Answer

Verified

401.7k+ views

**Hint:**In this question, we need to determine the deviation of a tiny fraction of the ray such that it enters the sphere, undergoes two internal reflections, and then refracts out into the air. For this, we will use the different properties of reflection of light rays and total internal reflections.

**Complete step by step answer:**

The refractive index of the sphere \[{\mu _2} = \sqrt 2 \]

Angle of incidence \[i = {45^ \circ }\]

As the light ray enters the sphere from the air hence refractive index of the air \[{\mu _1} = 1\]

Now by using Snell’s law, we will find the angle of reflection

\[

{\mu _1}\sin i = {\mu _2}\sin r \\

1 \times \sin {45^ \circ } = \sqrt 2 \sin r \\

\sin r = \dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\sqrt 2 }} \\

= \dfrac{1}{2} \\

\]

Therefore the of reflection

\[

\sin r = \dfrac{1}{2} \\

r = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right) \\

= {30^ \circ } \\

\]

Since the angle of reflection is equal to the angle of refraction, hence\[r = r' = {30^ \circ }\]

It is given that the light ray enters the sphere and undergoes two internal reflections, so the first angle of deviation will be

\[{\delta _1} = i - r' = 45 - 30 = {15^ \circ }\]

Now, when the light ray refracted from the surface of the sphere and touches the other face of the sphere, the light ray undergoes total internal reflection where the angle of incidence and reflection will be equal to \[{30^ \circ }\].

Therefore the second angle of deviation will be

\[{\delta _2} = 180 - 30 - 30 = {120^ \circ }\]

When the light ray goes the total internal reflection, then the light reflects back and it incident on the internal surface of a sphere where the light again goes to total internal reflection where the angle of incidence and reflection will be equal to\[{30^ \circ }\].

Therefore the third angle of deviation will be

\[{\delta _3} = 180 - 30 - 30 = {120^ \circ }\]

Now it is said that after light undergoes two internal reflections, the light refracts out into the air, so we will again use the Snell’s law to find the angle of the refraction

\[

{\mu _2}\sin i = {\mu _1}\sin r \\

\sqrt 2 \times \sin {30^ \circ } = 1\sin r \\

\sqrt 2 \times \dfrac{1}{2} = \sin r \\

\sin r = \dfrac{1}{{\sqrt 2 }} \\

\]

Therefore the of reflection

\[

\sin r = \dfrac{1}{{\sqrt 2 }} \\

r = {\sin ^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right) \\

= {45^ \circ } \\

\]

So the fourth angle of deviation will be

\[{\delta _4} = 45 - 30 = {15^ \circ }\]

Hence the total angle of deviation will be

\[

\delta = {\delta _1} + {\delta _2} + {\delta _3} + {\delta _4} \\

= 15 + 120 + 120 + 15 \\

= {270^ \circ } \\

\]

So the deviation of a tiny fraction of the ray \[ = {270^ \circ }\]

**Option 1 is correct.**

**Note:**It is interesting to note here that when light enters into a medium at a certain angle then, the light rays cannot go out in the initial medium; then, the reflection is known as Total Internal Reflection.

Recently Updated Pages

In Mendels experiments the color of the seed coat nature class 12 biology CBSE

In 1943 the causal organism and host of Bengal famine class 12 biology CBSE

Inner lining of gut stomach and liver is composed of class 12 biology CBSE

In addition to directional terms there are specific class 12 biology CBSE

In which of the following areas the subirrigation is class 12 biology CBSE

Synaptonemal complex is formed during A Meiosis B Amitosis class 12 biology CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Mention the different categories of ministers in the class 10 social science CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Who is the executive head of the Municipal Corporation class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which monarch called himself as the second Alexander class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE