A thin convex lens is made of two materials with refractive indices and , as shown in figure. The radius of curvature of the left and right spherical surface are equal. is the focal length of the lens when . [2019]

Question

A thin convex lens is made of two materials with refractive indices $n_{1}$ and $n_{2}$ , as shown in figure. The radius of curvature of the left and right spherical surface are equal. $f$ is the focal length of the lens when $n_{1} = n_{2} = n$ . The focal length is $f + Δ f$ when $n_{1} = n$ and $n_{2} = n + Δ n$ . Assuming $Δ n ≪ (n - 1)$ and $1 < n < 2$ , the correct statement(s) is/are. [2019]

Smart Achievers · Accepted Answer

(1, 2, 3)

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

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

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

$∴$ $\frac{f + Δ f}{f} = \frac{R}{2 (n - 1) + Δ n} \times \frac{2 (n - 1)}{R} = \frac{2 (n - 1)}{[2 (n - 1) + Δ n]}$

$∴$ $\frac{Δ f}{f} = \frac{2 n - 2 - 2 n + 2 - Δ n}{[2 (n - 1) + Δ n]}$

$= \frac{- Δ n}{2 n - 2 + Δ n}$ $= \frac{- Δ n}{2 (n - 1)}$ $\dots (i)$

$[∵ Δ n ≪ (n - 1)]$

From equation (i), $if \frac{Δ n}{n} < 0, then \frac{Δ f}{f} > 0 .$

Also, $| \frac{Δ f}{f} |$ $>$ $| \frac{Δ n}{n} |$

The relation between $\frac{Δ f}{f}$ and $\frac{Δ n}{n}$ remains unchanged if convex surfaces are replaced by concave surfaces of the same radius of curvature.

For $n = 1.5, Δ n = 10^{- 3}, f = 20 cm,$

equation (i) gives

$\frac{Δ f}{20} = \frac{- 10^{- 3}}{2 (1.5 - 1)}$ $\Rightarrow Δ f = - 0.02 cm$

or, $| Δ f |$ $= 0.02 cm$

Solution

1 If $\frac{Δ n}{n} < 0,$ then $\frac{Δ f}{f} > 0$

2 The relation between $\frac{Δ f}{f}$ and $\frac{Δ n}{n}$ remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.

3 For $n = 1.5$ , $Δ n = 10^{- 3}$ and $f = 20 cm$ , the value of $| Δ f |$ will be 0.02 cm (round off to $2^{nd}$ decimal place).

4 $| \frac{Δ f}{f} |$ $<$ $| \frac{Δ n}{n} |$

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Solution

1 If Δnn<0, then Δff>0

2 The relation between Δff and Δnn remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.

3 For n=1.5, Δn=10-3 and f=20 cm, the value of |Δf| will be 0.02 cm (round off to 2nd decimal place).

4 |Δff|<|Δnn|

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1 If $\frac{Δ n}{n} < 0,$ then $\frac{Δ f}{f} > 0$

2 The relation between $\frac{Δ f}{f}$ and $\frac{Δ n}{n}$ remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.

3 For $n = 1.5$ , $Δ n = 10^{- 3}$ and $f = 20 cm$ , the value of $| Δ f |$ will be 0.02 cm (round off to $2^{nd}$ decimal place).

4 $| \frac{Δ f}{f} |$ $<$ $| \frac{Δ n}{n} |$