Four combinations of two thin lenses are given in List-I. The radius of curvature of all curved surfaces is $r$ and the refractive index of all the lenses is 1.5. Match lens combinations in List-I with their focal length in List-II and select the correct answer using the code given below the lists. [2014]

List - I List - II

P. 1. $2 r$

Q. 2. $\frac{r}{2}$

R. 3. $- r$

S. 4. $r$

Codes:

Question

Four combinations of two thin lenses are given in List-I. The radius of curvature of all curved surfaces is $r$ and the refractive index of all the lenses is 1.5. Match lens combinations in List-I with their focal length in List-II and select the correct answer using the code given below the lists. [2014]

	List - I		List - II
P.		1.	$2 r$
Q.		2.	$\frac{r}{2}$
R.		3.	$- r$
S.		4.	$r$

Codes:

	List - I		List - II
P.		1.	$2 r$
Q.		2.	$\frac{r}{2}$
R.		3.	$- r$
S.		4.	$r$

Codes:

Smart Achievers · Accepted Answer

(2)

For double convex lens, (P), $\frac{1}{f} = (μ - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})$

$\Rightarrow (1.5 - 1) (\frac{1}{r} - \frac{1}{r}) = (1.5 - 1) (\frac{2}{r}) = \frac{1}{r}$ $\Rightarrow f = r$

$\frac{1}{F_{eq .}} = \frac{1}{f_{1}} + \frac{1}{f_{2}} = \frac{1}{r} + \frac{1}{r} = \frac{2}{r}$

$∴$ $F_{eq .} = \frac{r}{2}$

For (Q) plano-convex lens, $\frac{1}{f} = (μ - 1) [\frac{1}{R_{1}} - \frac{1}{R_{2}}]$

$= (1.5 - 1) (\frac{1}{\infty} - \frac{1}{- r}) = \frac{0.5}{r} = \frac{1}{2 r}$ $∴$ $f = 2 r$

$\frac{1}{F_{eq .}} = \frac{1}{f_{1}} + \frac{1}{f_{2}} = \frac{1}{2 r} + \frac{1}{2 r} = \frac{2}{2 r} = \frac{1}{r}$ $∴$ $F_{eq .} = r$

For (R) plano-concave lens,

$\frac{1}{f} = (1.5 - 1) (\frac{1}{- r} - \frac{1}{\infty})$ $\Rightarrow f = - 2 r$

$\frac{1}{F_{eq .}} = \frac{1}{f} + \frac{1}{f} = \frac{1}{- 2 r} + \frac{1}{- 2 r}$ $\Rightarrow F_{eq .} = - r$

For (S), combination of one double convex and one plano-concave lens,

$\frac{1}{F_{eq .}} = \frac{1}{r} + \frac{1}{- 2 r} = \frac{1}{2 r}$ $\Rightarrow F_{eq .} = 2 r$

Solution

1 P-1, Q-2, R-3, S-4

2 P-2, Q-4, R-3, S-1

3 P-4, Q-1, R-2, S-3

4 P-2, Q-1, R-3, S-4

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