physics-class-12-unit-10-chapter-04-optics-interference-with-coherent-and-incoherent-waves-l-4-9-liq0hwkslaw

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Young's Experiment </primary>
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- $S_1$ and $S_2$ are point sources, drawn from the same wavefront (or phasefront )

- $\Rightarrow S_1$ and $S_2$ are. "in phase" 
 
 - $\psi_1=a_1 \cos \omega t, \psi_2=a_2 \cos \omega t$

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<footer style = "font-size: 18px"> $\rightarrow$ Optics Interference with Coherent and Incoherent Waves $\rightarrow$ <span style = "color:blue"> Young's Experiment </span> $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Condition for Bright and Dark Fringes </primary>
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- At $P$,

- $ \psi_1=a_1 \cos \left(k r_1-\omega t\right) $

- $ \psi_2=a_2 \cos \left(k r_2-\omega t\right)$

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

- $\left(r_2-r_1\right) \rightarrow \text { Path Difference }$

- $\left(r_2-r_1\right)= \pm n \lambda \rightarrow \text { Bright fringes } $

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<footer style = "font-size: 18px">Optics Interference with Coherent and Incoherent Waves $\rightarrow$ Young's Experiment $\rightarrow$ <span style = "color:blue"> Condition for Bright and Dark Fringes </span> $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes </footer>

- $ \pm\left(n+\frac{1}{2}\right) \lambda \rightarrow \text { Dark fringes } $

& n=0,1,2...

- At the point $O, r_1 \rightarrow r_{1}0, r_2 \rightarrow r_{2}0 ; r_{1}0=r_{2}0$

- $\Rightarrow$ Path Difference $=0$

- Zeroth order bright fringe.

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<footer style = "font-size: 18px">Young's Experiment $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ <span style = "color:blue"> Condition for Bright and Dark Fringes </span> $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes </footer>

- If ' $S$ ' is slightly off-axis:

- At $P$,

- $ \psi_1=A_1 \cos \left(k r_1-\omega t\right) $

- $ \psi_2=A_2 \cos \left(k_2-\omega t+\Delta \phi\right)$

- $\Delta \phi$ is the initial phase difference.

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

- At $O, r_1=r_2$ but $\delta \neq 0$; $\Rightarrow$ Zeroth order bright fringe will not appear at $O$.

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<footer style = "font-size: 18px">Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ <span style = "color:blue"> Condition for Bright and Dark Fringes </span> $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ Path Difference </footer>

- $S_1 = cos \omega t$

- $S_2 = cos \omega (t- \Delta t)$

- [ $\omega t - \omega \Delta t$]

- [$\omega \Delta t = \Delta \phi$]

- $\omega t \- (\omega t - \Delta \phi)$

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<footer style = "font-size: 18px">Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ <span style = "color:blue"> Condition for Bright and Dark Fringes </span> $\rightarrow$ Path Difference $\rightarrow$ Path Difference </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Path Difference </primary>
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- For $\delta = 0 , k(r_2-r_1) =-\Delta \phi$

- $\Rightarrow r_2<r_1$

- $\delta = 0 $ at the point O' such that

- $S' S_1 + S_1 O' = S' S_2 + S_2 O'$

- The position of the zeroth order maxima the central fringe will be shifted

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<footer style = "font-size: 18px">Condition for Bright and Dark Fringes $\rightarrow$ Condition for Bright and Dark Fringes $\rightarrow$ <span style = "color:blue"> Path Difference </span> $\rightarrow$ Path Difference $\rightarrow$ Fringe width </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Fringe width </primary>
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- For the $n^{\text {n }}$ Bright fringe, if $\Delta \phi$ is a constant, Path Difference $\left(r_2-r_1\right)=\frac{x_n^{\prime} d}{D}=n \lambda-C$, --(i)

- where $C=\left(\frac{\Delta \phi}{2 \pi}\right) \lambda$ is a constant.

- For the $(n+1)^{\text {th }}$ fringe, $\frac{x_{n+1}^{\prime} d}{D}=(n+1) \lambda-c-(ii)$

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

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Fringe Shift </primary>
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- Fringe Shift, $\Delta x$ (due to Phase Difference $\Delta \phi$ )

- If $x_n^{\prime}$ is the new position of the $n^{\text {th }}$ order Bright fringe, in the presence of a constant phase difference $\Delta \phi$ between $S_1$ and $S_2$,

- $\frac{x_n^{\prime} d}{D}=n \lambda-C=\frac{x_n d}{D}-c $

- $ \therefore\left(x_n-x_n^{\prime}\right) \frac{d}{D}=C=\left(\frac{\Delta \phi}{2 \pi}\right)^\lambda $

- or Fringe shift (of the $n^n$ fringes) , $\Delta x_n = \frac{\Delta \phi}{2 \pi}) \frac{\lambda D}{d}=\frac{\Delta \phi}{2 \pi} \beta$

- which is independent of $n$.

- Thus, Fringe shift $\Delta x=\left(\frac{\Delta \phi}{2 \pi}\right) \beta$

-depends on $\Delta \phi$

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

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Central Fringe </primary>
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- (Geometry of the problem)

- $ s s^{\prime}=l $

- $ O O^{\prime}=x^{\prime} $

- $ s_1 s_2=d$

- For the central fringe(path difference =0)

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<footer style = "font-size: 18px">Fringe width $\rightarrow$ Fringe Shift $\rightarrow$ <span style = "color:blue"> Central Fringe </span> $\rightarrow$ Central Fringe $\rightarrow$ Fringe Shift due to Source Offset </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Central Fringe </primary>
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- $s^{\prime} s_1+s_1 O^{\prime}=s^{\prime} s_2+s_2 O^{\prime}$

- $\rightarrow s_2 O^{\prime}-s_1 O^{\prime}=s^{\prime} s_1-s^{\prime} s_2$

- $ \Rightarrow \frac{x'}{D}=\frac{l}{L} \rightarrow \theta^{\prime}=\theta$

- $\rightarrow \mid r_2-r_1=q_1-q_2 $

- $\frac{x^{\prime} d}{D}=\frac{l}{L} d $

- $ \rightarrow s^{\prime} o^{\prime}$ is a straight line passing through M.

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<footer style = "font-size: 18px">Fringe Shift $\rightarrow$ Central Fringe $\rightarrow$ <span style = "color:blue"> Central Fringe </span> $\rightarrow$ Fringe Shift due to Source Offset $\rightarrow$ Fringe Shift due to Source Offset </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Fringe Shift due to Source Offset </primary>
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- $l$ - Source offeat $\Delta x=x^{\prime}-0-$ Fringe shift
- $\frac{x^{\prime}}{D}=\frac{l}{L}$
- or $\Delta x=\left(\frac{D}{L}\right) l$ is the fringe shift doe to source offset by $l$.

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<footer style = "font-size: 18px">Central Fringe $\rightarrow$ Central Fringe $\rightarrow$ <span style = "color:blue"> Fringe Shift due to Source Offset </span> $\rightarrow$ Fringe Shift due to Source Offset $\rightarrow$ Fringe Shift due to Source Offset </footer>

- **Example:** $D=100 \mathrm{cm}$, L $=10 \mathrm{cm}$ For an offset of $1 \mathrm{mm}, \Delta x=\frac{109}{10} \times 1 \mathrm{mm}=10 \mathrm{mm}$

- **Ans:** Fringe shift will also occur if the aperture Plate QQ' is offset with respect to 50 . This type of fringe shift is called Lateral shift

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<footer style = "font-size: 18px">Central Fringe $\rightarrow$ Fringe Shift due to Source Offset $\rightarrow$ <span style = "color:blue"> Fringe Shift due to Source Offset </span> $\rightarrow$ Fringe Shift due to Source Offset $\rightarrow$ Fringe Shift </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Fringe Shift </primary>
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- **Summary: Fringe Shift $\Delta x $ due to $\Delta \phi $**

- **1.** If there is a constant Phase Difference $\Delta \phi$ (including $\Delta \phi=0$ ) between the interfering waves, then there can be sustained (observable) Interference fringes.

- **2.** Waves with a constant $\Delta \phi $ among them are called "Coherent Waves"

- $\Delta \phi=0 \Rightarrow $ Waves are in phase"

- $\Delta \phi = \pi \rightarrow $ Waves are "out of phase"

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<footer style = "font-size: 18px">Fringe Shift due to Source Offset $\rightarrow$ Fringe Shift due to Source Offset $\rightarrow$ <span style = "color:blue"> Fringe Shift </span> $\rightarrow$ Problem-1 $\rightarrow$ Problem-2 </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Problem-1 </primary>
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- **2.** The "central fringe" ( and all other fringes) would shift by an amount $\Delta x \propto \Delta \phi$, but the "fringe pattern" and "fringe width" does not change.

- $\Delta x=\left(\frac{\Delta \phi}{2 \pi}\right) \beta$

- **Q:** Then how to measure the fringe shift, in practice? (Particularly when $\Delta \phi$ is several times $2 \pi$ )

- **Ans:** Use of "white light interference"

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<footer style = "font-size: 18px">Fringe Shift due to Source Offset $\rightarrow$ Fringe Shift $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Problem-2 $\rightarrow$ Interference </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Interference </primary>
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<footer style = "font-size: 18px">Problem-2 $\rightarrow$ Interference $\rightarrow$ <span style = "color:blue"> Interference </span> $\rightarrow$ Extended Source $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Interference Between Two Different wave length </primary>
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- $\psi_1 = A_1 sin \omega_1 t$

- $\psi_2 = A_2 sin \omega_2 t$

- $\omega_2 = 2 \pi f_2$

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<footer style = "font-size: 18px">Extended Source $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Interference Between Two Different wave length </span> $\rightarrow$ Interference Between Two Different wave length $\rightarrow$ Interference </footer>

- $ A_1 sin \omega_1 t$

- $ A_2 sin \omega_2 t$

- $\Delta \phi (t)$

- $\Delta \phi =(\omega_2 - \omega_1) $ t

- $f_1 , f_2 \approx 10^{14} H_2 \Delta \phi (t)$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Interference Between Two Different wave length $\rightarrow$ <span style = "color:blue"> Interference Between Two Different wave length </span> $\rightarrow$ Interference $\rightarrow$ Interference with a Source </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Interference with a Source </primary>
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- **Emitting Multiple Wavelengths**

- Consider 3 wavelengths:

- $ \lambda_1=400 \mathrm{nm} \text { (near Blue) } $

- $ \lambda_2=500 \mathrm{nm} \text { (near Green) } $

- $ \lambda_3=600 \mathrm{nm} \text { (near Orange) }$

- Reveal :
'Blue' interferes with'Blue'. only,

- Interference in the Young's double -hole arrangement,

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<footer style = "font-size: 18px">Interference Between Two Different wave length $\rightarrow$ Interference $\rightarrow$ <span style = "color:blue"> Interference with a Source </span> $\rightarrow$ Interference with a Source $\rightarrow$ Interference with Multiple Wavelengths </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Interference with a Source </primary>
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- $\beta=\left(\frac{D}{d}\right)^\lambda$. Assume $D=1 \mathrm{m}, d=1 \mathrm{mm}$

- For,

- $ \lambda_1=400 \mathrm{nm}  $

- $ \lambda_2=500 \mathrm{km}  $

- $ \lambda_3=600 \mathrm{nm}, $

- $ \beta_1=0.4 \mathrm{mm} $

- $ \beta_2=0.5 \mathrm{mm} $

- $ \beta_3=0.6 \mathrm{mm} $

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<footer style = "font-size: 18px">Interference $\rightarrow$ Interference with a Source $\rightarrow$ <span style = "color:blue"> Interference with a Source </span> $\rightarrow$ Interference with Multiple Wavelengths $\rightarrow$ White Light Interference Pattern </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> Interference with Multiple Wavelengths </primary>
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- $\lambda $ = 400nm

- $\lambda $ = 500nmn

- $\lambda $ = 600nm

- Maxima and minima occur at different positions for different wave length, except the central maximum - some position for all colours.

- What to expect if the source is white light?

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<footer style = "font-size: 18px">Interference with a Source $\rightarrow$ Interference with a Source $\rightarrow$ <span style = "color:blue"> Interference with Multiple Wavelengths </span> $\rightarrow$ White Light Interference Pattern $\rightarrow$ White Light Interference </footer>

### <Success style="font-size:25px"> Optics Interference With Coherent And Incoherent Waves L-4 </Success>
### <primary style="font-size:24px"> White Light Interference Pattern </primary>
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<footer style = "font-size: 18px">Interference with a Source $\rightarrow$ Interference with Multiple Wavelengths $\rightarrow$ <span style = "color:blue"> White Light Interference Pattern </span> $\rightarrow$ White Light Interference $\rightarrow$ Thank You </footer>