Question

The refractive index of a material M1 changes by 0.014 and that of another material M2 changes by 0.024 as the colour of the light is changed from red to violet. Two thin prisms, one made of M1(A = 5.3°) and the other made of M2(A = 3.7°) are combined with their refracting angles oppositely directed. (a) Find the angular dispersion produced by the combination. (b) The prisms are now combined with their refracting angles similarly directed. Find the angular dispersion produced by the combination.

Answer 1

(a) The angular dispersion produced by the combination is 0.0146°.

(b) The prisms are now combined with their refracting angles similarly directed is 0.163°.

Explanation :  

$If \mu_{\mathrm{v}}^{\prime} and \mu_{\mathrm{r}}^{\prime}, \mathrm{M}_{1} are Material Refractive Indices$

$\mu_{\mathrm{v}}^{\prime}-\mu_{\mathrm{r}}^{\prime}=0.014$

$If \mu_{\mathrm{v}} and \mu_{\mathrm{r}}, \mathrm{M}_{2} are material refractive indices$

$\mu_{\mathrm{V}}-\mu_{\mathrm{r}}=0.024$

Now,

$For \ M_{1}, Angle of prism, A^{\prime}=5.3^{\circ}$

$For\ M_{2}, Angle of prism, \mathrm{A}=3.7^{\circ}$

(a) When the prisms are focused contrary, the angular dispersion

We know that,

$\delta_{1}=\left(\mu_{v}-\mu_{r}\right) A-\left(\mu_{v}^{\prime}-\mu_{r}^{\prime}\right) A^{\prime}$

When the values are replaced we get :

$\delta_{1}=0.024 \times 3.7^{\circ}-0.014 \times 5.3^{\circ}$

   $=0.0146^{\circ}$

The angular dispersion is, therefore, 0.0146 °.

(b) The angular dispersion is similar when the prisms are directed

We know that ,

$\delta_{2}=\left(\mu_{v}-\mu_{r}\right) A-\left(\mu_{v}^{\prime}-\mu_{r}^{\prime}\right) A^{\prime}$

When the values are replaced we get :

$\delta_{2}=0.024 \times 3.7^{\circ}+0.014 \times 5.3^{\circ}$

   $=0.163^{\circ}$

The angular dispersion is, therefore, 0.163°.