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

Rajasthan 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
$\rightarrow$ Figure $(a)$ shows the geometry of image formation by a convex lens.
$\rightarrow$ 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 $f$.
$\rightarrow$ 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 $I$ .
$\rightarrow$ For refraction at interface $\text{ABC} $,
$\frac{n_1}{ OB }+\frac{n_2}{ BI _1}=\frac{n_2-n_1}{ BC _1}......(1)$
$\rightarrow $ A similar procedure applied to the interface $\text{ADC}$ gives,
$-\frac{n_2}{ DI _1}+\frac{n_1}{ DI }=\frac{n_2-n_1}{ DC _2}$
$\rightarrow$ For a thin lens,
$BI _1= DI _1$
$\therefore -\frac{n_2}{ BI _1}+\frac{n_1}{ DI }=\frac{n_2-n_1}{ DC _2}......(2)$
$\rightarrow$ Adding equations $(1)$ and $(2),$
$\frac{n_1}{ OB }+\frac{n_1}{ DI }=\frac{n_2-n_1}{ BC _1}+\frac{n_2-n_1}{ DC _2}......(3)$
$\rightarrow$ Suppose the object is at infinity
i.e. $OB \rightarrow \infty$ and $DI \rightarrow f \ ($focal length$)$
$\rightarrow$ from equation $(3),$
${c}0+\frac{n_1}{f}=\frac{n_2-n_1}{ BC _1}+\frac{n_2-n_1}{ DC _2}$
$\therefore \frac{n_1}{f}=\left(n_2-n_1\right)\left(\frac{1}{ BC _1}+\frac{1}{ DC _2}\right)$
$\rightarrow$ 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$)$.
$\therefore \frac{n_1}{f}=\left(n_2-n_1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
$\therefore \frac{1}{f}=\left(\frac{n_2-n_1}{n_1}\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
$\therefore \frac{1}{f}=\left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
$\therefore \frac{1}{f}=\left(n_{21}-1\right)\left(\frac{1}{ R _1}-\frac{1}{ R _2}\right)$
$\rightarrow$ This equation is known as lensmaker's formula.
$\rightarrow$ 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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