A camera’s close-up lens is aimed at a butterfly 200 mm in front of the lens, creating a focused on the detector 50 mm behind the lens. A proper exposure requires an f-number of F8.0. What is the correct diameter of the lens aperture?

검증된 단계별 안내

Understand the relationship between the f-number (F) and the aperture diameter (D). The f-number is defined as the ratio of the focal length (f) to the diameter of the aperture:

F = \frac{f}{D}

Rearrange the formula to solve for the aperture diameter (D):

D = \frac{f}{F}

Identify the focal length (f) of the lens. From the lens equation,

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

, where

d_o

is the object distance (200 mm) and

d_i

is the image distance (50 mm). Substitute these values to calculate

f

Substitute the calculated focal length (f) and the given f-number (F = 8.0) into the formula

D = \frac{f}{F}

to find the aperture diameter.

Ensure the units are consistent (e.g., mm) throughout the calculation and simplify the expression to determine the correct aperture diameter.

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주요 개념

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Lens Aperture and f-number

The lens aperture is the opening in a camera lens that allows light to enter. The f-number (or f-stop) is a ratio that describes the size of the aperture relative to the focal length of the lens. A lower f-number indicates a larger aperture, allowing more light to hit the sensor, while a higher f-number indicates a smaller aperture, reducing light intake.

Focal Length and Object Distance

Focal length is the distance from the lens to the image sensor when the subject is in focus. In this scenario, the butterfly is positioned 200 mm from the lens, and the image is formed 50 mm behind the lens. Understanding the relationship between object distance, image distance, and focal length is crucial for determining the lens's optical properties.

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens, expressed as 1/f = 1/v + 1/u. This formula is essential for calculating the focal length based on the distances provided in the problem. By applying this formula, one can derive the necessary parameters to find the correct aperture diameter for the specified f-number.