A ray of light (ray a) in air strikes a flat piece of glass at an angle of φ0 = 54° with respect to the normal, as shown in the figure. The index of refraction of the glass is 1.5. What is the angle θ between the reflected ray (ray b) and the refracted ray (ray c)? The refractive index of air , nair = 1)

Question

A ray of light (ray a) in air strikes a flat piece of glass at an angle of φ0 = 54° with respect to the normal, as shown in the figure. The index of refraction of the glass is 1.5. What is the angle θ between the reflected ray (ray b) and the refracted ray (ray c)? The refractive index of air , nair = 1)

BlackTom AI · Accepted Answer

To tackle this problem, I’ll parse the geometry and apply the standard laws of optics step by step.
Option A (87°): The angle between the reflected ray in air and the refracted ray in glass is the sum of the angle of incidence (54°) and the angle of refraction within the glass. First, use Snell's law: n1 sin(i) = n2 sin(r). With n1 = 1 (air), n2 = 1.5 (glass), and i = 54°, sin(r) = (1/1.5) sin(54°) ≈ 0.539. Therefore r ≈ arcsin(0.539) ≈ 32.7°. The reflected ray in air makes 54° with the normal, while the refracted ray in glass makes about 32.7° with the normal on the glass side. The angle between these two rays is the sum of these two angles relative to the normal, i.e., θ ≈ 54° + 32.7° ≈ 86.7°, which rounds to 87°. So this option is consistent with the geometry.
Option B (24°): This would imply a much smaller separation between the rays than physically occurs here. It doesn’t match the sum of the incidence and refraction angles calculated from Snell’s law, so it’s incorrect.
Option C (93°): This is close to the computed 86.7°, but it represents a value that would arise if you incorrectly subtracted instead of added, or if you used rough rounding in a wrong way. It does not align with the actual sum of 54° and ~32.7°.
Option D (33°): This corresponds approximately to the refracted angle alone (≈32.7°) or a misinterpretation of the geometry, rather than the full angle between the two rays. It’s not the angle between ray b and ray c.
Option E (54°): This equals the angle of incidence, not the angle between the reflected and refracted rays. It ignores the refraction angle and the fact that the two rays lie on opposite sides of the normal.
In summary, the correct geometric relation is that θ equals the sum of the incidence angle in air (54°) and the refracted angle in glass (approximately 32.7°), giving about 86.7°, i.e., 87° when rounded.

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