Theory of Newton's rings

Theory of Newton's rings:-

3)Newton's rings with bright centre ( due to reflected light ).

If the plano-convex lens and glass plate are of different materials of refractive indices μ₁ and μ₂ and a liquid of refractive index μ
 ( lying between μ₁ and μ₂ ) is introduced between them. Then both the reflections from the film would be similar. So the centre of Newton's rings will be bright. This was shown by scientist young, using oil between crown and flint glass,
( μflint >μoil>μcrown ). If μ₁>μ>μ₂, then reflection from top and bottom surface of the liquid film takes place under similar conditions. In both the cases reflection takes place from the surface of optically rarer medium I (path difference=2t only )
Since at a point t=0; the condition of path difference=nλ =0 satisfies. So , central spot appears bright. If μ₁<μ<μ₂ then the central spot will also be bright. Here, also reflections from top and bottom surface of the liquid film takes place under similar conditions. In both the cases reflection takes place from the surface of optically denser medium. So additional phase change of π radius should be considered at interfering rays at the point of contact satisfies the condition for brightness. So the central spot appears bright.

4) Newton's rings by transmitted light

Newton's rings observed by transmitted light are exactly opposite to those observed by reflected light. In this transmitted light, the central spot is bright . By transmitted light:

The diameter of the dark ring,
Dₙ = √2(2n-1)λR

And the diameter for the bright ring,
Dₙ= 2√nλR