Rank (from smallest to largest) the angle of refraction for a light ray in air entering each of the following substances with an angle of incidence equal to \(30^{\circ}\) : (i) water; (ii) benzene; (iii) dense flint glass; (iv) diamond.

To calculate the angle of refraction, we first need to know the index of refraction for each substance. We are given the indices of refraction for air, water, benzene, dense flint glass, and diamond as follows: - Air: \(n_1 = 1\) - Water: \(n_2 = 1.33\) - Benzene: \(n_2 = 1.50\) - Dense flint glass: \(n_2 = 1.66\) - Diamond: \(n_2 = 2.42\)

Using the index of refraction for air and each substance, we can now apply Snell's law to calculate the angle of refraction for each substance. Let's plug in the given angle of incidence (\(\theta_1 = 30^{\circ}\)) and indices of refraction for air and each substance into Snell's law: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\) For the four substances: 1. Water: \(\sin \theta_2 = \frac{1 \sin 30^{\circ}}{1.33}\) 2. Benzene: \(\sin \theta_2 = \frac{1 \sin 30^{\circ}}{1.50}\) 3. Dense flint glass: \(\sin \theta_2 = \frac{1 \sin 30^{\circ}}{1.66}\) 4. Diamond: \(\sin \theta_2 = \frac{1 \sin 30^{\circ}}{2.42}\) Now we compute the angle of refraction, \(\theta_2\), for each substance using the inverse sine function: 1. Water: \(\theta_2 = \arcsin \left(\frac{1 \sin 30^{\circ}}{1.33}\right) \approx 22^{\circ}\) 2. Benzene: \(\theta_2 = \arcsin \left(\frac{1 \sin 30^{\circ}}{1.50}\right) \approx 19^{\circ}\) 3. Dense flint glass: \(\theta_2 = \arcsin \left(\frac{1 \sin 30^{\circ}}{1.66}\right) \approx 17^{\circ}\) 4. Diamond: \(\theta_2 = \arcsin \left(\frac{1 \sin 30^{\circ}}{2.42}\right) \approx 12^{\circ}\)

Now that we have the angles of refraction for each substance, we can rank them from smallest to largest: 1. Diamond: \(12^{\circ}\) 2. Dense flint glass: \(17^{\circ}\) 3. Benzene: \(19^{\circ}\) 4. Water: \(22^{\circ}\) Thus, the ranking of the angle of refraction for the given substances, from smallest to largest, is diamond, dense flint glass, benzene, and water.