A beam of light has a small wavelength spread $δ λ$ above a central wavelength $λ$ . The beam travels in vacuum until it enters a glass plate at an angle $θ$ relative to the normal to the plate as shown in the figure. The index of refraction of the glass is given by $n (λ)$ . The angular spread $δ θ$ ’ of the refracted beam is given by

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$δ θ'$ remains constant for different values of $θ$ in the range $\frac{π}{6} \leq θ \leq \frac{π}{3}$

$δ θ'$ increases with the increase in $θ$

$δ θ' = | \frac{\tan θ'}{n} \frac{d n (λ)}{d λ} δ λ |$

$δ θ' = | \frac{\sin θ}{\sin θ'} \frac{δ λ}{λ} |$

answer is A, B.

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Detailed Solution

$(1) \sin θ = n (λ) \sin θ' S o, 0 = \sin θ' \frac{d n (λ)}{d λ} + n (λ) \cos θ' \frac{d θ'}{d λ} ∴ d θ' = - \frac{\sin θ'}{\cos θ} \frac{1}{n (λ)} \frac{d n (y)}{d λ} d λ ∴ δ θ' = - \frac{\tan θ'}{n (λ)} \frac{d n (λ)}{d λ} δ λ$

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