Two monochromatic light waves of equal intensities produce an interference pattern. At a point in the pattern, the phase difference between the interfering waves is `pi//2` rad. Express the intensity at this point as a fraction of the maximum intensity in the pattern.

Explanation

Data: `I_(1) = I_(2) = I_(0),phi = pi//2` rad
The resultant intensity at the point in the pattern is
`I= I_(1) + I_(2) + 2sqrt(I_(1)I_(2)) cosphi`
`=I_(0) + I_(0) = 2sqrt(I_(0)I_(0))cosphi`
`= 2I_(0)(1+1) = 4I_(0)`...................(i)
Also, `1+ cosphi = 2 cos^(2) phi/2`................(2)
Also, `1+cosphi = 2 cos^(2) phi/2`.........(3)
From Eqs. (1) and (3),
`I= 2I_(0)(2cos^(2)phi/2)` [From Eq. (2)]
`I=2I_(0) ( 2cos^(2)phi/2)`
`=I_("max")(cospi/4)^(2) = I_("max")(1/sqrt(2))^(2)`
`therefore I= 1/2I_("max")`