Class 12 Physics Wave Optics | Fringe Pattern |

**Fringe Pattern**

- The alternate dark and red bands which are obtained on the screen are known as fringe pattern and the alternate dark and bright bands are known as fringes.
__Bright Bands:-__- Bright bands are formed as a result of constructive interference and they are the positions of maximum intensity.
- Condition for maximum intensity:-
- Path difference =n λ.
- =>(xd/D) = nλ using Equation(2)
- =>
**x**where x_{n}= ((n λ D)/d)_{n}=position of n^{th}bright band.- When n=0 then it will be
__central bright band__.

- When n=0 then it will be
__Dark Bands:-__- Dark bands are formed by the destructive interference and they are the positions of minimum intensity.
- Condition for destructive interference:-
- Path difference =(n+(1/2))λ
- => (xd/D) = (n+ (1/2)) λ.
- =>
**x**_{n}=(n+ (1/2)) (λD/d)- Where x
_{n}=position of n^{th}dark band.

- Where x

__Graphical representation of fringe pattern__

__Fringe width__:-

- Fringe width is the distance between consecutive dark and bright fringes.
- It is denoted by ‘β’.
- In case of constructive interference fringe width remains constant throughout.
- It is also known as
__linear fringe width__.

__Angular Fringe width:-__

- It is the angle subtended by a dark or bright fringe at the centre of the 2 slits.
- It is denoted by ‘θ’.

- Mathematical Expression for
__fringe width(__β):-- x
_{n}=((nλD)/d) - x
_{n+1 }=(((n+1)λD)/d) - β =x
_{n+1 }- x_{n} - =(((n+1) λD)/d) – ((nλD)/d)
- =>
**β = (λD/d)** - Therefore fringe width depends on:-
- (λ)Wavelength of the light used, (D) distance of the screen from the slits and (d) distance between two slits.

- Mathematical Expression for
__angular fringe width(__θ):-- θ =(β/D)
- θ = (λD)/(d D)
**θ = (λ/d)**

- x

__Conclusion of Young’s double slit experiment__

- Central fringes gets shifted by – θ if the source gets shifted by θ.
- If the source S is shifted by some angle θ, there will be no change in fringe pattern. The central fringe will get shifted in the opposite direction.

- Intensity of the fringes increase if point sources are replaced with slits.
- If there are slits instead of point source then more light waves will be able to pass through the slits.
- As a result stronger wavefronts are formedwhich give rise to even greater intensity fringes.

** Problem:- **In a Young’s double-slit experiment, the slits are separated by0.28 mm and the screen is placed 1.4 m away. The distance betweenthe central bright fringe and the fourth bright fringe is measuredto be 1.2 cm. Determine the wavelength of light used in theexperiment.

** Answer:- **Distance between the slits, d = 0.28 mm = 0.28 × 10

Distance between the slits and the screen, D = 1.4 m

Distance between the central fringe and the fourth (n = 4) fringe,

u = 1.2 cm = 1.2 × 10^{−2} m

In case of a constructive interference, we have the relation for the distance between thetwo fringes as:

u =(n λD)/(d)

Where, n = Order of fringes

= 4 λ = Wavelength of light used

Therefore, λ= (ud/nD)

= (1.2x10^{-2}x0.28x10^{-3})/ (4x1.4)

=6x10^{-7}

=600nm

Hence, the wavelength of the light is 600 nm.

** Problem:- **In Young’s double-slit experiment using monochromatic light of wavelengthλ, theintensity of light at a point on the screen where path difference is λ, is K units. What isthe intensity of light at a point where path difference is λ /3?

** Answer:- **Let I

Where,

φ = Phase difference between the two waves

For monochromatic light waves,

I_{1}=I_{2}

Therefore I’ = I_{1} + I_{2} +2√ I_{1} I_{2} cosφ

=2 I_{1} + 2I_{1}cosφ

Phase difference =(2π/λ) x Path difference

Since path difference = λ,

Phase difference,φ =2π

Therefore, I’ =2 I_{1} + 2I_{1} = 4I_{1}

Given,

I’ = K

Therefore, I_{1} = (k/4) (equation (1))

When path difference= λ /3,

Phase differenceφ = (2π/3),

Hence, resultant intensity,

I_{R} =I_{1} + I_{1}+2√ I_{1} I_{2} cos(2π/3),

=2I_{1}+2I_{1}(-1/2) =I_{1}

Using equation (1), we can write:

I_{R} = I_{1} = (k/4)

Hence, the intensity of light at a point where the path difference is λ /3 units is (k/4) units.

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