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

Hint: 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 \[\sin \dfrac{A}{2} = \dfrac{A}{2}\]
\[\mu = \dfrac{{\dfrac{{A + \delta }}{2}}}{{\dfrac{A}{2}}} \\
\Rightarrow \mu A = A + \delta \\ \]
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.