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Question

A ray of light travels from a medium of refractive index $$ \mu $$ to air . Its angle of incidence in the medium is $$ \theta $$ , measured from the normal to the boundary and its angle of deviation is $$ \delta $$ with the angle of refraction $$ \phi $$. A graph is plotted between $$ \delta $$ and $$ \theta $$ which is shown in the figure. Choose the correct relation from the following

A
$$ sin \, \phi = \dfrac{1}{2} $$
B
$$ f = \dfrac{\pi}{2} - sin^{-1} \left (\dfrac{1}{\pi} \right ) $$
C
$$ \dfrac{\delta_2}{\delta_1} = \mu $$
D
$$ \dfrac{\delta_2}{\delta_1} = 2 $$
Solution
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Correct option is B. $$ \dfrac{\delta_2}{\delta_1} = 2 $$
Given: refractive index of medium u,
angle of incidence ,θ.
angle of deviation, $$\delta$$
angle of refraction,ϕ .
For θ < c, $$\delta$$ = ϕ - θ with sinϕ / sinθ = μ or ϕ = sin-1 (μsin θ).
$$\delta$$ = sin-1 (μsin θ) - θ This is a nonlinear relation.
The maximum value of $$\delta$$ is $$\delta$$1 = π/2 - c for θ = c and ϕ = π/2.
For θ > c, $$\delta$$ = π - 2θ $$\delta$$ decreases linearly with θ.
$$\delta$$max = π - 2c = $$\delta$$2 = 2$$\delta$$1.
therefore, $$\delta$$2$$\div$$$$\delta$$1=2.

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