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Q.
What is the required condition, if the light incident on one face of a prism, does not emerge from the other face?
NTA AbhyasNTA Abhyas 2020Ray Optics and Optical Instruments
Solution:
Consider the situation shown in figure.
Light rays $P Q$ is incident on the face $A B$ . It goes along $Q R$ and then along $R S$ .
Here,
$i=$ angle of incidence at the face $AB$
$r_{1}=$ angle of refraction through $AB$
$r_{2}=$ angle of incidence at $AC$
$n=$ refractive index of the material of the prims.
For the prism, we have
$r_{1}+r_{2}=A$ .....(i)
Now for light not to emerge from the face $AC,$
$r_{2}>C$
Similarly for no emergence from the face $AB,$
$r_{1}>C$
$\Rightarrow \, \, r_{1}+r_{2}>2C$
Here, $C$ is critical angle.
$\Rightarrow \, A>2C$ $\left[\because \, r_{1} + r_{2} = A\right]$
$\Rightarrow \, \, \frac{A}{2}>C$
$\Rightarrow \, sin \frac{A}{2}>sin C$
$\Rightarrow sin \frac{A}{2}>\frac{1}{n}$
$\Rightarrow \, \, n>\frac{1}{sin \frac{A}{2}}$
$\Rightarrow \, n>cosec\left(\frac{A}{2}\right)$
