A Spherical Fish Bowl. A small tropical fish is at the center of a water-filled, spherical fish bowl 28.0 cm in diameter. 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.

검증된 단계별 안내

Determine the relevant parameters for the problem: The diameter of the spherical fish bowl is 28.0 cm, so the radius (R) is half of that, R = 14.0 cm. The refractive index of water (n₁) is approximately 1.33, and the refractive index of air (n₂) is approximately 1.00. The fish is located at the center of the sphere, so the object distance (s) is equal to the radius, s = 14.0 cm.

Use the formula for refraction at a spherical surface: \( \frac{n_2}{s'} - \frac{n_1}{s} = \frac{n_2 - n_1}{R} \), where \( s' \) is the image distance (apparent position of the fish). Substitute the known values: \( n_1 = 1.33 \), \( n_2 = 1.00 \), \( s = 14.0 \ \text{cm} \), and \( R = 14.0 \ \text{cm} \).

Rearrange the formula to solve for \( s' \) (the image distance): \( s' = \frac{n_2 s R}{n_1 R - (n_1 - n_2)s} \). Substitute the values into this equation to calculate \( s' \).

To find the magnification (M), use the magnification formula for refraction at a spherical surface: \( M = \frac{s'}{s} \). Substitute the value of \( s' \) obtained in the previous step and the given \( s = 14.0 \ \text{cm} \) to calculate the magnification.

Interpret the results: The value of \( s' \) will indicate the apparent position of the fish relative to the surface of the sphere, and the magnification \( M \) will indicate how much larger or smaller the fish appears to the observer outside the bowl.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 travels from water (denser medium) to air (less dense medium), resulting in the apparent displacement of the fish's position. This phenomenon is governed by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.

Lens Maker's Equation

The Lens Maker's Equation describes the relationship between the focal length of a lens and the radii of curvature of its surfaces, as well as the refractive indices of the lens material and surrounding medium. For a spherical fish bowl, it can be used to determine how the curvature of the bowl affects the focal point and the magnification of the image seen by an observer outside the bowl.

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, the magnification can be calculated using the distances involved in the refraction process, allowing the observer to understand how the fish appears through the curved surface of the bowl.