Case Study
Passage with linked questions
Case Set 1
Case AnalysisPassage
Riya is standing at the edge of a swimming pool filled with water (refractive index 1.33) to a depth of 2.5 m. She notices that the tiles at the bottom appear much closer to the surface than they actually are. Her physics teacher explains that this is due to refraction — when light travels from a denser medium (water) to a rarer medium (air), it bends away from the normal, causing the apparent position of the object to shift upward. The teacher also points out that if Riya were to look at the tiles from an oblique angle rather than straight down, the apparent depth would be even less than what is calculated using the standard formula. This phenomenon is routinely used by divers and underwater photographers to estimate distances.
Question 1: Calculate the apparent depth of the tiles as seen by Riya looking straight down into the pool.
- Apparent depth = Real depth / Refractive index = 2.5 / 1.33 ≈ 1.88 m.
- The tiles appear to be approximately 1.88 m below the surface instead of the actual 2.5 m, because light from the tiles bends away from the normal as it exits the water, making the source appear closer.
Question 2: If the water is replaced by a liquid of refractive index 1.63 to the same depth, by what distance would the apparent depth change compared to the water-filled pool?
- Apparent depth in new liquid = 2.5 / 1.63 ≈ 1.53 m; apparent depth in water ≈ 1.88 m.
- The apparent depth decreases by 1.88 − 1.53 = 0.35 m, so the tiles appear 35 cm closer to the surface in the denser liquid than in water.
Question 3: Explain why the apparent depth formula (apparent depth = real depth / n) is valid only for near-normal viewing and not for oblique viewing. What happens to the apparent position as the viewing angle increases?
- The formula apparent depth = real depth / n is derived using the small-angle (paraxial) approximation, where tan r ≈ sin r ≈ r. This approximation holds only when the angle of refraction is small, i.e., the observer views nearly perpendicularly to the surface.
- At oblique angles, the angle of refraction is large and the paraxial approximation breaks down; the actual apparent depth must be calculated using the full Snell's law geometry, which gives a smaller apparent depth than the formula predicts.
- As the viewing angle increases further toward the critical angle (ic ≈ 48.75° for water), the refracted ray grazes the surface; beyond the critical angle, no light reaches the observer from that point, so the object becomes invisible — this is related to the total internal reflection phenomenon observed from inside the denser medium.