Question

Parallel beam containing light of $\lambda =400nm$ and $500nm$ is incident on a prism as shown in figure. The refractive index $\mu$ of the prism is given by the relation, $\mu (\lambda )=1.20+\frac{0.8\times 10^{-14}}{{\lambda }^2}$ Which of the following statement is correct?

• A. Light of $\lambda =400nm$ undergoes
• B. Light of $\lambda =500nm$ undergoes total internal reflection
• C. Neither of the two wavelengths undergoes total intemal reflection
• D. Both wavelengths undergoes total internal reflection

Answer 1

Answer: (A)

We have, ${\mu }_1=1.20+\frac{0.8\times 10^{-14}}{{\left(400\times 10^{-9}\right)}^2}$
or ${\mu }_1=1.20+\frac{0.8\times 10^{-14}}{400\times 400\times 10^{-18}}$
or ${\mu }_1=1.20+\frac{0.8}{16}$
or ${\mu }_1=1.20+0.05$
or ${\mu }_1=1.25$
or $\sin i_c=\frac{1}{1.25}=0.8$
$\Rightarrow i_c=53.13^{\circ }$
Again, ${\mu }_2=1.20+\frac{0.8\times 10^{-14}}{{\left(500\times 10^{-9}\right)}^2}$

or ${\mu }_2=1.20+\frac{0.8}{25}$
or ${\mu }_2=1.20+0.32$
or ${\mu }_2=1.232$
Or $\sin i_c=\frac{1}{1.232}=0.81$
$\Rightarrow i_c={\sin }^{-1}0.81=54.26^{\circ }$
Now, $\sin \theta =0.8$ or $\theta =53.13^{\circ }$
This angle is clearly greater than critical angle corresponding to wavelength $400nm$.
So, light of $400nm$ wavelength undergoes total internal reflection.