The focal length of a simple magnifier is 8.00 cm. Assume the magnifier is a thin lens placed very close to the eye. If the object is 1.00 mm high, what is the height of its image formed by the magnifier?

검증된 단계별 안내

Identify the given values: The focal length of the lens \( f = 8.00 \; \text{cm} \) and the object height \( h_o = 1.00 \; \text{mm} \). The goal is to find the height of the image \( h_i \).

Use the lens formula to find the image distance \( v \): \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \), where \( f \) is the focal length, \( u \) is the object distance, and \( v \) is the image distance. Rearrange to solve for \( v \): \( \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \).

Determine the magnification \( M \) of the lens using the formula \( M = \frac{h_i}{h_o} = \frac{v}{u} \). Rearrange to solve for \( h_i \): \( h_i = M \cdot h_o \).

Substitute the values of \( v \), \( u \), and \( h_o \) into the magnification formula to calculate the image height \( h_i \). Ensure that all units are consistent (e.g., convert \( h_o \) to cm if necessary).

Interpret the result: The sign of \( h_i \) will indicate whether the image is upright or inverted, and the magnitude will give the size of the image formed by the magnifier.

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주요 개념

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

Focal Length

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge. For a simple magnifier, a positive focal length indicates that it is a converging lens. The focal length is crucial for determining how the lens will magnify an object and affects the size of the image formed.

Magnification

Magnification is the ratio of the height of the image to the height of the object. It can be calculated using the formula: magnification (M) = image height (h') / object height (h). In the case of a magnifier, the magnification also depends on the focal length and the distance of the object from the lens, allowing us to determine how much larger the image appears compared to the actual object.

Thin Lens Formula

The thin lens formula relates the object distance (d_o), image distance (d_i), and focal length (f) of a lens: 1/f = 1/d_o + 1/d_i. This formula is essential for understanding how lenses form images and can be used to find the image distance when the object distance and focal length are known. It helps in calculating the position and size of the image produced by the magnifier.