Theory of Newton's rings

Theory of Newton's rings:

2) Expression for the diameter of Newton's rings by reflected light:

                Let a plano-convex lens be placed on a plane glass plate. Let R is the radius of curvature of the lens and C is the centre of curvature. The thickness of the air film at point B is MO(=t) at a distance of ON=rₙ from the point of contact O.

                For interference due to reflected light,

The path difference between
two interfering rays  = 2μtcos( r+ϴ  ) +𝞴 /2

Here,     μ= 1 (for air film)
               r= 0 (for normal incidence)

              Geometrical Optics and Interference
And ϴ = 0 ( for large value of R )

.•. path difference=2t + 𝞴 /2  ...(1)

From figure A) ,
                 R² = rₙ²+(R-t)²
                 rₙ²= R²-(R-t)²
                 rₙ²= 2Rt-t²

                Since R >>t, t² becomes very small, so it can be neglected.

                 rₙ² = 2R.t
                 2t = rₙ²/R

Putting this value of 2t in eq.(1) we get,

Path difference= rₙ²/R+𝞴 /2    ....(2)

Where, rₙ = radius of the nth Newton's ring.

For dark rings:

The condition of dark ring is

Path difference = (2t+1)𝞴 /2

.•. rₙ²/R+𝞴 /2= (2n+1)𝞴 /2

.•. rₙ²/R=n𝞴

If Dₙ is the diameter of the nth Newton's rings then,

rₙ=Dₙ/2      ⇒    rₙ²=Dₙ²/4

.•. Dₙ²/4R =n𝞴  i.e. Dₙ²=4Rn𝞴

Dₙ = 2⎷nR𝞴

.•. Diameter of dark ring is the proportional to the square root of natural number
( Dₙ ∝ ⎷n)

For bright ring:

For condition for bright ring is that, path difference=n𝞴

.•. rₙ²/R + 𝞴/2=n𝞴

rₙ²/R= (2n-1)𝞴/2    where n= 1,2,3,......

Putting rₙ² = Dₙ²/4

.•. Dₙ²/4R = (2n-1) 𝞴/2

Dₙ² = 4R(2n-1) 𝞴/2

       =2𝞴R(2n-1)

Dₙ = ⎷2(2n-1)𝞴R

Diameter of the bright ring is proportional to the square root of odd natural numbers.

Dₙ ∝  (2n-1)