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# 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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