BIO The Lens of the Eye. The crystalline lens of the human eye is a double-convex lens made of material having an index of refraction of 1.44 (although this varies). Its focal length in air is about 8.0 mm, which also varies. We shall assume that the radii of curvature of its two surfaces have the same magnitude. Find the radii of curvature of this lens.

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Step 1: Recall the lens maker's formula, which relates the focal length \( f \), the index of refraction \( n \), and the radii of curvature \( R_1 \) and \( R_2 \) of a lens: \( \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \).

Step 2: Since the problem states that the radii of curvature of the two surfaces have the same magnitude, we can set \( R_1 = R \) and \( R_2 = -R \) (the negative sign accounts for the opposite curvature direction). Substitute these into the lens maker's formula: \( \frac{1}{f} = (n - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) \).

Step 3: Simplify the expression inside the parentheses: \( \frac{1}{R} - \frac{1}{-R} = \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \). The formula now becomes \( \frac{1}{f} = (n - 1) \cdot \frac{2}{R} \).

Step 4: Rearrange the equation to solve for \( R \): \( R = 2(n - 1)f \).

Step 5: Substitute the given values: \( n = 1.44 \) and \( f = 8.0 \ \text{mm} \). Plug these into the equation to calculate \( R \): \( R = 2(1.44 - 1)(8.0) \). Perform the arithmetic to find the radius of curvature.

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주요 개념

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

Lens Formula

The lens formula relates the focal length (f) of a lens to the object distance (u) and the image distance (v) through the equation 1/f = 1/v - 1/u. For a double-convex lens, the focal length is positive, indicating that it converges light. Understanding this formula is essential for calculating the radii of curvature based on the lens's focal length.

Index of Refraction

The index of refraction (n) is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. For the crystalline lens, with an index of refraction of 1.44, this property affects how light bends when entering and exiting the lens, influencing its focal length and curvature.

Radii of Curvature

The radii of curvature (R1 and R2) of a lens are the distances from the lens's surfaces to its center of curvature. For a double-convex lens, both surfaces have the same magnitude of curvature but opposite signs. These radii are crucial for applying the lens maker's equation, which relates the focal length of the lens to its radii of curvature and the index of refraction.