Question

A transparent slab of thickness \[d\] as a refractive index \[n\left( z \right)\]that increases with \[z\]. Here \[z\] is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices \[{n_1}\] and \[{n_2}\]. As shown in the figure. A ray of light incident with angle \[{\theta _1}\] from medium \[1\] and emerges in medium \[2\] with refraction angle\[{\theta _r}\]with lateral displacement \[1\].
Which of the following statement(s) is (are) true?
A. \[1\]is dependent on \[n\left( z \right)\]
B. \[{n_1}\]\[\sin {\theta _1}\]=\[\left( {{n_2} - {n_1}} \right)\sin {\theta _f}\]
C. \[{n_1}\]\[\sin {\theta _i}\]= \[{n_2}\]\[\sin {\theta _f}\]
D. \[1\] is independent of\[{n_2}\]

Answer 1

Hint:As we know that the Law of refraction – (a) The incident ray, reflected ray and the normal, to the interface of given two mediums, all lie in the same plane.
(b) The ratio of sine of angle of incidence and sine of angle refraction is constant.
 \[{n_1}\]\[\sin {\theta _i}\]= \[{n_2}\]\[\sin {\theta _f}\]
Where, \[{n_1}\] is incident index,\[{n_2}\] is refractive index, \[{\theta _i}\] is incident angle and \[{\theta _r}\] is refracted angle.

Complete step by step solution:
\[{n_1}\]\[\sin {\theta _i}\]= \[{n_2}\]\[\sin {\theta _f}\]
Where,\[{n_1}\] is incident index,\[{n_2}\] is refractive index,\[{\theta _i}\]is incident angle and \[{\theta _r}\] is refracted angle.

As we know that lateral displacement occurs before the light enters on medium \[2\].So,\[1\] is independent of \[{n_2}\]. Since lateral displacement depends upon the refractive index. So, \[1\] is dependent on \[n\left( z \right)\].

Hence, options A,C and D are correct.

Additional information:
Refraction of light is one of the most commonly observed phenomena, but other waves like sound waves and water waves also experience refraction. Refraction makes it possible for us to have optical instruments such as magnifying glasses, lenses and prisms. It is also because of the refraction of light that we are able to focus light on to our retina.

Note:In this question, remember l is independent of \[{{\text{n}}_{\text{2}}}\] ​since the lateral displacement occurs even before the light enters medium 2. We can directly write that I is dependent on n(z) since lateral displacement depends on the refractive index.