Question

Two prisms $ A $ and $ B $ are in contact with each other and have angular dispersions of $ {2^0} $ and $ {4^0} $ respectively. The dispersive power of $ 'A' $ is $ 0.002 $ . If the combination produces dispersion without deviation, the dispersive power of ' $ B $ ' is
(A) $ 0.001 $
(B) $ 0.004 $
(C) $ 0.002 $
(D) $ 0.006 $

Answer 1

Hint: Dispersive power of prism is defined as the measure of the difference refraction of light between the highest wavelength of light and the lowest wavelength entering the prism. In terms of the angle between the two extreme wavelengths, the dispersive power of prism can also be expressed. The angle between them increases as the dispersive power increases. Watt is the SI unit of prism dispersive power.

Complete Step-by-Step Solution
According to the question, it is given that the two prisms $ A $ and $ B $ have angular dispersions of $ {2^0} $ and $ {4^0} $ respectively
The dispersive power of prism $ A $ is $ 0.002 $
Now, the dispersive power of prism $ A $ is given by the following relation
 $ \dfrac{{{{\left( {{\delta _v} - {\delta _R}} \right)}_A}}}{{{\delta _A}}} = 0.002 $
The above equation can be rewritten as
 $ \dfrac{{{2^ \circ }}}{{{\delta _A}}} = 0.002 $
On rearranging the terms, we get
 $ {\delta _A} = \dfrac{{{2^ \circ }}}{{0.002}} $
Similarly, the dispersive power of prism $ B $ can be calculated by
 $ \dfrac{{{{\left( {{\delta _v} - {\delta _R}} \right)}_B}}}{{{\delta _B}}} $
Since, the combination of prisms $ A $ and $ B $ produces no net deviation, the dispersive powers of prism $ A $ and prism $ B $ will be equal.
That is
 $ {\delta _A} = {\delta _B} $
So, the dispersive power of prism $ B $ is given by
 $ \dfrac{{{2^ \circ }}}{{{4^ \circ }}} \times 0.002 = 0.004 $
Hence, the correct option is (B).

Additional Information
A transparent, polished flat optical element that reflects light is defined as a prism in physics. These can be made from any transparent material for which they are designed to have wavelengths. Glass, fluorite, and plastic are the most commonly used materials.

Note
The difference in the refraction of lights at the highest and lowest speeds is the dispersion. Dispersion relies on the refractive indexes of both media. Thus, depending on the material of the prism, the dispersive power varies. These are used to break the light into the spectral colours of their constituents. Amici prism, a triangular prism, are a few examples of the dispersive prism.