Chapter 23, Problem 90GP

To determine

The allowable range of index of refraction of a prism ( n ).

Answer to Problem 90GP

Solution:Allowable range of prism index n is greater than 1.881.

Explanation of Solution

According to the question, when there is no liquid covering the hypotenuse of the prism,a total internal reflection of the light beam happens. For the total internal reflection,the incidence angle should be greater than the critical angle. So, the maximum allowable range of refractive index can be related to the critical angle.

Given: The angle of a prism is $45°$ and the prism is right-angled.The Refractive index of a prism is n .

Formula used:

1. Snell’s law $n_1\sin {\theta }_1=n_2\sin {\theta }_2$ where $n_1$ and $n_2$ are the refractive index of two mediums and ${\theta }_1$ and ${\theta }_2$ are incident angle and refracted angle.
2. Angles in an isosceles right-angled triangle are $45°,45°$ and $90°$ .

Calculation:

  

For deviation of the light beam at the right angle, the beam should undergo total internal reflection at the hypotenuse of the prism. For this, the critical angle for glass- liquid interface must be less than the angle of incidence that is $45°$ as described in the above diagram. The incidence angle and reflected angle is $45°$ by the angle concurrency theorem and perpendicular bisector theorem.

  ${\theta }_c<45°\mathrm{...}\left(1\right)$

According to Snell’s law,

  $n\sin {\theta }_c=n_L\sin 90°$

  $\begin{array}{l}\sin {\theta }_c=n_L/n \left(\because \sin 90°=1\right)\hfill \\ {\theta }_c={\sin }^{-1}\left(n_L/n\right) \mathrm{...}\left(\text{2}\right)\hfill \end{array}$

According to equation 1 and equation 2,

  $\begin{array}{l}{\theta }_c<{45}^0\hfill \\ {\sin }^{-1}\left(n_L/n\right)<{45}^0\hfill \\ n_L/n<\sin {45}^0\hfill \\ n_L/n<1/\sqrt{2}\hfill \\ n>\sqrt{2}{n}_L\hfill \\ n>1\cdot 414n_L\hfill \end{array}$

So, if a liquid filled in the washing machine is water then the refractive index of water is 1·33.

So refractive index of the prism $n>\left(1\cdot 414\times 1\cdot 33\right)$

  $n>\left(1.881\right)$

Conclusion:

So,the allowable range of prism index n is greater than 1.881.