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Experiment-4

To measure the radius of curvature of a spherical surface of a plano-convex lens.

Experimental set up: Plano-convex Lens, a flat, Sodium Lamp, Glass beam splitter, Travelling Microscope

Theory: When a plano-convex lens is placed on the flat surface as shown in the figure, there is a point of contact and around this an air-film of increasing thickness.

Light is reflected from the lower surface of plano-convex lens and top surface of the flat. These two waves will interfere to produce a fringe pattern. There is a phase change of p on reflection from the flat surface (reflection from rare to denser media). Since the thickness of the air-film is variable, the condition for constructive interference at any point on the film is , where t(x,y) is the thickness of the film (origin of the coordinate system is at the point of contact) and m is the order of interference. At the point of contact t(x,y)=0 and hence the two waves are out of phase by p (anti-phase), resulting in dark spot. The fringes are localised in the air-film itself. The fringes are loci of constant thickness.
The thickness t(x,y) is related to the radius of curvature R of the spherical surface as
(x2 + y2) +[ R - t(x,y)]2 = R2

Under the approximation that R>> t(x,y), we obtain
2t (x,y) =  x2 + y2

R

The condition of interference thus becomes
x2 + y2 = (2m-1)λR

2
The interference fringes are circular with center at the point of contact. The fringes are loci of constant thickness. The radius of the circular fringes increases as the square-root of the natural number. The radius rm of the mth bright fringe is given as
rm2 = (2m-1)λR

2
Similarly the radius of (m+n)th fringe will be
rm+n2 = [(2m+n)-1] λR

2
From these two equations we can obtain the radius of curvature R of the surface of plano-convex lens as
R = rm+n2 - rm2

n λ
This can be rewritten by writing the diameters Dm and Dm+n of mth and (m+n)th fringes as
R = Dm+n2 - Dm2

4n λ

Given the refractive index of the material (glass) of the plano-convex lens, its focal length is calculated from the relation

  1 =  (μ-1) 1 or      f = R



f R (μ-1)

Further this experiment can be modified to measure the thickness of a thin sheet/paper, either transparent or opaque. Simplest method would be to use two flats with their flat surfaces in contact. In an ideal situation, there should be a single black fringe. If the sheet whose thickness is to be measured is inserted between the two plates. This would create an air-film of linearly varying thickness. Since the fringes are loci of constant thickness, straight line fringes are formed. The spacing of the fringes is related to the thickness.

 

Procedure:
Travelling Microscope is used for the measurement. Experimental set-up for Newton’s ring experiment is given below:

 
Light from the sodium lamp illuminates the plano-convex lens and flat arrangement via a beam-splitter. Ravelling microscope is focussed on to the film so that fringes with high contrast are visible. The cross-wire of the eye-piece can be centered on the fringe center and traverse of the microscope is so arranged that one line of the cross-wire always passes through the center of the fringe pattern. Since the diameters of the fringes are to be measured, this requirement meets this condition. Measurement is made from one end of the fringe pattern and the fringes are lebeled properly.

S.No.

Fringe order

Reading from left of center

Reading to the right of center

Diameter

Dm+n2 - Dm2

1

m+n

 

 

 

 

2

m+n-1

 

 

 

 

3

m+n-2

 

 

 

 

 

 

 

 

 

 

 

m+n-n

 

 

 

 

 

m-1

 

 

 

 

 

m-2

 

 

 

 

 

m-3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1          ↑↓

 

 

 

 

 

 


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