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STD 12 - 9. Ray optics abd optical instruments — NEET STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceNEETSTD 12 - 9. Ray optics abd optical instruments4 Marks
Question
Light guidance in an optical fiber can be understood by considering a structure comprising of thin solid glass cylinder of refractive index $n_1$ surrounded by a medium of lower refractive index $n_2$. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media $n_1$ and $n_2$ as shown in the figure. All rays with the angle of incidence $i$ less than a particular value $i_m$ are confined in the medium of refractive index $n_1$. The numerical aperture ($NA$) of the structure is defined as $\sin i_m$ - $Image$

$1.$ For two structures namely $S_1$ with $n_1=\sqrt{45} / 4$ and $n_2=3 / 2$, and $S_2$ with $n_1=8 / 5$ and $n_2=7 / 5$ and taking the refractive index of water to be $4 / 3$ and that of air to be 1 , the correct option$(s)$ is(are)

$(A)$ $NA$ of $S_1$ immersed in water is the same as that of $S_2$ immersed in a liquid of refractive index $\frac{16}{3 \sqrt{15}}$

$(B)$ $NA$ of $S _1$ immersed in liquid of refractive index $\frac{6}{\sqrt{15}}$ is the same as that of $S _2$ immersed in water

$(C)$ $NA$ of $S_1$ placed in air is the same as that of $S_2$ immersed in liquid of refractive index $\frac{4}{\sqrt{15}}$.

$(D)$ $NA$ of $S_1$ placed in air is the same as that of $S_2$ placed in water

$2.$ If two structures of same cross-sectional area, but different numerical apertures $N A_1$ and $NA _2\left( NA _2< NA _1\right)$ are joined longitudinally, the numerical aperture of the combined structure is

$(A)$ $\frac{ NA _1 NA _2}{ NA _1+ NA _2}$ $(B)$ $NA _1+ NA _2$ $(C)$ $NA _1$ $(D)$ $NA _2$

Give the answer question $1$ and $2.$

✓

Answer

$1.$ $\theta \geq c$

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

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

$\Rightarrow \cos r \geq \sin c$

using $\frac{\sin i }{\sin r }=\frac{ n _1}{ n _{ m }}$ 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 options as $A \& C$

$2.$ For total internal reflection to take place in both structures, the numerical aperture should be the least one for the combined structure \& hence, correct option is $D$.

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