A Spherical Fish Bowl. A small tropical fish is at the center of a water-filled, spherical fish bowl 28.0 cm in diameter. (a) Find the apparent position and magnification of the fish to an observer outside the bowl. The effect of the thin walls of the bowl may be ignored.

Verified step by step guidance

1. Identify the relevant variables and constants: The diameter of the spherical fish bowl is 28.0 cm, so the radius (r) is half of that, which is 14.0 cm. The refractive index of water (n) is approximately 1.33.
2. Understand the concept of refraction at a spherical surface: When light passes from a medium of higher refractive index (water) to a lower refractive index (air), it bends away from the normal. Use the formula for refraction at a spherical surface to find the image distance (i) from the center of the sphere: \( \frac{n_1}{p} + \frac{n_2}{i} = \frac{n_2 - n_1}{R} \), where n1 is the refractive index of water, n2 is the refractive index of air (approximately 1.0), p is the object distance (which is -14.0 cm, as the object is inside the sphere), and R is the radius of curvature of the sphere.
3. Calculate the apparent position of the fish: Substitute the known values into the refraction formula and solve for the image distance (i). This will give the position of the fish as seen by an observer outside the bowl.
4. Determine the magnification of the fish: Magnification (m) can be calculated using the formula \( m = -\frac{n_1}{n_2} \times \frac{i}{p} \). Substitute the values of n1, n2, i, and p to find the magnification, which tells you how much larger or smaller the fish appears to an observer outside the bowl.
5. Interpret the results: If the magnification is greater than 1, the fish appears larger; if it is less than 1, the fish appears smaller. The sign of the magnification also indicates whether the image is upright or inverted.

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 the context of the fish bowl, light rays emanating from the fish travel through water and then into air, resulting in a change of direction. This bending alters the perceived position of the fish to an observer outside the bowl.

Lens Maker's Equation

The Lens Maker's Equation relates the focal length of a lens to the radii of curvature of its surfaces and the refractive indices of the materials involved. For a spherical interface like the fish bowl, this equation helps determine how the curvature affects the light paths and the resulting image characteristics, such as position and size.

Magnification

Magnification is the ratio of the height of the image to the height of the object, indicating how much larger or smaller the image appears compared to the actual object. In this scenario, calculating the magnification involves understanding how the refraction of light through the spherical bowl alters the size of the image of the fish as seen by the observer.