91Ó°ÊÓ

Make use of the Snell's Law to determine the angle of refraction within the glass slab. Snell's Law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction for a given pair of media is equal to the ratio of the indices of refraction: \[n_1 \times \sin{\theta_1} = n_2 \times \sin{\theta_2}\] Here, \(n_1 = 1\) (since the light ray is entering from air), and \(\theta_1 = 45^{\circ}.\) We need to find \(\theta_2.\) \[1 \times \sin{45^{\circ}} = n_2 \times \sin{\theta_2}\] Now, let's call the minimum index of refraction for the glass slab \(n_{min}.\) Using this notation, \[\sin{45^{\circ}} = n_{min} \times \sin{\theta_2}\]

Total internal reflection occurs when the angle of incidence within the denser medium (glass slab) is greater than the critical angle. The critical angle formula is: \[\sin{\theta_c} = \frac{n_1}{n_2}\] In this case, \(\theta_c = \theta_2,\) and \[\sin{\theta_c} = \frac{1}{n_{min}}\]

Combine the results obtained from Step 1 and Step 2: \[\sin{45^{\circ}} = n_{min} \times \sin{\theta_c}\] Now we substitute \(\sin{\theta_c}\) with its expression in terms of \(n_{min}\): \[\sin{45^{\circ}} = n_{min} \times \frac{1}{n_{min}}\] \[\sin{45^{\circ}} = 1\] To achieve total internal reflection, we must have \(1 \ge \frac{1}{n_{min}},\) which implies: \[n_{min} \ge 1\] But this is a trivial inequality, so we consider the condition of equality: \[1 = \frac{1}{n_{min}}\] This leads us to the conclusion that total internal reflection is not possible in this scenario, since the index of refraction should be greater than 1 for total internal reflection to occur and the equation above shows that it should be equal to 1. However, since we need to choose an answer, we'll choose the one that is closest to the trivial inequality: Among the options, (A) \(\frac{\sqrt{3}}{2} \approx 0.87\), (B) \(\sqrt{\frac{3}{2}} \approx 1.22\), (C) \(\frac{3}{2} = 1.5\), and (D) \(\frac{3}{\sqrt{2}} \approx 2.12\), the closest to the trivial inequality is: Answer: (B) \(\sqrt{\frac{3}{2}}\)