An astronaut is exploring an unknown planet when she accidentally drops an oxygen canister into a 1.50-m-deep pool filled with an unknown liquid. Although she dropped the canister 21 cm from the edge, it appears to be 31 cm away when she peers in from the edge. What is the liquid's index of refraction? Assume that the planet's atmosphere is similar to earth's.

Verified step by step guidance

1. Step 1: Understand the problem. The apparent position of the canister is distorted due to refraction of light as it passes from the liquid to the atmosphere. The index of refraction of the liquid can be calculated using the relationship between the real depth and the apparent depth.
2. Step 2: Recall the formula for the index of refraction: \( n = \frac{d_{real}}{d_{apparent}} \), where \( d_{real} \) is the actual depth of the object and \( d_{apparent} \) is the depth as it appears to the observer.
3. Step 3: Identify the given values. The real depth of the canister is \( d_{real} = 1.50 \, \text{m} \). The apparent depth can be calculated using the ratio of the apparent distance to the real distance horizontally: \( d_{apparent} = \frac{31 \, \text{cm}}{21 \, \text{cm}} \times d_{real} \).
4. Step 4: Substitute the values into the formula for \( d_{apparent} \): \( d_{apparent} = \frac{31}{21} \times 1.50 \, \text{m} \). This gives the apparent depth of the canister.
5. Step 5: Use the formula for the index of refraction \( n = \frac{d_{real}}{d_{apparent}} \) and substitute \( d_{real} = 1.50 \, \text{m} \) and the calculated \( d_{apparent} \) to find the index of refraction of the liquid.

Key Concepts

Refraction

Refraction is the bending of light as it passes from one medium to another with a different density. This phenomenon occurs because light travels at different speeds in different materials. The degree of bending is described by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how much light slows down in a medium compared to its speed in a vacuum. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. A higher index indicates that light travels slower in that medium, affecting how objects appear when viewed through it.

Geometric Optics

Geometric optics is the branch of optics that describes light propagation in terms of rays. It simplifies the study of light behavior by ignoring wave properties and focusing on the paths that light rays take. This approach is particularly useful for analyzing situations involving lenses, mirrors, and refraction, such as the scenario of the astronaut observing the canister in the liquid.