For two structure namely S₁\;{with}\;n₁={{45}}{4}\; and \;n₂=3/2, and \;S₂{with} n₁=8/5 and n₂=7/5 and taking refractive index of water to be 4.3 and that of air to be 1,the correct option(s) is (are) [JEE Adv 2015 P2]

NA of S₁ immersed in water is the same as that of S₂ immersed in a liquid of refractive index {16}{3{15}}

NA of S₁ immersed in liquid of refaractive index {6}{{15}} is the same as that of S₂ immersed in water

NA of S₁ placed is the same as that of S₂ immersed in liquid of refractiv index {4}{{15}}

NA of S₁ placed in air is the same as that of S₂ placed in water

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JEE Advanced 2015 Paper 2

Paragraph for next 2 Questions

Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glasscylinder of refractive index

n_1

surrounded by a medium of lower refractive index

n_2

. The light guidance in thestructure takes place due to successive total internal reflections at the interface of the media

n_1

and

n_2

as shown inthe figure. All rays with the angle of incidence i less than a particular value

i_{m}

are confined in the medium ofrefractive index

n_1

. The numerical aperture (NA) of the structure is defined as sin

i_{m}

Section: Physics

Question : 17 of 60

Marks: +1, -0

S_1\;\text{with}\;n_1=\frac{\sqrt{45}}{4}\;

\;n_2=\frac{3}{2},

\;S_2\text{with} n_1=\frac{8}{5}

n_2=\frac{7}{5}

NA of $S_1$ immersed in water is the same as that of $S_2$ immersed in a liquid of refractive index $\frac{16}{3\sqrt{15}}$
NA of $S_1$ immersed in liquid of refaractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S_2$ immersed in water
NA of $S_1$ placed is the same as that of $S_2$ immersed in liquid of refractiv index $\frac{4}{\sqrt{15}}$
NA of $S_1$ placed in air is the same as that of $S_2$ placed in water

Solution:

\theta \geq c

\Rightarrow 90^{\circ} - r \geq c

\Rightarrow \sin(90^{\circ} - r) \geq c

\Rightarrow \cos r \geq \sin c

using

\frac{\sin i}{\sin r} = \frac{n_1}{n_m}\;\text{and}\;\sin c = \frac{n_2}{n_1}

we get,

\sin^2 i_m = \frac{n_1^2 - n_2^2}{n_m^2}

Putting values ,we get,correct option as A & C

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