Question

A ray of light strikes a plane mirror at an angle of incidence $45^\circ$as shown in fig. 1.364. After reflection, the ray passes through a prism of reflection index $1.5$ whose apex angle is$4^\circ$. If the mirror is rotated by $x^\circ$then the total deviation of the ray becomes $90^\circ$. Find $x$

Answer 1

When there are many optical instruments involved then the total angle of deviation will be the sum of the angle of deviation of every optical instrument. The angle of the prism and apex angle represent the same thing which is the topmost angle of the prism.
Formula used:
For mirror
$\delta = 180^\circ - 2i$
For prism with small apex angle
$\delta = A(n - 1)$
Where $\delta$ is the angle of deviation, $A$is the apex angle,$i$ is the angle of incidence, and$n$ is the refractive index of prism.

Complete step-by-step answer:
For the initial case when the angle of incidence on mirror is $45^\circ$(before rotation)
We know that,
For mirror
$\delta = 180^\circ - 2i$
For prism with small apex angle
$\delta = A(n - 1)$
Where $\delta$ is the angle of deviation, $A$is the apex angle,$i$ is the angle of incidence, and$n$ is the refractive index of prism.
Hence
The angle of deviation of ray due to mirror be ${\delta _1}$
$\Rightarrow {\delta _1} = 180^\circ - 2(45^\circ )$
$\Rightarrow {\delta _1} = 90^\circ$
The angle of deviation of ray due to prism be ${\delta _2}$
It is given that the apex angle is$4^\circ$ and the refractive index of the prism is $1.5$
$\Rightarrow {\delta _2} = 4(1.5 - 1)$
$\Rightarrow {\delta _2} = 2^\circ$
Let the total angle of deviation be ${\delta _3}$
We know that the total angle of deviation from the given setup will be ${\delta _1} + {\delta _2}$
$\Rightarrow {\delta _3} = {\delta _1} + {\delta _2}$
$\Rightarrow {\delta _3} = 90^\circ + 2^\circ$
$\Rightarrow {\delta _3} = 92^\circ$
Now solving for the second case
For the second case when the mirror is rotated by $x^\circ$(let’s assume that we have turned the mirror in an anticlockwise direction) hence angle of incidence upon the mirror is $(45 + x)^\circ$
We know that,
For mirror
$\delta = 180^\circ - 2i$
For prism with small apex angle
$\delta = A(n - 1)$
Where $\delta$ is the angle of deviation, $A$is the apex angle,$i$ is the angle of incidence, and$n$ is the refractive index of prism.
Hence
The angle of deviation of ray due to mirror be ${\delta _1}$
$\Rightarrow {\delta _1} = 180^\circ - 2((45 + x)^\circ )$
$\Rightarrow {\delta _1} = (90 - 2x)^\circ$
The angle of deviation of ray due to prism be ${\delta _2}$
It is given that the apex angle is$4^\circ$ and the refractive index of the prism is $1.5$
$\Rightarrow {\delta _2} = 4(1.5 - 1)$
$\Rightarrow {\delta _2} = 2^\circ$
Let the total angle of deviation be ${\delta _3}$ and it’s given that in this case ${\delta _3} = 90^\circ$
We know that the total angle of deviation from the given setup will be ${\delta _1} + {\delta _2}$
$\Rightarrow {\delta _3} = {\delta _1} + {\delta _2}$
$\Rightarrow {\delta _3} = (90 - 2x)^\circ + 2^\circ$
$\Rightarrow 90^\circ = (92 - 2x)^\circ$
$\Rightarrow 2x = 2$
$\Rightarrow x = 1^\circ$
As the value of $x$is positive our assumption of anticlockwise rotation was correct.

Therefore the correct answer to the above question is $1^\circ$in an anticlockwise direction.

Note:
When the apex angle of the prism is small then we can use the trigonometric approximation which is $\sin \theta \approx \theta$. Due to this, the formula derived for the angle of deviation of the prism is independent of the angle of incidence which made our solution simpler.