Question

If the refractive indices for crown glass for red, yellow and violet colors are 1.5140, 1.5170, and 1.5318, and 1.6434, 1.6499 and 1.6852 respectively, then the dispersive power for crown and flint glass are respectively-

Answer 1

The dispersive power of the glass is the ratio of the angular dispersion to the mean deviation. The angular separation is the difference in the refractive indices of widely separated colors in the spectrum that is violet and red. Recall the formula for the dispersive power and find out the dispersive power of crown glass and flint glass.

Formula used:
${\omega _c} = \dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y} - 1}}$
Here, ${\mu _v}$ is the refractive index for the violet color, ${\mu _r}$ is the refractive index for the red color, ${\mu _y}$ is the refractive index for the yellow color.

Complete step by step answer:
We have given that for a crown glass, the refractive index of red color is ${\mu _r} = 1.5140$, for yellow color ${\mu _y} = 1.5170$ and for violet color ${\mu _v} = 1.5318$.
Also, for flint glass, the refractive index of red color is ${\mu _r} = 1.6434$, for yellow color ${\mu _y} = 1.6499$ and for violet color ${\mu _v} = 1.6852$.

We have the dispersive power of the glass is the ratio of the angular dispersion to the mean deviation. The angular dispersion is the difference in the refractive index of violet and red color. The mean deviation is ${\mu _y} - 1$. Therefore, we can express the dispersive power for crown glass as,
${\omega _c} = \dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y} - 1}}$
Substituting ${\mu _r} = 1.5140$, ${\mu _v} = 1.5318$ and ${\mu _y} = 1.5170$ in the above equation, we get,
${\omega _c} = \dfrac{{1.5318 - 1.5140}}{{1.5170 - 1}}$
$\Rightarrow {\omega _c} = \dfrac{{0.0178}}{{0.517}}$
$\Rightarrow {\omega _c} = 0.034\,{\text{W}}$
Therefore, the dispersive power of the crown glass is 0.034 Watt.

We can express the dispersive power for flint glass as,
${\omega _f} = \dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y} - 1}}$
Substituting ${\mu _r} = 1.6434$, ${\mu _v} = 1.6852$ and ${\mu _y} = 1.6499$ in the above equation, we get,
${\omega _c} = \dfrac{{1.6852 - 1.6434}}{{1.6499 - 1}}$
$\Rightarrow {\omega _f} = \dfrac{{0.0418}}{{0.6499}}$
$\therefore {\omega _f} = 0.064\,{\text{W}}$

Therefore, the dispersive power of the flint glass is 0.064 Watt.

Note: The angular separation is the difference between the refractive index of the widely separated colors in the spectrum and therefore, it is the difference between the refractive indices of violet and red colors. We know that the refractive index has no unit but the dispersive power has a unit of watt. Make sure that the refractive index of all the colors is greater than 1 because the speed of any colored light is less than the speed of light.