} = {I_1} + {I_2} + 2{I_1}{I_2}\cos \frac{\pi }{2}\) \( = 2I = \frac{{{I_0}}}{2}\)