Question

A spectrum is formed by a prism of dispersive power $'\omega '$ . If the angle of deviation is $'\delta '$ then the angular dispersion is:
A. $\dfrac{\omega }{\delta }$
B. $\delta /s$
C. $\dfrac{1}{{\omega \delta }}$
D. $\omega \delta $

Answer 1

Hint:As we know when a ray of white light passes through a prism it gets split into a spectrum of light with red and violet at the two extremes.And we know that each colour deviates at different angles since each colour experience a different value of refractive index in the medium.Now, Dispersive power is nothing but just the property of the material of the prism which tells us about how greater is the angular dispersion by the prism.

Complete step by step answer:
Here, in the question it is given that the dispersive power of the prism is $\omega $.
And the angle of deviation is $\delta $ .
Angular dispersion = angular deviation of violet – angular deviation of red
i.e. Angular dispersion = ${\delta _V} - {\delta _R}$
Dispersive power is given by the ratio of angular dispersion to the angle of deviation of the mean colour between the two extremes (I.e. yellow).
\[
{\text{Dispersive power = }}\dfrac{{{\text{Angular dispersion}}}}{{{\text{mean deviation}}}} \\
\Rightarrow \omega = \dfrac{{{\delta _V} - {\delta _R}}}{{{\delta _{mean}}}} - - - - - - - - - - - - - [1] \\
\]
From eq.1 we can find that:
Angular dispersion = ${\delta _V} - {\delta _R} = \omega {\delta _{mean}}$
The angle of deviation given in the question must be mean deviation (i.e. ${\delta _{mean}} = \delta $ )
Hence, the answer is: Angular dispersion = $\omega \delta $

Therefore, option D is correct.

Additional information:
Angular dispersion is the difference in deviations of the two extreme frequencies in the spectrum produced by the prism. Dispersive power of the prism is given by the ratio of angular dispersion to the mean angle of deviation. So, we can find the formula of angular dispersion from the above formula. Using this information, we can find out the correct answer.

Note:As we know when a ray of white light passes through a prism it gets split into a spectrum of light with red and violet at the two extremes and we know that each colour deviates at different angles since each colour experience a different value of refractive index in the medium. Now, Dispersive power is nothing but just the property of the material of the prism which tells us about how greater is the angular dispersion by the prism. Hence, the dispersive power is found for these two extremes and mean between them (yellow).