Monochromatic light passes through an orifice in a scrren `Sc` (Fig.) and being reflected from a thin transparent plate `P` produces fringes of equal inclinaiton on the screen. The thickness of the plate is equal to `d`, the distance between the plate and the screen is `l`, the radii of the `ith` and `kth`dark rings are `r_(i)` and `r_(k)`. find the wavalength of light taking into account that `r_(i,k) lt lt l`.

Explanation

For small angles `theta` we write fro dark fringes
`2d sqrt(n^(2) - sin^(@) theta) = 2d (n - (sin^(2) theta)/(2n)) = (k + 0) lambda`
For the first fringe `theta = 0` and
`2dn = (k_(0) +0) lambda`
For the `i^(th)` dark fringe
`2d (n - (sin^(2) theta_(i))/(2n)) = (k_(0) - i + 1) lambda`
or `sin^(2) theta_(i) = (n lambda)/(d) (i - 1) = (r_(i)^(2))/(4l^2)`

Finally `(n lambda)/(d) (i - k) = (r_(i)^(2) - r_(k)^(2))/(4l^(2))`
so `lambda = (d(r_(i)^(2) - r_(k)^(2)))/(4l^(2) n(i - k))`