A light ray passes from air into a liquid with i=50° and r=32°. The problem also states sound speeds of 340 m s⁻¹ in air and 1500 m s⁻¹ in the liquid. Find (a) optical refractive index and (b) speed of light in the liquid.

Solution & explanation

1. Given: i = 50°, r = 32°; take optical c = 3.0×10⁸ m s⁻¹.

2. The stated sound speeds do not determine optical refraction; do not use 340/1500.

3. Apply Snell’s law for air to liquid.

   $$n=\frac{\sin \left(\frac{50\times \pi }{180}\right)}{\sin \left(\frac{32\times \pi }{180}\right)}$$

4. Evaluate both sines and divide.

   $$n=\frac{0.7660444431}{0.5299192642}$$

5. This gives approximately 1.44595.

6. Rearrange n = c/v to v = c/n.

   $$v=\frac{c}{n}$$

7. Use the unrounded refractive index.

   $$v=\frac{300000000\times \sin \left(\frac{32\times \pi }{180}\right)}{\sin \left(\frac{50\times \pi }{180}\right)}$$

8. State the result with units: n ≈ 1.446; v ≈ 2.075×10⁸ m s⁻¹.

n ≈ 1.446; v ≈ 2.08×10⁸ m s⁻¹