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 of `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)`
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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)

Options

A. NA of `S_(1)` immersed in water is the same as that of `S_(2)` immersed in liquid of refractive index `(4)/(sqrt(5))`
B. NA of `S_(1)` immersed in liquid of refractive index `(6)/(sqrt(5))` 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 `(4)/(sqrt(15))`
D. NA of `S_(1)` placed in air is the same as that of `S_(2)` placed in water .

Answer

A,C

Explanation

Let the whole structure is placed in a medium of refractive index n , then
n sini = `n_(1) "sin"(90- theta)`
` " n sini" = n_(1) "cos" theta …..(i)`
Here for `i_(m) , theta = C "and " "sin" C = (n_(2))/(n_(1))`
from eq. (i) ,
`n"sin"i_(m) = n_(1) sqrt((1 - n_(0)^(2))/(n_(1)^(2))) = sqrt(n_(1)^(2) - n_(2)^(2))`
`implies "sin"i_(m) = sqrt(n_(1)^(2) - n_(2)^(2))/(n)`
Now , for (A)
`(NA)_(s_(1)) = (3)/(4)sqrt((45)/(16) - (9)/(4)) = (3)/(4) xx (3)/(4) = (9)/(16)`
`(NA)_(s_(2)) = (3sqrt(15))/(16) sqrt(64)/(25) - (49)/(25)) = (3sqrt(15))/(16) (1)/(5)sqrt(15) = (9)/(16)`
For (B)
`(NA)_(s_(1)) = sqrt(15)/(6) xx (3)/(4) = sqrt((15))/(8)`
`(NA)_(s_(2)) = (3)/(4) = sqrt(15)/(5)` Not equal
For (C) `" " (NA)_(s_(1)) = 1 xx (3)/(4) = (3)/(4)`
`(NA)_(s_(2)) = sqrt(15)/(4) xx sqrt((15))/(5) = (15)/(4 xx 5) = (3)/(4)`
For (D) `" " (NA)_(s_(1)) = (3)/(4)`
`(NA)_(s_(2)) = (3)/(4) sqrt(15)/(5)` Not equal