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# The focal length of a lens is in ratio 13:8 when it is immersed in two different liquids of refractive indices 1.3 and 1.2 respectively. The reflective index of the material of the lens is:(A) 1.25(B) 1.56(C) 1.5(D) 0.5

Last updated date: 15th Sep 2024
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Hint: We should know that for a thin lens in air, the focal length is the distance from the centre of the lens to the principal foci or focal points of the lens. For a converging lens for example a convex lens, the focal length is positive, and is the distance at which a beam of collimated light will be focused to a single spot. It should also be known that focal length can also change the perspective and scale of our images. A lens with a shorter focal length expands perspective, giving the appearance of more space between the elements in our photo. Meanwhile, telephoto lenses tend to stack elements in the frame together to compress perspective.

Let us consider that the refractive index of the material of the lens is $\mu$.
Using the formula mentioned below, we get:
$\dfrac{1}{f}=({{\mu }_{rel}}-1)[1/{{R}_{1}}-1/{{R}_{2}}]$
It is given that the ratio of the focal length of the lens is in the ratio 13:8.
So, the focal length of the first lens is given as: 13f
The focal length of the second lens is given as: 8f
So, for the first lens we can say that:
$\dfrac{1}{13f}=(\mu /1.3-1)[1/{{R}_{1}}-1/{{R}_{2}}]......(1)$
For the second lens we can write as:
$\dfrac{1}{8f}=(\mu /1.2-1)[1/{{R}_{1}}-1/{{R}_{2}}].......(2)$
From the equations (1) and (2) we get that:
$8/13=\dfrac{(\mu /1.3-1)}{(\mu /1.2-1)}$
$\Rightarrow 8/13=\dfrac{1.2(\mu -1.3)}{1.3(\mu -1.2)}$
$\Rightarrow 8(\mu -1.2)=12(\mu -1.3)$
$\Rightarrow 2\mu -2.4=3\mu -3.9$
$\Rightarrow \mu =1.5$

Hence, the correct answer is Option C.

Note: Refractive index, also called index of refraction, measure of the bending of a ray of light when passing from one medium into another. Refractive index has a large number of applications. It is mostly applied for identifying a particular substance, confirming its purity, or measuring its concentration. Generally, it is used to measure the concentration of a solute in an aqueous solution.
It should be known to us that the higher the refractive index the slower the light travels, which causes a correspondingly increased change in the direction of the light within the material. We should know that this means for lenses that a higher refractive index material can bend the light more and allow the profile of the lens to be lower.