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**1. Interference**

By superposition of waves of same frequency and constant phase difference (coherent waves) redistribution of energy takes place so intensity is maximum at certain points and minimum at others. This phenomenon is called interference.

**2. Principle of superposition**

y = y_{1} + y_{2} + …….. + y_{n}

For two waves y_{1} = a sin ωt

and y_{2} = b sin (ωt + Φ)

Resultant Amplitude A = (a^{2} + b^{2} + 2ab cos Φ)^{1/2}

tan θ = \(\frac{b \sin \phi}{a+b \cos \phi}\)

(θ → phase difference between y_{1} & y_{2})

Intensity I ∝ A^{2}

∴ I = I_{1} + I_{2} + 2\(\sqrt{I_{1} I_{2}}\) cosΦ

If a = b = a_{0}, then I_{1} = I_{2} = I_{0}

∴ A = 2a_{0} cos \(\left(\frac{\phi}{2}\right)\)

and I = 4I_{0} cos^{2}\(\left(\frac{\phi}{2}\right)\)

**3. Constructive interference**

(a) Condition – Phase difference Φ = 0, 2π,…. 2nπ

path difference x = 0, λ, 2λ ……… nλ.

A = A_{max} = (a + b),

I = I_{max} = \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}\) = (a + b)^{2}

(b) If a = b = a_{0}, I_{1} = I_{2} = I_{0}

A_{max} = 2a_{0} and I_{max} = 4I_{0}

**4. Destructive interference**

(a) Condition – Phase difference

Φ = π, 3π, = (2n+ 1)π

path difference

x = \(\frac{\lambda}{2}, \frac{3 \lambda}{2}, \ldots . .=(2 n+1) \frac{\lambda}{2}\)

A = A_{min} = (a – b)

I = I_{min} = \(\left(\sqrt{\mathrm{I}_{1}}-\sqrt{\mathrm{I}_{2}}\right)^{2}\) = (a – b)^{2}

(b) If a = b = a_{0}, I_{1} = I_{2} = I_{0}

A_{min} = 0, I_{min} = 0

**5. Coherent source**

Two identical source which have

- Same amplitude.
- Same frequency.
- Constant phase difference, are called coherent source.

**6. Methods of obtaining coherent sources**

- By division of wavefront and
- By division of amplitude

**7. Conditions for sustained interference**

- Coherent waves
- Amplitudes nearly equal for better contrast and
- Same polaristion

**8. Formation of fringes and fringe width (young’s double slit exp.)**

(a) Fringe width

β = \(\frac{\lambda \mathrm{D}}{\mathrm{d}}\) (same for bright and dark fringes)

(b) Position of n^{th} bright fringe

x_{n} = n \(\frac{\lambda \mathrm{D}}{\mathrm{d}}\)

(c) Position of n^{th} dark fringe

x_{n}‘ = (2n – 1) \(\frac{\lambda \mathrm{D}}{\mathrm{d}}\)

(d) Angular width of fringe

θ = \(\frac{\beta}{D}=\frac{\lambda}{d}\)

(e) Fringe visibility or contrast

Q = \(\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}=\frac{2 \sqrt{I_{1} I_{2}}}{I_{1}+I_{2}}\)

(f) Shape of fringes in space is hyperbolic and the intercept on screen gives straight line fringes.

(g) If whole exp. is done in water then p reduces, (β ∝ \(\frac{1}{\mu}\))

**9. Effect of thin sheet placed in the path of one of the waves**

Thin sheet introduces an extra path difference = (µ – 1) t.

If central fringe is shifted to position of n^{th} bright fringe then

(µ – 1) t = nλ

If central fringe is shifted to position of n,h dark fringe then

(µ – 1) t =(2n – 1)\(\frac{\lambda}{2}\)

**10. Fresnel’s biprism**

(a) Coherent sources produced by refraction from the two parts of biprism

(b) d = 2a(µ – 1)α

D = (a + b), µ is refractive index of prism of angle α.

(c) d = \(\sqrt{\mathrm{d}_{1} \mathrm{d}_{2}}\) , d_{1} and d_{2} are distances between images of the coherent sources in the two conjugate positions of lens.

(d) β = \(\frac{\lambda \mathrm{D}}{\mathrm{d}}\), λ = \(\frac{\beta d}{D}\)

(e) If whole apparatus is placed in a medium of refractive index _{a}µ_{m} then both d and λ are affected-

β_{m} = \(\frac{\lambda_{m} D}{d_{m}}\)

λ_{m} = \(\frac{\lambda_{\mathrm{air}}} { }_{\mathrm{a}} \mu_{\mathrm{m}}}\)

d_{m} = 2a(_{m}µ_{g} – 1)α

(f) If whole exp. is done in water the β increases.

**11. Interference by thin films**

(a) Interference occurs between the waves reflected from the upper and lower surfaces of the film

(b) Effective path difference introduced in reflected waves

x = 2µt cos r + \(\frac{\lambda}{2}\) = nλ (max)

= (2n – 1) \(\frac{\lambda}{2}\) (min)

∴ 2µt cos r = (2n – 1) \(\frac{\lambda}{2}\) (max)

= nλ (min)

(c) If t ≈ 0, very thin film, x ≈ 0, film appears dark in reflected light.

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