Problem 16 - University Physics with Modern Physics

Question

A tank whose bottom is a mirror is filled with water to a depth of 20.0 \(\mathrm{cm} .\) A small fish floats motionless 7.0 \(\mathrm{cm}\) under the surface of the water. (a) What is the apparent depth of the fish when viewed at normal incidence? (b) What is the apparent depth of the image of the fish when viewed at normal incidence?

Step-by-Step Solution

Answer

(a) 5.26 cm, (b) 20.30 cm

Step 1: Understanding the Concept

When an object is viewed through a different medium, such as water, it appears at a different position than its actual position due to refraction. The apparent depth is given by the formula \( d' = \frac{d}{n} \), where \( d \) is the actual depth and \( n \) is the refractive index of water (approximately 1.33).

Step 2: Calculating Apparent Depth of the Fish

The fish is actually 7 cm below the water surface. Using the formula \( d' = \frac{d}{n} \), substitute \( d = 7 \) cm and \( n = 1.33 \). The apparent depth \( d' = \frac{7}{1.33} \approx 5.26 \) cm.

Step 3: Understanding the Image in the Mirror

The bottom of the tank is a mirror, so an image of the fish will be formed at an additional depth equal to the distance from the fish to the mirror. The mirror creates a virtual image at the same distance below it as the original fish above.

Step 4: Calculating Path to Image in the Mirror

The total distance from the water surface to the image in the mirror is the sum of the depth of the fish (7 cm) and the distance from the mirror to the water surface (20 cm). This is 27 cm.

Step 5: Calculating Apparent Depth of Mirror Image

The apparent depth of the image is calculated by using the entire 27 cm distance as the actual depth, leading to \( d'' = \frac{27}{1.33} \approx 20.30 \) cm.

Key Concepts

Refraction

When light passes through different materials, its speed changes. This bending of light is known as refraction. If you're looking at something underwater, like a fish, refraction makes it seem like it's in a different spot than it really is. Imagine bending a pencil by placing part of it in a glass of water, it looks broken at the water's surface, due to the light bending.

• Refraction causes objects to look as if they are at a shallower depth than they actually are.
• This happens because light travels slower in water compared to air, altering its path.

Understanding refraction is key to explaining why a fish looks closer to the water surface than it really is.

Refractive Index

The refractive index is a number that tells us how much light slows down in a material. Each substance has its own refractive index. For example, the refractive index of water is around 1.33.

• This means light travels 1.33 times slower in water than in a vacuum.
• The apparent depth can be calculated using the formula: \( d' = \frac{d}{n} \), where \( d \) is the actual depth and \( n \) is the refractive index.

For the fish at 7 cm depth, using this formula gives us an apparent depth of approximately 5.26 cm when viewed from above, showing how much closer the fish seems to be.

Virtual Image

A virtual image is formed when light rays appear to meet at a location, but do not actually converge there. In our scenario with the fish and the mirror at the tank's bottom, the virtual image looks real to the observer, even though no fish is present at that location in the water.

• The mirror at the bottom of the tank creates an image of the fish as if below the mirror.
• The fish's apparent image is then deeper than the actual mirrored image due to refraction.

By calculating the apparent depth of the fish and mirror image, we find that the fish appears to be at a depth of approximately 20.30 cm, considering the combined effects of the mirror and refraction.