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RAY OPTICS AND OPTICAL INSTRUMENTS — Physics STD 12 Science — Question

Bihar BoardEnglish MediumSTD 12 SciencePhysicsRAY OPTICS AND OPTICAL INSTRUMENTS4 Marks
Question
Explain how the image is formed by thin convex lens and derive
$-\frac{1}{u}+\frac{1}{v}=\left(n_{21}-1\right)\left[\frac{1}{R_1}-\frac{1}{ R _2}\right]$###
Derive lens maker's formula for thin convex lens.
✓

Answer

Image
Image
→Figure (a) shows the geometry of image formation by a convex lens.
→A point object $O$ is placed at a distance $u$ from the optical centre. On the other side of the lens there is image I. Here image distance is $v$. The radii of curvature of both surfaces of the lens are $R_1$ and $R_2$ respectively and the focal length of the lens is $f$.
→The image formation can be seen in terms of two steps :
(i) The first refracting surface forms the image $I_1$ of the object $O$. (figure b)
(ii) The image $I _1$ acts as a virtual object for the second surface. (figure c) that forms image at I .
→For refraction at interface ABC ,
$\frac{n_1}{ OB }+\frac{n_2}{ BI _1}=\frac{n_2-n_1}{ BC _1}......(1)$
→A similar procedure applied to the interface ADC gives,
$-\frac{n_2}{ DI _1}+\frac{n_1}{ DI }=\frac{n_2-n_1}{ DC _2}$
→For a thin lens,
$\begin{aligned}
& BI _1= DI _1 \\\therefore & -\frac{n_2}{ BI _1}+\frac{n_1}{ DI }=\frac{n_2-n_1}{ DC _2}......(2)\end{aligned}$
→Adding equations (1) and (2),
$\frac{n_1}{ OB }+\frac{n_1}{ DI }=\frac{n_2-n_1}{ BC _1}+\frac{n_2-n_1}{ DC _2}......(3)$
→Suppose the object is at infinity
i.e. $OB \rightarrow \infty$ and $DI \rightarrow f$ (focal length)
→from equation (3),
$\begin{array}{c}0+\frac{n_1}{f}=\frac{n_2-n_1}{ BC _1}+\frac{n_2-n_1}{ DC _2} \\\therefore \quad \frac{n_1}{f}=\left(n_2-n_1\right)\left(\frac{1}{ BC _1}+\frac{1}{ DC _2}\right)\end{array}$
→Now substituting $BC _1= R _1$ and $DC _2=- R _2$ in above equation.
(Positive and negative signs are determined according to the sign convention). $\begin{array}{l}\therefore \quad \frac{n_1}{f}=\left(n_2-n_1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right) \\\therefore \quad \frac{1}{f}=\left(\frac{n_2-n_1}{n_1}\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right) \\\therefore \quad \frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right) \\ \therefore \quad \frac{1}{f}=\left(n_{21}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)\end{array}$
→This equation is known as lensmaker's formula.
→Note that the formula is true for a concave lens also. For concave lens $R_1$ is negative, $R_2$ positive and therefore $f$ is negative.

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