Question

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

Answer 1

When a ray of light travels through a glass prism, it undergoes refraction and gets deviated from its original path. The deviation produced by a small angled prism is always a constant. Additionally, it is given when the mirror is rotated by X degrees then the total deviation of the ray becomes ${90^ \circ }C$. Substitute these values to the basic formal and solve for the answer.

Complete answer:

When light ray enters through a glass prism, the emergent ray is not parallel to the incident ray after refraction. Rather, the emergent ray deviates from its original direction by a certain angle, known as the angle of deviation.

${\text{In case of a prism the deviation, }}{\delta _m}{\text{ of the emergent ray is given by:}}$

$\mu {\text{ = }}\dfrac{{\dfrac{{A + {\delta _m}}}{2}}}{{\sin \dfrac{A}{2}}}$

${\text{If the angle of prism A is small,}}$

${\delta _m}{\text{ is also small}}{\text{. So the equation becomes:}}$

${\delta _m} = \left( {\mu - 1} \right)A$

So, the deviation produced via a small angled prism is always, given by

${\delta _1} = \left( {\mu - 1} \right)\alpha = \left( {1.5 - 1} \right){4^ \circ }$

${\delta _1} = {2^ \circ }{\text{ }}\left[ {Always} \right]$

Deviation caused by mirror will be:

${\delta _2} = {180^ \circ } - 2i$

${\delta _2} = {180^ \circ } - 2 \times {45^ \circ }$

${\delta _2} = {90^ \circ }$

Thus, the net deviation produced by the system will be

${\delta _1} + {\delta _2} = {2^ \circ } + {90^ \circ }$

${\delta _1} + {\delta _2} = {92^ \circ }$

Clearly, the total deviation is more than ${90^ \circ }$.

If the angle of incidence on the mirror is greater than its associated deviation will be smaller. Let X be the angle of rotation of mirror in clockwise direction done to increase the angle of incidence, so deviation produced by the mirror will be:

${180^ \circ } - 2\left( {{{45}^ \circ } + X} \right) = {90^ \circ } - 2X$

Hence, total deviation produced ${90^ \circ } - 2X + {2^ \circ } = {92^ \circ } - 2X$

But,

${92^ \circ } - 2X = {90^ \circ }{\text{ }}\left[ {given} \right]$

$\Rightarrow X = {1^ \circ }$

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Therefore, the mirror is rotated by ${1^ \circ }$ degree then the total deviation of ray becomes ${90^ \circ }C$.

Note: Draw a well-labeled diagram of the given reflection and refraction scenario for a better understanding of the given question as visual cues will help. Formulas and universal facts like the deviation produced by a small angled prism are always a constant must be learned by the students beforehand.