physics-class-12-unit-09-chapter-05-refraction-at-spherical-surfaces-and-by-lenses-ray-optics-l-5-9-vl-0ayt-jsi

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Refraction at a Spherical Surface </primary>
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- Assume 'small aperture' $\Rightarrow$ Paraxial  is valid.

- i.e. $M$ is close to $P$

- $\Rightarrow \alpha, \beta, \gamma, i$ and r are small

- $\Rightarrow \tan \alpha \approx \sin \alpha \approx \alpha$.

- $\tan \beta \approx \sin \beta \approx \beta$, etc.

-  Refracted light can ba minimized by using Anti-reflection coatings.

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<footer style = "font-size: 18px"> $\rightarrow$ Refraction at Spherical Surfaces and by Lenses Ray Optics $\rightarrow$ <span style = "color:blue"> Refraction at a Spherical Surface </span> $\rightarrow$ Paraxial Approximation $\rightarrow$ Refraction at a Spherical Surface </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Paraxial Approximation </primary>
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<footer style = "font-size: 18px">Refraction at Spherical Surfaces and by Lenses Ray Optics $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ <span style = "color:blue"> Paraxial Approximation </span> $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ Refraction at a Spherical Surface </footer>

- Angles

- $ i=\alpha+\beta $

- $ r=\beta-\gamma $

- $ \alpha \approx \tan \alpha=\frac{M D}{O D} \approx \frac{M D}{O P} $

- $ \beta \approx \tan \beta-\frac{M D}{C D} \approx \frac{M D}{C P} $

- $ \gamma \approx \tan \gamma=\frac{M D}{I D} \approx \frac{M P}{I P}$

- $\therefore  i = \frac{MD}{OP} + \frac{MD}{CP}  $-----(3)

- $r = \frac{MD}{CP} \frac{MD}{IP}  $-----(4)

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<footer style = "font-size: 18px">Refraction at a Spherical Surface $\rightarrow$ Paraxial Approximation $\rightarrow$ <span style = "color:blue"> Refraction at a Spherical Surface </span> $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ Refraction at a Spherical Surface </footer>

- Using Snell's Law

- $\frac{\sin i}{ \sin r} = \frac{n_2}{n_1} \Rightarrow n_1 \sin i = n_2 \sin r $

- For small i,r : $\sin i \approx i, \sin r \approx r \Rightarrow n_1 i = n_2 r $-----(5)

- $\therefore n_1 (\frac{MD}{OP} + \frac{MD}{CP} = n_2 (\frac{MD}{CP} \frac{MD}{IP}) $-----(6)

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<footer style = "font-size: 18px">Paraxial Approximation $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ <span style = "color:blue"> Refraction at a Spherical Surface </span> $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ Sign Convention </footer>

- **Refraction at a spherical surface :**

- $n_1 (\frac{MD}{OP} + \frac{MD}{CP} = n_2 (\frac{MD}{CP} \frac{MD}{IP}) \rightarrow (eqn-6)$

- $\frac{n_1}{OP} + \frac{n_2}{IP} = \frac{(n_2-n_1)}{CP} \rightarrow (eqn-7)$

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<footer style = "font-size: 18px">Refraction at a Spherical Surface $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ <span style = "color:blue"> Refraction at a Spherical Surface </span> $\rightarrow$ Sign Convention $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Sign Convention </primary>
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- Applying 'sign convention',

- OP = -u, CP = R, IP = v

- $\frac{n_1}{-u} + \frac{n_2}{v} = \frac{n_2 - n_1}{R}$

- $\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \rightarrow (eqn-8)$

- Gives a relation between object distance and image distance in terms of r.i and R.O.C of the spherical surface.

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<footer style = "font-size: 18px">Refraction at a Spherical Surface $\rightarrow$ Refraction at a Spherical Surface $\rightarrow$ <span style = "color:blue"> Sign Convention </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Determine the position of the image when the point object is at a distance of (i) $100 \{cm}$,
- (ii) $50 \{cm}$ (iii) $25 \{cm}$

- in front of the  spherical surface.

- Solution: $\frac{n_2}{v}-\frac{n_1}{u}=\frac{(n_2-n_1)}{R}$---(1)

- $ u_1=-100 \{cm}, R=25 \{cm} $

- $n_1=1.0, n_2=1.5 $

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<footer style = "font-size: 18px">Refraction at a Spherical Surface $\rightarrow$ Sign Convention $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Refraction by a Lens </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $ \frac{1.5}{v}-\frac{1.0}{(-100)}=\frac{1.5-1.0}{2.5}=\frac{1  }{50}$

- gives $v=150 \{cm}$

- i.e Real image in glass medium.

- (2)  $u=-25 \{cm} $

- $ \frac{1.5}{v}-\frac{1.0}{(-25)}=\frac{1}{50} $

- $ \frac{1.5}{v}=\frac{1}{50}-\frac{1}{25}=-\frac{1}{50} $

- $ v=-75 \{cm} $, virtual image

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<footer style = "font-size: 18px">Sign Convention $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Refraction by a Lens $\rightarrow$ Successive Refraction at Interface 1,2 </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Refraction by a Lens </primary>
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- "Thin Lens" $OA \approx OP \approx OB$

- $IA \approx IP \approx IB$

- 't' is small

- AB = thickness of the lens.

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<footer style = "font-size: 18px">Example $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Refraction by a Lens </span> $\rightarrow$ Successive Refraction at Interface 1,2 $\rightarrow$ Successive Refraction at Interface 1,2 </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Successive Refraction at Interface 1,2 </primary>
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- medium.1 $\rightarrow$ Air $n_1 = 1$

- medium.2 $\rightarrow$ Glass material of r.i $n_2$

- $\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1} \rightarrow (1)$

- $\frac{n_1}{v} - \frac{n}{u} = \frac{n_1 - n_2}{R_2} \rightarrow (2)$

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<footer style = "font-size: 18px">Example $\rightarrow$ Refraction by a Lens $\rightarrow$ <span style = "color:blue"> Successive Refraction at Interface 1,2 </span> $\rightarrow$ Successive Refraction at Interface 1,2 $\rightarrow$ Lens Immersed in Liquid </footer>

- **Adding eq(1) and eq(2)**

- $\frac{n_1}{v}-\frac{n_1}{u} = (n_2-n_1)(\frac{1}{R_1}-\frac{1}{R_2})$

- $\frac{1}{v}-\frac{1}{u}=(\frac{n_2}{n_1}-1)(\frac{1}{R_1}-\frac{1}{R_2}) \rightarrow  (4) $ constant for a given lens

- For large object distances, $\frac{1}{u} \rightarrow 0 \therefore \frac{1}{v} = constant$.

- $\Rightarrow$ Image point is fixed, independent of $u$, and is called principal focus F. The corresponding image distance is called the focal length, $f$.

- $\therefore \frac{1}{v}=\frac{1}{f}=(\frac{n_2}{n_1}-1)(\frac{1}{R_1}-\frac{1}{R_2})$

- From (4) & (5).

- $\frac{1}{v}-\frac{1}{u}=\frac{1}{f} \rightarrow(6)$

- Lens formula

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<footer style = "font-size: 18px">Refraction by a Lens $\rightarrow$ Successive Refraction at Interface 1,2 $\rightarrow$ <span style = "color:blue"> Successive Refraction at Interface 1,2 </span> $\rightarrow$ Lens Immersed in Liquid $\rightarrow$ Lens Maker's Formula </footer>

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### <primary style="font-size:24px"> Lens Immersed in Liquid </primary>
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- $\frac{1}{v}-\frac{1}{u}=\frac{1}{f} \text { (Lens formula) }$

- with $\frac{1}{f}=(\frac{n_2}{n_1}-1)(\frac{1}{R_1}-\frac{1}{R_2})$

- for $u \rightarrow \infty$, rays from object are almost parallel to the axis, and $v=f$, the focal length.

- **For a lens immersed in a liquid**

- of $r . i \cdot n_l, \frac{1}{f_l}=(\frac{n_2}{n_l}-1)(\frac{1}{R_1}-\frac{1}{R_2})$

- Since $n_l>n_{air}(=1)$,

- $f_e >f_{air}$

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<footer style = "font-size: 18px">Successive Refraction at Interface 1,2 $\rightarrow$ Successive Refraction at Interface 1,2 $\rightarrow$ <span style = "color:blue"> Lens Immersed in Liquid </span> $\rightarrow$ Lens Maker's Formula $\rightarrow$ Converging and Diverging Lenses </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
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- For common applications of

- a lens, $n_1=n_{air}=1$ $n_2=n$, r.i. of the material of lens

- $\therefore \frac{1}{f}=(n-1)(\frac{1}{R_1}-\frac{1}{R_2}) \rightarrow$ (Lens-maker's formula.)

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- The formula indicates to the choice of $R_1$ & $R_2$ to obtain a desired focal length.

- For a symmetric biconvex lens, $R_1 = -R_2 = R$

- $\Rightarrow \frac{1}{f}$ = $(n-1)(\frac{1}{R}$ - $\frac{1}{(-R)})$ = $\frac{2}{R} (n-2)$

- n>1 $\Rightarrow$ f>0 $\Rightarrow$ "converying lens"

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<footer style = "font-size: 18px">Successive Refraction at Interface 1,2 $\rightarrow$ Lens Immersed in Liquid $\rightarrow$ <span style = "color:blue"> Lens Maker's Formula </span> $\rightarrow$ Converging and Diverging Lenses $\rightarrow$ Various Situations </footer>

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### <primary style="font-size:24px"> Converging and Diverging Lenses </primary>
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- For a symmetric biconvex lens,

- $\frac{1}{f} =\frac{2}{R}(n-1) $

- $f =\frac{R}{2} \frac{1}{(n-1)} \rightarrow(1)$

- For a symmetric biconcave lens,

- $R_1=-R, R_2=R$ gives

- $ f=-\frac{R}{2} \frac{1}{(n-1)} \rightarrow(2)$

- since n>1, f< 0  in Equation(2)

- $ \Rightarrow$ Diverging lens

- for n=1.5 , f=R and for n=2, f=$\frac{R}{2}$

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<footer style = "font-size: 18px">Lens Immersed in Liquid $\rightarrow$ Lens Maker's Formula $\rightarrow$ <span style = "color:blue"> Converging and Diverging Lenses </span> $\rightarrow$ Various Situations $\rightarrow$ Light Incident from Right Side? </footer>

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### <primary style="font-size:24px"> Various Situations </primary>
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- Normally $n_2>n_1$. if $n_1>n_2$?

- for $n_2<n_1$

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<footer style = "font-size: 18px">Lens Maker's Formula $\rightarrow$ Converging and Diverging Lenses $\rightarrow$ <span style = "color:blue"> Various Situations </span> $\rightarrow$ Light Incident from Right Side? $\rightarrow$ Focal Lengths of a Lens </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
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- Is |$f_2$| = |f|?

- $\frac{1}{f} = (\frac{n_2}{n_1} - 1)(\frac{1}{R_1} - \frac{1}{R_2})$

- $\frac{1}{f_2} = (\frac{n_2}{n_1} - 1)(\frac{1}{R_2} =- \frac{1}{R_1})$

-$\frac{1}{f}$

- $\therefore$ |$f_2$| = |f|, is long as

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- $n_1$ is the same (medium) on both sides of the lens, even through $R_1 \neq R_2$.

- Thus, a lens has two principal focus.

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<footer style = "font-size: 18px">Converging and Diverging Lenses $\rightarrow$ Various Situations $\rightarrow$ <span style = "color:blue"> Light Incident from Right Side? </span> $\rightarrow$ Focal Lengths of a Lens $\rightarrow$ Image formation by a Lens </footer>

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### <primary style="font-size:24px"> Focal Lengths of a Lens </primary>
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- $F_1$ is the first principal focus, $f_1 \rightarrow$ first focal length

- $F_2$ is the second principal focus,

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- $f_2 \rightarrow$ second focal length

- $F_1$ and $F_2$ are equidistant from the lens.

- Normally, when we refer to the. "focus of a lens", wa refer to the point $F_2$, and the focal length $f_2$ Then, what is the importance of $F_1$ ?

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<footer style = "font-size: 18px">Various Situations $\rightarrow$ Light Incident from Right Side? $\rightarrow$ <span style = "color:blue"> Focal Lengths of a Lens </span> $\rightarrow$ Image formation by a Lens $\rightarrow$ Imaging by a Bioconcave Lens </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Image formation by a Lens </primary>
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- $\triangle A B P$ and $\triangle A'B' P$ are similar.

- $\frac{A B}{B P}=\frac{A' B'}{P' B'}$ or $\frac{A' B'}{AB}=\frac{B'P}{B P}$:

- Applying sign convention :

- $-\frac{h'}{h}=\frac{v}{-u}$ or $ m=\frac{h'}{h}=\frac{v}{u}$

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<footer style = "font-size: 18px">Light Incident from Right Side? $\rightarrow$ Focal Lengths of a Lens $\rightarrow$ <span style = "color:blue"> Image formation by a Lens </span> $\rightarrow$ Imaging by a Bioconcave Lens $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Refraction At Spherical Surfaces And By Lenses Ray Optics L-5 </Success>
### <primary style="font-size:24px"> Imaging by a Bioconcave Lens </primary>
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- $\triangle A B P$ and $\triangle A'B'P$ are similar;

- $\therefore \frac{A'B'}{A B} =\frac{B'P}{B P}$

- i.e $\frac{h'}{h}=\frac{-v}{-u}=\frac{v}{u}$

- or $m = \frac{v}{u}$ as before.

- Rays 1,2 and 3 appear to come from point A

- $\therefore$ A'B' is a virtual image.

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<footer style = "font-size: 18px">Focal Lengths of a Lens $\rightarrow$ Image formation by a Lens $\rightarrow$ <span style = "color:blue"> Imaging by a Bioconcave Lens </span> $\rightarrow$ Thank You $\rightarrow$  </footer>