physics-class-12-unit-10-chapter-03-optics-youngs-interference-experiment-l-3-9-s8ifq4mxdng

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Interference </primary>
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* In Optics, it generally refers to 'Two-Beam' or 'Two-Wave' interference, which results in some perceptible and sustained 'fringe pattern:

* 'Fringe Pattern' refers to the light intensity pattern comprising of alternate bright and dark regions in the form of lines or rings.

* Formation of fringes can be explained in terms of superposition of the two waves.

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<footer style = "font-size: 18px"> $\rightarrow$ Optics Youngs Interference Experiment $\rightarrow$ <span style = "color:blue"> Interference </span> $\rightarrow$ Interference $\rightarrow$ Requirements of Interference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
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- To be able to see a sustained fringe pattern,

- 1. The two waves interfering must have the same frequency (or wavelength).

- 2. There should be a constant phase difference between the two waves.

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<footer style = "font-size: 18px">Interference $\rightarrow$ Interference $\rightarrow$ <span style = "color:blue"> Requirements of Interference </span> $\rightarrow$ Schematic of Young's Experimental Setup $\rightarrow$ Superposition of Two Waves </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Schematic of Young's Experimental Setup </primary>
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* $S, S_1$ and $S_2$ and small holes in an opaque screen

* Superposition of waves occurs at each point P

* d = difference between $S_1$ & $S_2$

* Longitudinal Section in the $x-z$ plane

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<footer style = "font-size: 18px">Interference $\rightarrow$ Requirements of Interference $\rightarrow$ <span style = "color:blue"> Schematic of Young's Experimental Setup </span> $\rightarrow$ Superposition of Two Waves $\rightarrow$ Superposition of Two Waves </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
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- At an arbitrary point $P$,

- $ \psi_1=\frac{A_1^{\prime}}{r_1} \cos \left(k r_1-\omega t\right),$

- $ \psi_2=\frac{A_1^{\prime}}{r_2} \cos \left(k r_2-\omega t\right)$

- Let $A_1=\frac{A_1^{\prime}}{r_1^{\prime}}$ and $A_2=\frac{A_2^{\prime}}{r_2}$

- $\psi_{\text {resultant}}=  \psi_1+\psi_2 $

- = $ A_1 \cos \left(k r_1-\omega t\right)+A_2 \cos \left(k r_2-\omega t\right) $

- = $ A_1\left[\cos k r_1 \cos \omega t+\sin k r_1 \sin \omega t\right] $

- $ +A_2\left[\cos k r_2 \cos \omega t+\sin k r_2 \sin \omega t\right]$

- (using $\cos (A-B)=\cos A \cos B+\sin A \sin B$ )

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<footer style = "font-size: 18px">Requirements of Interference $\rightarrow$ Schematic of Young's Experimental Setup $\rightarrow$ <span style = "color:blue"> Superposition of Two Waves </span> $\rightarrow$ Superposition of Two Waves $\rightarrow$ Superposition of Two Waves </footer>

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- $\psi_{\text {res. }}=  \cos \omega t \left(A_1 \cos k r_1+A_2 \cos k r_2\right)+\sin \omega t\left(A_1 \sin k r_1+A_2 \sin k r_2\right)$

- Setting $\left(A_1 \cos k r_1+A_2 \cos k r_2\right)=A \cos \phi$ and $\left(A_1 \sin k r_1+A_2 \sin k r_2\right)=A \sin \phi$

- $\text { with } \tan \phi=\frac{\left(A_1 \sin k r_1+A_2 \sin k r_2\right)}{\left(A_1 \cos k r_1+A_2 \cos k r_2\right)}$

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<footer style = "font-size: 18px">Schematic of Young's Experimental Setup $\rightarrow$ Superposition of Two Waves $\rightarrow$ <span style = "color:blue"> Superposition of Two Waves </span> $\rightarrow$ Superposition of Two Waves $\rightarrow$ Superposition of Two Waves </footer>

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- $\psi_{\text {vas. }}=A \cos \omega t \cos \phi+A \sin \omega t \sin \phi$

- or $\psi_{\text {res. }}= A \cos {\omega t - \phi}$ at the point P

- with $A=\left(A^2 \cos ^2 \phi+A^2 \sin ^2 \phi\right)^{1 / 2}$

- Superposition of Two Waves (contd)

- Intensity at the point P  $ I=\left|\psi_{\text {res }}\right|^2$

- $\quad=A^2 \cos ^2(\omega t-\phi)$

- $\left\langle\cos ^2(\omega t-\phi)\right\rangle-1 / 2$

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<footer style = "font-size: 18px">Superposition of Two Waves $\rightarrow$ Superposition of Two Waves $\rightarrow$ <span style = "color:blue"> Superposition of Two Waves </span> $\rightarrow$ Superposition of Two Waves $\rightarrow$ Superposition of Two Waves </footer>

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- $ \omega=2 \pi \nu \longrightarrow 10^{14}-10^{15} \mathrm{Hz} \text { for light- } $

- $ \cos ^2(\omega t-\phi)$

- $ \<\cos ^2(\omega t-\phi)\>$ time average

- = $\frac {1}{2}$

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- Intensity at the point  P  I= $\left|\psi_{\text {res. }}\right|^2$

- $=A^2 \cos ^2(\omega t-\phi)$

- $\left\langle\cos ^2(\omega t-\phi)\right\rangle-1 / 2$

- time average

- If only the source $S_1$ war present, then intensity of the point $P$ due to $S_1$ is given by

- $I_1=\left|\psi_1\right|^2=A_1^2\left\langle\cos ^2\left(k_1 r_1-\omega t\right)\right\rangle=\underline{\frac{A_1^2}{2}}$

- parallely, Intensity at $p$ due to only $S_2$,

- $I_2=\left|\Psi_2\right|^2=\frac{A_2^2}{2} .$

- Superposition of Two Waves (contd.)

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- where $\delta=k\left(r_2-r_1\right) \rightarrow$ the "phase difference" between the two waves at $P$.

- $\left(r_2-r_1\right)$ is the "path difference" between the two waves, or $I=I_1+I_2+2 \sqrt{I_1} \sqrt{I_2} \cos \delta$ "Interference Equation"

- Superposition of Two Waves (contd.)

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- $\psi_{res.}=  \cos \cos \left(A_1 \cos k r_1+A_2 \cos k r_2\right)$
 - $+\sin \omega t\left(A_1 \sin k r_1+A_2 \sin k r_2\right)$

- setting $\left(A_1 \cos k r_1+A_2 \cos k r_2\right)=A \cos \phi$

- $\text { and }\left(A_1 \sin k r_1+A_2 \sin k r_2\right)=A \sin \phi \text {, }$

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- $\text { with } \tan \phi=\frac{\left(A_1 \sin k r_1+A_2 \sin k r_2\right)}{\left(A_1 \cos k r_1+A_2 \cos k r_2\right)}$

- $\psi_{\text {var. }}=A \cos \omega t \cos \phi+A \sin \omega t \sin \phi$

- or $\psi_{\text {res. }}=A \cos (\omega t-\phi)$ of the point $P$.

- with $A=\left(A^2 \cos ^2 \phi+A^2 \sin ^2 \phi\right)^{1 / 2}$

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<footer style = "font-size: 18px">Superposition of Two Waves $\rightarrow$ Superposition of Two Waves $\rightarrow$ <span style = "color:blue"> Superposition of Two Waves </span> $\rightarrow$ Intensity Distribution(on the Screen) $\rightarrow$ Intensity Distribution(on the Screen) </footer>

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- $I=I_1+I_2+2 \sqrt{I_1} \sqrt{I_2} \cos \delta$

- For $\delta=0, \pm 2 \pi, \pm 4 . \pi, \ldots$.

- $I=I_{\max }=\left(\sqrt{I_1}+\sqrt{I_2}\right)^2$

- For $\delta= \pm \pi, \pm 3 \pi$,

- $I=I_{\min }=\left(\sqrt{I_1}-\sqrt{I_2}\right)^2$

- If $A_1=A_2, I_1=I_2=I_2$ (say),
- then

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<footer style = "font-size: 18px">Superposition of Two Waves $\rightarrow$ Superposition of Two Waves $\rightarrow$ <span style = "color:blue"> Intensity Distribution(on the Screen) </span> $\rightarrow$ Intensity Distribution(on the Screen) $\rightarrow$ Intensity Variation with δ </footer>

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- $ I_{\max }=4 I_0-\varepsilon=0, \pm 2 \pi, 3,4 \pi $

- $ I_{\min }=0-\delta= \pm \pi, \pm 3 \pi, .$

& $I=2 I_0(1+\cos \delta)=1+I_0 \cos ^2\left(\frac{\delta}{2}\right)$

- Schematic of Young's
- Experimental Setup

- $S, S_1$ and $S_2$ are small holes in an opaque screen

- Superposition of waves occurs at each point P

- d = difference between $S_1$ & $S_2$

- Longitudinal Section in the $x-z$ plane

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<footer style = "font-size: 18px">Superposition of Two Waves $\rightarrow$ Intensity Distribution(on the Screen) $\rightarrow$ <span style = "color:blue"> Intensity Distribution(on the Screen) </span> $\rightarrow$ Intensity Variation with δ $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
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- $ I(\delta)=4 I \cdot \cos ^2\left(\frac{\delta}{2}\right) $

- $ \delta=k\left(r_2-r_1\right) $

- Path Difference $\left(r_2-r_1\right)$
- For a point $P$ on the $x$-axis,

- $ r_2-r_1=S_2 p-S_1 p $

- $ S_2 p^2=D^2+(x+d / 2)^2 $

- $ S_1 p^2=D^2+(x-d / 2)^2 $

- $ \therefore S_2 p^2-S_1 p^2=2 x d $

- $ \therefore S_2 p-S_1 p=2 x d /\left(S_2 p+S_1 p\right)$

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<footer style = "font-size: 18px">Intensity Distribution(on the Screen) $\rightarrow$ Intensity Distribution(on the Screen) $\rightarrow$ <span style = "color:blue"> Intensity Variation with δ </span> $\rightarrow$ Path Difference $\rightarrow$ Intensity Maxima and Minima </footer>

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- Path difference $\left(r_2-r_1\right)=\frac{2 x d}{\left(S_1 p+S_2 p\right)}$

- In practice, $D$ : $50-100 \mathrm{~cm}, x$ : far mm d: $0.1-1 \mathrm{mm}$

- Noting that $x, d \ll D, S_1 p+S_2 p \cong 2 D$

- $\therefore$ Path Difference $r_2-r_1=\frac{x d}{D}$

- $\left(r_2-r_1\right) \propto x$, position coordinate of $P$.

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<footer style = "font-size: 18px">Intensity Distribution(on the Screen) $\rightarrow$ Intensity Variation with δ $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Intensity Maxima and Minima $\rightarrow$ Intensity Maxima and Minima </footer>

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- RECALL:

- $I=4 I_0 \cos ^2(\delta / 2)$

- Intensity Maxima are given by -

- $\text { Phase Difference } \delta=\frac{2vv \pi}{\lambda}\left(r_2-r_1\right)= \pm n, 2 \pi$

- with $n=0,1,2, \ldots \ldots$

- or Path Difference $\left(r_2-r_1\right)= \pm n \lambda$ '

- $n$ ' is called 'order' of the maxima.

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<footer style = "font-size: 18px">Intensity Variation with δ $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Intensity Maxima and Minima </span> $\rightarrow$ Intensity Maxima and Minima $\rightarrow$ Path Difference </footer>

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- Intensity Minima ara given by -

- Phase Difference $\delta=\frac{2 \pi}{\lambda}\left(r_2-r_1\right)= \pm(n+1 / 2) 2 \pi$ with $n=0,1,2, \cdots$

- or Path Difference $\left(r_2-r_1\right)= \pm(n+1 / 2) x$

' + 'sign gives positions of maxima on one site of the point $\delta$=0, while. - 'sign given the other ride.

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Intensity Maxima and Minima $\rightarrow$ <span style = "color:blue"> Intensity Maxima and Minima </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

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- Path Difference $\left(r_2-r_1\right)=\frac{2 x d}{\left(S_1 p+S_2 p\right)}$

- In practice, $D$ : $50-100 \mathrm{~cm}, x$ : far mm d: $0.1-1 \mathrm{mm}$

- Noting that $x, d \ll D, S_1 p+S_2 p \cong 2 D$

- $\therefore$ Path Difference $r_2-r_1=\frac{x d}{D}$

- $\left(r_2-r_1\right) \propto x$, position coordinate of $P$.

- Intensity Maxima and Minima

- RECALL:

- $I=4 I_0 \cos ^2(\delta / 2)$

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<footer style = "font-size: 18px">Intensity Maxima and Minima $\rightarrow$ Intensity Maxima and Minima $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

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- $I=4 I_0 \cos ^2(\delta / 2)$

- Intensity Maxima are given by -

- $\text { Phase Difference } \delta=\frac{1 \pi}{\lambda}\left(r_2-r_1\right)= \pm n, 2 \pi$

- with $n=0,1,2, \ldots \ldots$

- or Path Difference $\left(r_2-r_1\right)= \pm n \lambda$ '

- $n$ ' is called 'order' of the maxima.

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<footer style = "font-size: 18px">Intensity Maxima and Minima $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Positions of Maxima and Minima </footer>

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- Intensity Minima ara given by -

- Phase Difference $\delta=\frac{2 \pi}{\lambda}\left(r_2-r_1\right)= \pm(n+1 / 2) 2 \pi$ with $n=0,1,2, \cdots$

- or Path Difference $\left(r_2-r_1\right)= \pm(n+1 / 2) x$

' + 'sign gives positions of maxima on one site of the point $\delta$=0$, while. - 'sign given the other ride.

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ Positions of Maxima and Minima </footer>

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- At the point $0,S_1 0 = S_2 0$ $\Rightarrow\left(r_2-r_1\right)=0, \rightarrow n=0$. It is a paint of maximum intensity, called "zeroth order maxima" or "Central maxima"

- $\rightarrow$ zero path difference point at nest maxima asur when it difference $\left(r_2-r_1\right)=\frac{x_1 d}{D}=\lambda ;(n-1)$

- Position of first order maximum, $x_1=\left(\frac{D}{d}\right)^\lambda$ antral, position of the order maxima is joining } $x_n=n(\frac {D}{d})^\lambda$

- Path Difference $\left(r_2-r_1\right)=\frac{2 x d}{\left(S_1 p+S_2 p\right)}$

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Positions of Maxima and Minima </span> $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ Positions of Maxima and Minima </footer>

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- In practice, $D$ : $50-100 \mathrm{~cm}, x$ :o far mm d: $0.1-1 \mathrm{mm}$

- Noting that $x, d \ll D, c_1 p+S_2 p \cong 2 D$

- $\therefore$ Path Difference $r_2-r_1=\frac{x d}{D}$

- $\left(r_2-r_1\right) \propto x$, position coordinate of $P$.

- Intensity Maxima are given by -

- $\text { Phase Difference } \delta=\frac{1 \pi}{\lambda}\left(r_2-r_1\right)= \pm n, 2 \pi$

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ <span style = "color:blue"> Positions of Maxima and Minima </span> $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ Positions of Maxima and Minima </footer>

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- with $n=0,1,2, \ldots \ldots$

- or Path Difference $\left(r_2-r_1\right)= \pm n \lambda$ '

- $n$ ' is called 'order' of the maxima.

- Intensity Minima ara given by -

- Phase Difference $\delta=\frac{2 \pi}{\lambda}\left(r_2-r_1\right)= \pm(n+1 / 2) 2 \pi$ with $n=0,1,2, \cdots$

- or Path Difference $\left(r_2-r_1\right)= \pm(n+1 / 2) x$

' + 'sign gives positions of maxima on one site of the paint $\delta$=0$, while. - 'sign given the other ride.

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- Separation between adjacent maxima $=x_{n+1}-x_n$ or

- $\beta=(n+1) \frac{D}{d} \lambda-n \frac{D}{d} \lambda=\left(\frac{D}{d}\right)^\lambda$

- Note that $\beta \propto D$ and $\propto \frac{1}{d}$ at a given wavelength.

- Typical: $D=100 \mathrm{~cm}, d=0.3 \mathrm{mm}, \lambda=600 \mathrm{nm}$

- $
\beta=\frac{100 \times 10^{-2} \times 600 \times 10^{-9}}{0.3 \times 10^{-8}} \mathrm{m}=2 \mathrm{mm}
- $

- Similarly, positions of minima are given by -
- $r_2-r_1=\frac{x_m d}{D}= \pm(m+1 / 2) \lambda, \quad m=0,1,2 ;$

- Thus, $\quad x_m=(m+1 / 2)\left(\frac{D}{d}\right)^\lambda$

- $m=0$ gives position of first minima, $m=1$ gives position of second minima

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<footer style = "font-size: 18px">Positions of Maxima and Minima $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ <span style = "color:blue"> Positions of Maxima and Minima </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

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- Path Difference at a point $Q$ in the $x-y$ plane

- Coordinates of positions Q,$ S_1 and S_2$

- Q : (x, y, 0)

- $S_1 : (\frac{d}{2}, 0, -D)$

- $S_2 : (\frac {-d}{2}, 0, -D)$

- Path Difference $\left(r_2-r_1\right)=S_2 Q-s, Q$.

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<footer style = "font-size: 18px">Positions of Maxima and Minima $\rightarrow$ Positions of Maxima and Minima $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

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- Path Difference at the point $Q=S_2 Q - S_1 Q,-\Delta$, say

- $ {[(x+d / 2)^2+y^2+D^2]^{1 / 2}}$ -${(x-d / 2)^2+y^2+D^2]^{1 / 2}-\Delta} $

- or $(x+d / 2)^2+y^2+D^2= [{\Delta+ [(x-d / 2)^2+y^2+D^2]^{1 / 2}\}]^2$

- on simplification, we get

- $\left(d^2-\Delta^2\right) x^2-\Delta^2 y^2=\Delta^2\left[D^2+\frac{1}{2}\left(A^2-\Delta^2\right)\right]$

- for a fixed value of $\Delta$, the above equation is of the form 
- $\quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1  $

- Equation of a Hyperbola.

- where $a$ \& $b$ are constants.

- For different values of $\Delta$, we get different Hyperbola.

- ie. the locus of all points with constant path differing are hyperbola.

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<footer style = "font-size: 18px">Positions of Maxima and Minima $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- d $\sim 0.1 - 1 mm \sim mn$

- $\Delta \rightarrow \lambda , 2\lambda, 3\lambda,$  for $\lambda$

- 600 nm $\rightarrow 0.6 \mu m \sim \mu m$

- d >> $\Delta$

- Path Difference at the point $Q=S_2 Q - S_1 Q,-\Delta$, say

- $ {[(x+d / 2)^2+y^2+D^2]^{1 / 2}}$

-${(x-d / 2)^2+y^2+D^2]^{1 / 2}-\Delta} $

- or $(x+d / 2)^2+y^2+D^2= [{\Delta+ [(x-d / 2)^2+y^2+D^2]^{1 / 2}\}]^2$

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- on simplification, we get

- $\left(d^2-\Delta^2\right) x^2-\Delta^2 y^2=\Delta^2\left[D^2+\frac{1}{2}\left(A^2-\Delta^2\right)\right]$

- for a fixed value of $\Delta$, the above equation is of the form 
- $\quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1  $

- Equation of a Hyperbola.

- where $a$ \& $b$ are constants.

- For different values of $\Delta$, we get different Hyperbola.

- ie. the locus of all points with constant path differing are hyperbola.

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<hr>
<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- Path Difference at the point $Q=S_2 Q - S_1 Q,-\Delta$, say

- $ {[(x+d / 2)^2+y^2+D^2]^{1 / 2}}$

-${(x-d / 2)^2+y^2+D^2]^{1 / 2}-\Delta} $

- or $(x+d / 2)^2+y^2+D^2= [{\Delta+ [(x-d / 2)^2+y^2+D^2]^{1 / 2}\}]^2$

- on simplification, we get

- $\left(d^2-\Delta^2\right) x^2-\Delta^2 y^2=\Delta^2\left[D^2+\frac{1}{2}\left(A^2-\Delta^2\right)\right]$

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Interference Fringes </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- for a fixed value of $\Delta$, the above equation is of the form 
- $\quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1  $

- Equation of a Hyperbola.

- where $a$ \& $b$ are constants.

- For different values of $\Delta$, we get different Hyperbola.

- ie. the locus of all points with constant path differing are hyperbola.

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Interference Fringes $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Interference Fringes </primary>
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- If $\Delta=n \lambda$, then the hyperbola would comprise of all points with intensity maxima, and if $\Delta=\left(n+ \frac {1}{2}\right) \lambda$, then the hyperbola would comprise of all points with intensity minima

- $\Rightarrow$ We will see alternate bright and dark hyperbola on the screen.

- These are called "interference fringes", and the pattern on the screen is called "fringe pattern"

- If the locus of all points with a constant path difference happens to he circles, we would see alternate bright and dark rings" or "circular fringes" on a screen.

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Interference Fringes </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- Path Difference $(r_2 - r_2)$ = $\frac {2xd}{(S_1 P + S_2 P)}$

- In practice D : 50-100 cm, x = few mm

- d : 0.1 - 1 mm

- Noting that x, d << D , $S_1 P + S_2 P$ = 2D

- Path difference $r_2 - r_1 = \frac {xd}{D}$

- $(r_2 - r_1) \propto x $ position coordinates of P.

- Interference Fringes in Young's Experiment

- $\left(d^2-\Delta^2\right) x^2-\Delta^2 y^2=\Delta^2\left[D^2+\frac{1}{4}\left(d^2-\Delta^2\right)\right] $

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Interference Fringes $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Fring Pattern </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- $\quad x={\frac{\Delta}{\left(d^2-\Delta^2\right)^{1 / 2}}\left[D^2+y^2+\frac{1}{4}\left(d^2-\Delta^2\right)\right]^{1 / 2}}$

- In practice, $y \ll D$ ( $y-$ few $\mathrm{mm}$ to $\mathrm{cm}, D \sim 100 \mathrm{~cm})$

- $\therefore y^2$ can be neglected in comparison to $D^2$.

-  or  x=$\frac{\Delta}{\left(d^2-\Delta\right)^{1 / 2}}\left[D^2+\frac{1}{4}\left(d^2-\Delta^3\right)\right]^{1 / 2}$

- $\therefore e . x=$ constant, for each (fixed) value of $\Delta$

- $\Rightarrow$ locus of constant path difference are straight lines parallel to the $y$-axis ($\perp$ to $x$-air)

- $\Rightarrow$ straight line interference fringes II

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<footer style = "font-size: 18px">Interference Fringes $\rightarrow$ Path Difference $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Fring Pattern $\rightarrow$ Fring Width </footer>

### <Success style="font-size:25px"> Optics Youngs Interference Experiment L-3 </Success>
### <primary style="font-size:24px"> Fring Width </primary>
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- The separation between adjacent bright (or dark) fringes is called "fringe width".

- Recall: $\beta=\left(\frac{D}{d}\right)^\lambda$ - separation between maxima on the x-axis

- $x  ={\frac{\Delta}{\left(d^2-\Delta^2\right)^{1 / 2}}}\left[D^2+\frac{1}{4}\left(d^2-\Delta^2\right)\right]^{1 / 2} $

- $ \approx \frac{\Delta D}{d} \text {, as } \Delta \ll d \ll D$

- For $\Delta=0, h^2=0$ - constant
- $\rightarrow y$-axis is the locus of all bright points

- ie. the "central fringe".

- $\Delta-\lambda, \quad x_1=\frac{\lambda D}{d}$, for the first order bright fringe

- $\Delta=2 \lambda, x_2= \frac {2 \lambda D} {d}, 2^{nd} $ bright fringe,

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<footer style = "font-size: 18px">Path Difference $\rightarrow$ Fring Pattern $\rightarrow$ <span style = "color:blue"> Fring Width </span> $\rightarrow$ Fring Width $\rightarrow$ Thank You </footer>