Question

Light takes time $t_1$ to travel a distance $x_1$ in rarer medium($M_1$) and the same light takes time $t_2$ to travel a distance $x_2$ in a relatively denser medium($M_2$). The value of critical angle when light goes from medium $M_2$ to medium $M_1$ is :

• A. ${\sin }^{-1}\left(\frac{x_2{t}_2}{x_1{t}_1}\right)$
• B. ${\sin }^{-1}\left(\frac{x_1{t}_2}{x_2{t}_1}\right)$
• C. ${\sin }^{-1}\left(\frac{x_1{t}_1}{x_2{t}_2}\right)$
• D. ${\sin }^{-1}\left(\frac{x_2{t}_1}{x_1{t}_2}\right)$

Answer 1

The correct option is D ${\sin }^{-1}\left(\frac{x_2{t}_1}{x_1{t}_2}\right)$

Speed of light in medium $M_1$,
$v_1=\frac{x_1}{t_1}$
So, refractive index, ${\mu }_1=\frac{c}{v_1}=\frac{c{t}_1}{x_1}$
Speed of light in medium $M_2$,
$v_2=\frac{x_2}{t_2}$
So, refractive index, ${\mu }_2=\frac{c}{v_2}=\frac{c{t}_2}{x_2}$
Now,
From formula,
$i_c={\sin }^{-1}\left(\frac{{\mu }_r}{{\mu }_d}\right)={\sin }^{-1}\left(\frac{{\mu }_1}{{\mu }_2}\right)$
$\Rightarrow i_c={\sin }^{-1}\left(\frac{c{t}_1/x_1}{c{t}_2/x_2}\right)$
$\Rightarrow i_c={\sin }^{-1}\left(\frac{x_2{t}_1}{x_1{t}_2}\right)$