1st Semester Notes for B.tech

Engineering Physics: Notes of Interference and Diffraction | Part 1

Light as the electromagnetic wave may not require any medium to travel. It has:

(i)  Electric Field(E)    

(ii) Magnetic Field(B)

Light contains 'quanta' of the photon which doesn't use any medium to travel.

WaveFront  of light

The locus of all points having the same phase at a given instant of time is known as a wavefront. It depends on the shape of the source of interference and always normal to the light rays. It doesn't propagate in the backward direction.

Types of wavefront:

1. Plane Wavefront

2. Spherical Wavefront

3. Cylindrical Wavefront

Huygen's Theory:

Huygen's Principle state that:

Each point of a given wavefront is the source of secondary wavelets (The new source from each point) spreads in all directions with the speed of the wave.

The tangent of secondary wavelets by drawing lines formed the new wavefront.

AB: the position of a wavefront at t=0

Circle around sources is secondary wavelets

Coherent Source:

When the waves emitted from two sources having the same frequency and constant phase difference are known as Coherent source.

It's of two types:

• Temporal coherence
• Spatial coherence

Interference of Light:

When two monochromatic(single frequency) light from different coherent sources proceeding in the same direction, superpose to form alternate bright and dark fringes is known as the interference of light.

Types of interference:

1. Division of wavefront: Young's Double Slit
2. Division of amplitude: Newton's ring

1. Young's Double Slit

S:  line source of monochromatic light(emit plane wavefront)

MN: Double Slit Instrument

S1S2: Two slits in MN

 d: Distance b/w S1 and S2

AB: Screen where interference pattern observe

D: distance b/w AB and MN

O: Point on AB which is equidistant from S1 and S2 

P: Point on AB where interference pattern will be tested

x: distance b/w O and P

The path difference (x) between two sources meeting at point P

    △x= S2P-S1P

    S1P= D² + (x-d/2)²

    S2P= D² + (x+d/2)²

∴   S2P²-S1P² = 2xd

    (S2P+S1P)(S2P-S1P)= 2xd

    S2P ≃S1P ≃D

    S2P-S1P = xd/D

Interference Fringes:

Distance between two consecutive bright and dark fringes

For Constructive Interference (bright fringe),

        β= xn+1 - xn 

          = (n+3/2)λD/d - (n+1/2)λD/d

        β= λD/d 

For Destructive Interference (dark fringe),

         β= xn+1 - xn 

          = (n+1)λD/d - nλD/d

        β= λD/d

Intensity Distribution:

I= I1+I2+2√I1I2 Cos(△Φ)   i.e△Φ = constant

For interference,

    I1=I2=Io 

Intensity of monochromatic light,

    I= 2Io(1+cos(△Φ)

     = 4Iocos(△Φ/2)

2. Newton's Ring

In 1717, Newton first demonstrated newton's ring

L: Plano-convex lens

S: Source of Monochromatic light

L1: Convex lens mounted vertically to L and P

P: Plane glass Plate 

O: Point of contact between L and P

Newtons ring for reflected light:

Path difference between two interfering reflected waves

        d= 2μt±λ/2

At P, t=0

        path difference(d)= λ/2

The central spot is dark

For bright fringe

    Path difference (d)= 2μt±λ/2= 2nλ/2

or,    d= 2μt=(2n±1)λ/2    where  n=1,2,3,....

For dark fringe

     Path difference (d)= 2μt±λ/2= (2n±1)λ/2 

or,    d= 2μt=2nλ/2    where  n=1,2,3,....

Now from the figure of Newton's ring for reflected light

    R²= rn² + (R-t)

    t= rn²/2R

Putting the value of t in bright fringe

    2μ(rn²/2R)=(2n±1)λ/2

    rn²= R(2n+1)λ/2μ 

The diameter of nth ring:

    Dn²= 2R(2n+1)λ/μ  

for air μ= 1 

    Dn²= 2R(2n+1)λ

The diameter of (n+m)th ring:

     Dn+m²= 2R(2n+2m+1)λ

     Dn+m² - Dn²=4Rmλ

      λ = Dn+m² - Dn²/4Rm

Putting the value of t in dark fringe

    2μ(rn²/2R)=nλ

     rn²=Rnλ/μ 

 The diameter of (n+m)th ring:

     Dn+m² - Dn²=4R(n+m)λ-4Rnλ

      λ = Dn+m² - Dn²/4Rm

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