Monochromatic light is incident on a glass prism of angle A. If the refractive index of the material of the prism is, a ray, incident at an angle , on the face AB would get transmitted through the face AC of the prism provided : Option 1) Option 2) Option 3) Option 4)

As we learnt in

Relation between angle of incidence and angle of refaction -

$\mu _{1}\sin i= \mu _{2}\sin r$

- wherein

$\mu _{1}=$ refractive index of medium of incidence.

$\mu _{2}=$  refractive index of medium where rays is refracted.

$i=$ angle of incidence.

$r=$ angle of refraction.

Where r2 = C     $\angle N_{2}Rc=90^{o}$

where c = critical angle

$sinc=\frac{1}{\nu}=sinr_{2}$

Applying Snell's law at R

$\mu sin r_{2}=1\ sin90^{o}$                        (i)

Snell's law at Q

$1+sin\theta=\mu sinr r_{1}$                    (ii)

$A=r_{1}+r_{2}\ \Rightarrow\ \; r_{1}=A-r_{2}$

$sin \theta=\mu sin(A-r_{2})\ \; \Rightarrow\ \; sin\theta=\mu sin A cos r_{2}-cosA$

$cos\ r_{2}=\sqrt{1-sin^{2}r_{2}}=\sqrt{1-\frac{1}{\mu^{2}}}$                                (iv)                    (iii)

$\therefore$    From equation (iii) and (iv)

$sin\theta=\mu sinA \sqrt{1-\frac{1}{\mu^{2}}}-cos A$

$theta=sin^{-1}\left[\mu sin(A-sin^{-1})\left(\frac{1}{\mu} \right ) \right ]$

Correct option is 1.

Option 1)

This is the correct option.

Option 2)

This is an incorrect option.

Option 3)

This is an incorrect option.

Option 4)

This is an incorrect option.

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