Question

The refractive indices of red, violet and yellow light are respectively $1.42$, $1.62$ and $1.50$ . The dispersive power of the medium will be?

Answer 1

Hint: Dispersive power of the medium is defined as the ratio of angular dispersion to mean deviation. This ratio can be found using the deviation angles through a prism. Also, we can find the same deviation if we know the deviation of red, violet and yellow light. We will use the formula which makes use of their deviation to find the dispersive power.

Formula used: $\text{Dispersive Power = }\dfrac{{{\mu }_{v}}-{{\mu }_{r}}}{{{\mu }_{y}}-1}$

Complete step by step answer:
We know that dispersive power is nothing but the ratio of angular dispersion to the mean deviation produced. This can be written as:
$\text{Dispersive Power = }\dfrac{\text{Angular Dispersion}}{\text{Mean Deviation}}$ -----(i)
We also know that the dispersive power can be calculated as from the deviation of different colors when passed through a prism. We can also write the formula of dispersive power in terms of the refractive indices of these colors as:
$\text{Dispersive Power = }\dfrac{{{\delta }_{v}}-{{\delta }_{r}}}{\delta }$
$\Rightarrow \text{Dispersive Power = }\dfrac{{{\mu }_{v}}-{{\mu }_{r}}}{\mu -1}$ -----(ii)
Where, ${{\mu }_{v}}$ is the refractive index of violet light, ${{\mu }_{r}}$ is the refractive index of red light and, $\mu $ is the mean refractive index.
Also, we know that yellow is considered to have a mean refractive index and we can put the refractive index of yellow light in place of the mean refractive index. So we can modify the equation (ii) as:
$\Rightarrow \text{Dispersive Power = }\dfrac{{{\mu }_{v}}-{{\mu }_{r}}}{{{\mu }_{y}}-1}$
Now, putting the corresponding values as mentioned in the question, we get:
$\text{Dispersive Power = }\dfrac{1.62-1.42}{1.5-1}$
$\Rightarrow \text{Dispersive Power = 0}\text{.4}$

Hence the dispersive power of the medium is 0.4

Note: The dispersive power of a body is nothing but the ability to disperse light incidents on that body. The measure of dispersive power of a transparent material for the given visible spectrum is termed as Reciprocal Dispersion. It is also unitless as dispersion.