Question

A thin prism ${{P}_{1}}$ with angle ${{4}^{o}}$ and made from glass of refractive index 1.54 is combined with another thin prism ${{P}_{2}}$ made from glass of refractive index 1.72 to produce dispersion without deviation. The angle of the prism ${{P}_{2}}$ is
$\text{A}\text{. }{{5.33}^{o}}$
$\text{B}\text{. }{{\text{4}}^{o}}$
$\text{C}\text{. }{{\text{3}}^{o}}$
$\text{D}\text{. 2}\text{.}{{\text{6}}^{o}}$

Answer 1

Hint: Use the formula for angle of deviation of light through a thin prism of refractive index $\mu $ and angle A , i.e. $\delta =(\mu -1)A$. No dispersion occurs when the angle of deviation of light is equal in both the prisms.

Complete step by step answer:
When light passes through a prism, due to refraction the light deviates. The angle of deviation is represented by $\delta $.

When the light from air passes through a thin prism the angle deviation is equal to $\delta =(\mu -1)A$….(i), where A is the angle of the prism and $\mu $ is the refractive index of the medium of the prism.

Light is made up of waves of different wavelengths and frequencies. The spectrum of visible light is known as VIBGYOR. The refractive index of a light wave depends on its wavelength. Waves with different wavelengths have different refractive indices.

From equation (i) we know that the deviation of light through thin prism is directly proportional to the refractive index of the light in the material of the prism.

Since, light consists of several wavelengths, all the different wavelengths deviate by different angles. And the white light splits into different colours.

The phenomenon of splitting of white light into its colours is called dispersion.

However, when another thin prism is placed upside down in front of the prism. The light deviates again but in the opposite direction such that the total angle of deviation is zero.

Hence, there is no dispersion.

It is given that prism ${{P}_{1}}$ has an angle of ${{4}^{o}}$ and it is made up of glass of refractive index 1.54.

Hence, the angle of deviation is ${{\delta }_{1}}=({{4}^{o}})(1.54-1)$.

For the second prism ${{P}_{2}}$, its refractive index is given as 1.72 and the angle of the prism is to be found. Let that angle be A’.

Hence, the angle of deviation is ${{\delta }_{2}}=A'(1.72-1)$.

For no dispersion ${{\delta }_{1}}={{\delta }_{2}}$

Therefore,

$({{4}^{o}})(1.54-1)=A'(1.72-1)$
$A'=\dfrac{({{4}^{o}})(0.54)}{(0.72)}={{3}^{o}}$

Therefore, the angle of the prism ${{P}_{2}}$ is ${{3}^{o}}$.

Hence, the correct option is C.

Note: Note that the formula $\delta =(\mu -1)A$ to find the angle of deviation of light rays works only when the light rays pass through a thin prism and the outside medium is air.
If the prism is a thin prism, the angle of deviation is independent of angle of incidence. It is constant for all angles of incidence.