Answer
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Hint: As here we have to find the focal length of a thin concavo – convex lens whose radii of curvatures are given so it can be solved by using lens maker’s formula for thin lenses.
Formula used: Lens maker formula for thin – lenses is given by
$\dfrac{1}{f} = \left( {\mu - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$
Where f is the focal length of a thin lens, $\mu $is its refractive index of lens, ${R_1}$and ${R_2}$ are the radii of curvature.
Complete step by step answer:
We know that a thin concavo convex lens with light from left to right is given by the figure below
For light from left to right, the radii of curvature of first surface $ = {R_1} = - R$
The radii of curvature of second surface $ = {R_2} = - 2R$
Refractive index of material of lens $ = \mu $
So, in this case, the focal length of the lens by lens maker formula is given by
$
\dfrac{1}{{{f_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right) \\
\dfrac{1}{{{F_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{ - R}}\dfrac{{ - 1}}{{ - 2R}}} \right) \\
= \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}} - \dfrac{1}{R}} \right) = \left( {\mu - 1} \right)\left( {\dfrac{{1 - 2}}{{2R}}} \right) \\
\dfrac{1}{{{f_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{{ - 1}}{{2R}}} \right) \\ $
$\dfrac{1}{{{f_1}}} = - \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}}} \right)$ … (i)
When the light is incident from right to left in this concavo-convex lens, then as the given figure below we have
For light from right to left,
The radii of curvature of first surface $ = {R_1} = R$
The radii of curvature of second surface $ = {R_2} = R$
So, the focal length, ${f_2}$, in this case by lens maker’s formula will be given by
$\dfrac{1}{{{f_2}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}} - \dfrac{1}{R}} \right)$
$\dfrac{1}{{{f_2}}} = - \dfrac{{\left( {\mu - 1} \right)}}{{2R}}$… (ii)
From (i) and (ii), it is clear that
$\dfrac{1}{{{f_1}}} = \dfrac{1}{{{f_2}}} = - \dfrac{{\left( {\mu - 1} \right)}}{{2R}}$
$ \Rightarrow $Focal length, \[{f_1} = {f_2} = \dfrac{{ - 2R}}{{\left( {\mu - 1} \right)}}\]
Hence, option (D) is the correct option.
Note:: In case of incident light from left to right, then
${R_1} = - R$and ${R_2} = - 2R$
It is negative because here the distances ${R_1}$and ${R_2}$are measured against the direction of incident light.
When light is incident from right to left, then
${R_1} = + 2R,{\text{ }}{{\text{R}}_2} = + R$
Both are positive as both being measured in direction of incident light.
Formula used: Lens maker formula for thin – lenses is given by
$\dfrac{1}{f} = \left( {\mu - 1} \right)\left[ {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right]$
Where f is the focal length of a thin lens, $\mu $is its refractive index of lens, ${R_1}$and ${R_2}$ are the radii of curvature.
Complete step by step answer:
We know that a thin concavo convex lens with light from left to right is given by the figure below
For light from left to right, the radii of curvature of first surface $ = {R_1} = - R$
The radii of curvature of second surface $ = {R_2} = - 2R$
Refractive index of material of lens $ = \mu $
So, in this case, the focal length of the lens by lens maker formula is given by
$
\dfrac{1}{{{f_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right) \\
\dfrac{1}{{{F_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{ - R}}\dfrac{{ - 1}}{{ - 2R}}} \right) \\
= \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}} - \dfrac{1}{R}} \right) = \left( {\mu - 1} \right)\left( {\dfrac{{1 - 2}}{{2R}}} \right) \\
\dfrac{1}{{{f_1}}} = \left( {\mu - 1} \right)\left( {\dfrac{{ - 1}}{{2R}}} \right) \\ $
$\dfrac{1}{{{f_1}}} = - \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}}} \right)$ … (i)
When the light is incident from right to left in this concavo-convex lens, then as the given figure below we have
For light from right to left,
The radii of curvature of first surface $ = {R_1} = R$
The radii of curvature of second surface $ = {R_2} = R$
So, the focal length, ${f_2}$, in this case by lens maker’s formula will be given by
$\dfrac{1}{{{f_2}}} = \left( {\mu - 1} \right)\left( {\dfrac{1}{{2R}} - \dfrac{1}{R}} \right)$
$\dfrac{1}{{{f_2}}} = - \dfrac{{\left( {\mu - 1} \right)}}{{2R}}$… (ii)
From (i) and (ii), it is clear that
$\dfrac{1}{{{f_1}}} = \dfrac{1}{{{f_2}}} = - \dfrac{{\left( {\mu - 1} \right)}}{{2R}}$
$ \Rightarrow $Focal length, \[{f_1} = {f_2} = \dfrac{{ - 2R}}{{\left( {\mu - 1} \right)}}\]
Hence, option (D) is the correct option.
Note:: In case of incident light from left to right, then
${R_1} = - R$and ${R_2} = - 2R$
It is negative because here the distances ${R_1}$and ${R_2}$are measured against the direction of incident light.
When light is incident from right to left, then
${R_1} = + 2R,{\text{ }}{{\text{R}}_2} = + R$
Both are positive as both being measured in direction of incident light.
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