Question 16: Light travels a distance x in time t_{1} in air and 10x in time t_{2} in another denser medium. What is the critical angle for this medium?
(1) \sin^{-1}\left(\frac{t_{1}}{10t_{2}}\right)
(2) \sin^{-1}\left(\frac{10t_{1}}{t_{2}}\right)
(3) \sin^{-1}\left(\frac{t_{2}}{t_{1}}\right)
(4) \sin^{-1}\left(\frac{10t_{2}}{t_{1}}\right)
Answer: Option (2)
Explanation:
The speed of light in air is given by
v_{\text{air}}=\frac{x}{t_{1}}The speed of light in the denser medium is
v_{\text{medium}}=\frac{10x}{t_{2}}The refractive index of the denser medium with respect to air is
n=\frac{v_{\text{air}}}{v_{\text{medium}}}Substituting the values,
n=\frac{\frac{x}{t_{1}}}{\frac{10x}{t_{2}}}=\frac{t_{2}}{10t_{1}}The critical angle C for the medium is given by
\sin C=\frac{1}{n}Substituting the value of n,
\sin C=\frac{10t_{1}}{t_{2}}Hence, the critical angle is
C=\sin^{-1}\left(\frac{10t_{1}}{t_{2}}\right)Therefore, the correct answer is Option (2).