A goldfish lives in a 50-cm-diameter spherical fish bowl. The fish sees a cat watching it. If the cat's face is 20 cm from the edge of the bowl, how far from the edge does the fish see it as being? (You can ignore the thin glass wall of the bowl.)

Verified step by step guidance

1. Determine the radius of the spherical fish bowl. Since the diameter is given as 50 cm, the radius \( r \) is half of the diameter: \( r = \frac{50}{2} = 25 \, \text{cm} \).
2. Identify the refractive index of water \( n \), which is approximately \( n = 1.33 \). The refractive index of air is \( n_{\text{air}} = 1.00 \).
3. Use the formula for refraction at a spherical surface: \( \frac{n_1}{s} + \frac{n_2}{s'} = \frac{n_2 - n_1}{R} \), where \( n_1 \) is the refractive index of the medium the light is coming from (air), \( n_2 \) is the refractive index of the medium the light is entering (water), \( s \) is the object distance (distance of the cat's face from the bowl), \( s' \) is the image distance (distance where the fish sees the cat's face), and \( R \) is the radius of curvature of the spherical surface.
4. Substitute the known values into the formula: \( \frac{1.00}{20} + \frac{1.33}{s'} = \frac{1.33 - 1.00}{25} \). Simplify the equation to solve for \( s' \), the image distance.
5. Rearrange the equation to isolate \( \frac{1.33}{s'} \), then take the reciprocal to find \( s' \). This will give the distance from the fish to the image of the cat's face as seen through the water.

Key Concepts

Refraction of Light

Refraction is the bending of light as it passes from one medium to another, caused by a change in its speed. In this scenario, light travels from air into the water of the fish bowl, altering its path. This bending affects how the fish perceives the position of the cat, making it appear at a different distance than it actually is.

Spherical Geometry

Spherical geometry deals with figures on the surface of a sphere, where traditional Euclidean rules do not apply. The fish bowl's spherical shape means that the angles and distances perceived by the fish are influenced by the curvature of the bowl, which must be considered when calculating how far the fish sees the cat.

Optical Illusion

An optical illusion occurs when the perception of an object differs from its physical reality. In this case, the fish perceives the cat's position differently due to the combined effects of refraction and the spherical shape of the bowl, leading to a misjudgment of distance that is crucial for understanding the fish's perspective.