When a plano-convex lens with its convex surface is placed on a plane glass sheet, an air film of gradually increasing thickness outward is formed between the lens and the sheet. The thickness of film at the point of contact is zero. If monochromatic light is allowed to fall normally on the lens, and the film is viewed in reflected light, alternate bright and dark concentric rings are seen around the point of contact. These rings were first discovered by Newton, that's why they are called NEWTON'S RINGS .
Newton's rings are formed due to interference between the light waves reflected from the top and bottom surfaces of the air film formed between the lens and glass sheet.
  The phenomenon of the formation of the Newton's rings can be explained on the basis of wave theory of
An air film of varying thickness is formed between the lens and the glass sheet.
When a light ray is incident on the upper surface of the lens, it is reflected as well as refracted.
When the refracted ray strikes the glass sheet, it undergo a phase change of 180O on reflection.
Interference occurs between the two waves which interfere constructively if path difference between     them is (m+1/2)l and destructively if path difference between them is ml producing alternate bright     and dark rings.
  Let the radius of curvature of the convex lens is R and the radius of ring is 'r'. Consider light of wave length 'l' falls on the lens. After refraction and reflection two rays 1 and 2 are obtained. These rays interfere each other producing alternate bright and dark rings. At the point of contact the thickness of air film
is zero and the path difference is also
zero and as a 180O phase change occurs, so they cancel each other and
a dark ring is obtained at the centre.
  As we move away from the central point , path difference is also changed and alternate dark and bright rings are obtained.
  Let us suppose that the thickness of air film is 't'.
By using the theorem of geometry,
    r2 = 2Rt.............. (1)
  In thin films, path difference for constructive interference is:
2nt = (m+1/2) l
  Where n= refractive index
For air n = 1
                  2t = (m+1/2) l .............. (2)
For first bright ring m = 0
For second bright ring m = 1
For third bright ring m = 2
For Nth bright ring m = N-1
Putting the value of m in equation (2)
     2t = (N-1+1/2) l
2t = (N-1/2)l
                             t =1/2 (N-1/2) l .............. (3)
Putting the value of 't' in equation (1)
r2 = 2Rt
r2 = 2R . 1/2 (N-1/2) l
r2 = R (N-1/2) l
This is the expression for the radius of Nth bright ring where
rn = radius of Nth bright ring
N = Ring number
R = radius of curvature of lens
l = Wave length of light
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