physics-class-12-unit-09-chapter-09-refraction-througha-prism-and-dispersion-ray-optics-l-9-9-ler-scvuzxg

### <Success style="font-size:25px"> Refraction Througha Prism And Dispersion Ray Optics L-9 </Success>
### <primary style="font-size:24px"> Refraction Through a Prism </primary>
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- Successive refractions at two planar interfaces at an angle.

- $\angle{A}$ - Angle of the prism

- $i -$ Angle of incidence

- $e -$ Angle of emergence

- $D -$ Angle of deviation

- $N_1 -$ Normal to the refracting interface. AB

- $N_2 -$Normal to the refracting interface. AC

- $B C -$ usually, ground surface

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<footer style = "font-size: 18px"> $\rightarrow$ Refraction Through a Prism and Dispersion Ray Optics $\rightarrow$ <span style = "color:blue"> Refraction Through a Prism </span> $\rightarrow$ Refraction Through a Prism $\rightarrow$ Refraction Through a Prism </footer>

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- Reversing the direction of the ray:

- $e \rightarrow i$

- $i \rightarrow e$

- No change in $D$.

- if: $D=(i+e)-A$

- I**f we reverse the direction of propagation of the ray, $i$ and $e$ would get interchanged; but $D$ remains the same.**

- $\Rightarrow$ For two different values of $i, D$ will be the same.

- $\therefore$ There must be a point of degeneracy (i.e. $i=e)$

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<footer style = "font-size: 18px">Refraction Through a Prism and Dispersion Ray Optics $\rightarrow$ Refraction Through a Prism $\rightarrow$ <span style = "color:blue"> Refraction Through a Prism </span> $\rightarrow$ Refraction Through a Prism $\rightarrow$ Refraction Through a Prism </footer>

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### <primary style="font-size:24px"> Refraction Through a Prism </primary>
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- $\frac{\sin i}{\sin r_1}=\frac{n_2}{n_1}-I^{st} \text{interface }$
 
- $\frac{\sin r_2}{\sin e}=\frac{n_1}{n_2}-II^{nd} \text{interface }$

- **For a given prism, $n_2$ and $A$ are known.**

- $\therefore$ For each angle $i, r_1$ and hence $r_2\left(=A-r_1\right)$, and then ' $e$ ' can be determined.

- $\Rightarrow$ D=(i+e)-A can be calculated.

- As discussed, by the reciprocity of light propagation, $i$ and $e$ are interchangeable.

- $\Rightarrow$ for each $D$, there will be two values of $i$.

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<footer style = "font-size: 18px">Refraction Through a Prism $\rightarrow$ Refraction Through a Prism $\rightarrow$ <span style = "color:blue"> Refraction Through a Prism </span> $\rightarrow$ Refraction Through a Prism $\rightarrow$ Refraction Through a Prism </footer>

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### <primary style="font-size:24px"> Refraction Through a Prism </primary>
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- $ D=i+e-A $

- $ D_m=2 i-A $

- $ i=\frac{A+D _m}{2}-(1) $

- $ \text { When } i=e, $

- $ r_1=r_2=r(\operatorname{ray}) $

- $ \Rightarrow r=\frac{A}{2} -(2) $

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- Qualitative-plot of D vs i for a typical prism of

- $A=60^{\circ}, n=1.5$

- ${\text {Using (1) and (2), }}$

- $ \frac{\sin i}{\sin r}=\frac{n_ 2}{n_ 1}=\frac{\sin \left(\frac{A+D_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}-(3)$

- Usually, $n=n_{\text {air }}=1$

- $n_2=n=$ refractive index of prism

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<footer style = "font-size: 18px">Refraction Through a Prism $\rightarrow$ Refraction Through a Prism $\rightarrow$ <span style = "color:blue"> Refraction Through a Prism </span> $\rightarrow$ Refractive Index of A Prism $\rightarrow$ Thin Prisms </footer>

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### <primary style="font-size:24px"> Refractive Index of A Prism </primary>
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- $ n=\frac{\sin \left(\frac{A+D_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

- $ \text { A - Angle of the Prism} $

- $ D_m \text { - Angle of minimum } $

- $ \text { Deviation } $

- Experiment with a Spectrometer

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<footer style = "font-size: 18px">Refraction Through a Prism $\rightarrow$ Refraction Through a Prism $\rightarrow$ <span style = "color:blue"> Refractive Index of A Prism </span> $\rightarrow$ Thin Prisms $\rightarrow$ Exercise </footer>

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### <primary style="font-size:24px"> Thin Prisms </primary>
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- $A$ is small, thickness of the medium is small

- $\therefore D$ is also small

- $n= \frac{\sin(\frac{A+D_m}{2})}{\sin (A / 2)}$

- $ \approx \frac{(A+D_m) / 2}{A / 2}=1+\frac{D_m}{A} $

- $ \text { or } D_m=(n-1) A$

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<footer style = "font-size: 18px">Refraction Through a Prism $\rightarrow$ Refractive Index of A Prism $\rightarrow$ <span style = "color:blue"> Thin Prisms </span> $\rightarrow$ Exercise $\rightarrow$ Solution </footer>

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### <primary style="font-size:24px"> Exercise </primary>
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- Consider a glass prism of equilateral triangular cross section and refractive index 1.56:

- (i) What should be the angle of incidence (on the refracting surface) for a ray so that $-$ angle of incidence = angle of emergence?

- (ii) If the prism is immersed in water $(n=1.33)$, what would be the angle of minimum deviation?

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<footer style = "font-size: 18px">Refractive Index of A Prism $\rightarrow$ Thin Prisms $\rightarrow$ <span style = "color:blue"> Exercise </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

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### <primary style="font-size:24px"> Solution </primary>
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- $n_2 = 1.56$

- $A = 60^o$

- $i= e\rightarrow r_1 = r_2 $

- $\frac{A}{2}=30^o$

- (1) $\frac{\sin i}{\sin r_ 1}= \frac{n_ 2}{n_1} = \frac{1.56}{1}$

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<footer style = "font-size: 18px">Thin Prisms $\rightarrow$ Exercise $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

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### <primary style="font-size:24px"> Solution </primary>
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- $n_2 = 1.56$

- $A = 60^o$

- $i= e\rightarrow r_1 = r_2 $

- $\frac{A}{2}=30^o$

- (1) $\frac{\sin i}{\sin r_ 1}= \frac{n_ 2}{n_1} = \frac{1.56}{1}$

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

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### <primary style="font-size:24px"> Dispersion </primary>
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- White light spectrum

- **VIBGYOR**

- $V\rightarrow\sim 400 ~nm$

- $R\rightarrow\sim 650 ~nm$

- $\text{Why does this happen?}$

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<footer style = "font-size: 18px">Solution $\rightarrow$ Dispersion $\rightarrow$ <span style = "color:blue"> Dispersion </span> $\rightarrow$ Dispersion $\rightarrow$ Refractive Index vs Wavelength </footer>

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### <primary style="font-size:24px"> Dispersion </primary>
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* Refractive index of a material depends on the wavelength of light
- $n \equiv n(\lambda)$

* Widely used materials in a glass prism -
  - Crown glass
  - Flint glass
  - Fused quart$_3(Si D_2)$

* Variation of $n$ with wavelength ( $\lambda$ )

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<footer style = "font-size: 18px">Dispersion $\rightarrow$ Dispersion $\rightarrow$ <span style = "color:blue"> Dispersion </span> $\rightarrow$ Refractive Index vs Wavelength $\rightarrow$ Refractive Index vs Wavelength </footer>

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| Colour  |  Wavelength (nm) |  Fused $Quart_3$|  Crown Glass  |  Flint Glass |
| --------| --------         | --------        | --------      | --------     |
| Violet  | 396.9            | 1.470           | 1.533         | 1.663        |
| Blue    | 486.1            | 1.463           | 1.523         | 1.689        |
| Yellow  | 589.3            | 1.458           | 1.517         | 1.627        |
| Red     | 656.3            | 1.456           | 1.515         | 1.622        |

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<footer style = "font-size: 18px">Dispersion $\rightarrow$ Refractive Index vs Wavelength $\rightarrow$ <span style = "color:blue"> Refractive Index vs Wavelength </span> $\rightarrow$ Cauchy's Formula $\rightarrow$ Dispersion Compensation </footer>

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### <primary style="font-size:24px"> Cauchy's Formula </primary>
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- $n$ vs.$\lambda$

- **Cauchy's formula:** $n(\lambda) = A + \frac{B}{\lambda^2}$

- A, B are constants (for given material)

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<footer style = "font-size: 18px">Refractive Index vs Wavelength $\rightarrow$ Refractive Index vs Wavelength $\rightarrow$ <span style = "color:blue"> Cauchy's Formula </span> $\rightarrow$ Dispersion Compensation $\rightarrow$ Formation of Rainbow </footer>

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### <primary style="font-size:24px"> Dispersion Compensation </primary>
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- Compensating for the dispersion effects due to a particular optical component or medium.

- By choosing a second prism of appropriate size and refractive index, it is possible to compensate for the dispersion of the first prism.

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<footer style = "font-size: 18px">Refractive Index vs Wavelength $\rightarrow$ Cauchy's Formula $\rightarrow$ <span style = "color:blue"> Dispersion Compensation </span> $\rightarrow$ Formation of Rainbow $\rightarrow$ Prism </footer>

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- **Example from Nature: Formation of Rainbow**

- Eye will observe red colour making a higher inclination with the horizon; will appear at the upper part of the rainbow.

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<footer style = "font-size: 18px">Cauchy's Formula $\rightarrow$ Dispersion Compensation $\rightarrow$ <span style = "color:blue"> Formation of Rainbow </span> $\rightarrow$ Prism $\rightarrow$ Thin Lenses </footer>

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### <primary style="font-size:24px"> Prism </primary>
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- $n=\frac{\sin \left(\frac{A+ D_m}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

- $n = n(\lambda)$

- $D_m = D_m(\lambda)$

- $n_{Blue} = D_m \text{~not blue}$

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<footer style = "font-size: 18px">Dispersion Compensation $\rightarrow$ Formation of Rainbow $\rightarrow$ <span style = "color:blue"> Prism </span> $\rightarrow$ Thin Lenses $\rightarrow$ Thank You </footer>

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- $D = (n-1) A$

- $D(\lambda) = (n(\lambda)-1)A$

- $(D_{Blue} - D_{Red})\rightarrow$ Very small

- **In the case of mirrors**

- There is no dispersion Light is reflected.

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<footer style = "font-size: 18px">Formation of Rainbow $\rightarrow$ Prism $\rightarrow$ <span style = "color:blue"> Thin Lenses </span> $\rightarrow$ Thank You $\rightarrow$  </footer>