Theory of Air Wedge

THEORY OF AIR WEDGE

Let us consider two plane surfaces OA and OB inclined at an angle θ, enclosing a wedge shaped air film as shown in Fig 8.1 Let μ be the refractive index of the enclosing film inbetween the two plane surfaces OA and OB. The thickness of the film increases from O to A.

When this thin air film is illuminated by the monochromatic source of light, interference occurs between the rays reflected from upper and lower surface of the film. The interfering rays do not enter the eye parallel but they appear to diverge from a point near the film.

Let 't_f' be thickness of the film at a distance x from the edge as shown in Fig. 8.1.

We know the path difference for the reflected light Δ=2μt_f cosr.

For normal incidence r=90°; cos90°=1

The path difference (Δ) = 2μt_f

If θ is very small then we can write t_f = xθ

The path difference (Δ) = 2μt_f = 2μxθ        …....(1)

We know the condition for the formation of bright fringes due to the reflected light is

The path difference (Δ) = (2n+1) λ/2         ………...(2)

From equations (1) and (2), we can write

The path difference 2μxθ = (2n+1) λ/2

Since the film enclosed is an air medium, the refractive index for air (μ)=1.

∴ The path difference 2xθ = (2n+1) λ/2   ……....(3)

Similarly, we know the condition for dark fringes due to reflected light is

The path difference Δ = nλ   ……....(4)

∴ From equations (1) and (4), we can write

The path difference 2μxθ = nλ

Since μ=1 (for air), we have

 2xθ = nλ                ....(5)

Fringe width

Case (i): Let us consider the dark fringe from the edge O. Let x_n be the distance of the n^th dark fringe of thickness (t_n) and x_(n+1) be the distance of the (n+1)^th dark fringe of thickness (t_n+1), as shown in Fig. 8.2. Then from eqn.(5), we can write,

x_n = nλ / 2θ            …………..(6)

and

x_n+1 = (n+1)λ / 2θ            …………..(7)

Fringe width (β), which is the distance between any two consecutive bright (or) dark fringes can be got by subtracting eqn.(6) from eqn.(7).

 ∴ β = x_(n+1) ‒ x_n

 = [ (n + 1)λ / 2θ ] ‒ [ nλ / 2θ ]

(or) Fringe width β = λ / 2θ          …………..(8)

Case(ii): Similarly if we consider any two consecutive bright fringes, then too the fringe width β will be the same.

i.e., From equation (3), we can write

β = λ / 2θ          …………..(9)