Question

Formula for dispersive power is (where symbols have their usual meanings) or
If the refractive indices of crown glass for red, yellow, and violet colours are respectively ${\mu _r}$ , ${\mu _y}$ and ${\mu _v}$ , then the dispersive power of this glass would be
A. $\dfrac{{{\mu _v} - {\mu _y}}}{{{\mu _r} - 1}}$
B. $\dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y} - 1}}$
C. $\dfrac{{{\mu _v} - {\mu _y}}}{{{\mu _y} - {\mu _r}}}$
D. $\dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y}}} - 1$

Answer 1

Before we get into the question, let's review some basics of dispersive power. The breaking of white light into its constituent colours is referred to as dispersion. A spectrum is a collection of colours. The separation of different colours of light by refraction is the dispersive power of a transparent medium.

Complete step by step answer:
Consider a glass prism; the refractive index of a prism is determined by the connection relationship,
$\mu = \dfrac{{\sin \dfrac{{A + D}}{2}}}{{\sin \dfrac{A}{2}}}$
where $A$ is the prism's central angle and $\delta$ is the deviation angle.

If $A$ is a small angled prism's refracting angle and is the angle of deviation $\delta$ . The prism formula is as follows:
$\mu = \dfrac{{\sin \dfrac{{A + \delta }}{2}}}{{\sin \dfrac{A}{2}}}$
Because we're talking about small angled prisms,
$\sin \dfrac{{A + \delta }}{2} = \dfrac{{A + \delta }}{2}$ and $\sin \dfrac{A}{2} = \dfrac{A}{2}$

\Rightarrow \mu A = A + \delta \\\ $$ Therefore, $$\delta = \left( {\mu - 1} \right)A$$. The corresponding wavelengths are $${\mu _v}$$and $${\mu _r}$$ if ${\delta _v}$ and $${\delta _r}$$ are the deviations of violet and red rays, respectively. As a result, the angular dispersion is expressed as, $$\delta v - \delta r = (\mu v - \mu r)A$$ The difference in deviation between extreme colours is known as angular dispersion. If ${\mu _y}$ and $${\delta _y}$$ are the refractive index and deviation of an intermediate wavelength yellow, then $${\delta _y} = \left( {{\mu _y} - 1} \right)A$$ When both equations are divided, we get $$\dfrac{{{\delta _v} - {\delta _r}}}{{{\delta _y}}} = \dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y}}}$$ The term $$\dfrac{{{\delta _v} - {\delta _r}}}{{{\delta _y}}}$$ stands for dispersive power of the prism's substance and is represented by the symbol $\omega $ , $$\omega = \dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y}}}$$ Thus, $$\omega = \dfrac{{(\mu y - 1)A}}{{(\mu v - \mu r)A}} \\\ \therefore \omega = \dfrac{{(\mu y - 1)}}{{(\mu v - \mu r)}}$$ Therefore, the dispersive power of this glass would be $$\dfrac{{{\mu _v} - {\mu _r}}}{{{\mu _y} - 1}}$$. **Hence, the correct option is B.** **Note:** Violet rays have a larger deviation and refractive index than red rays, so keep that in mind. As a result, violet light passes through glass at a slower rate than red rays. The refractive index and deviation of yellow rays are utilised as mean values.