A sharp image is located 373 mm behind a 235-mm-focal-length converging lens. Find the object distance by calculation.

검증된 단계별 안내

Start by identifying the lens formula, which relates the focal length \( f \), the object distance \( d_o \), and the image distance \( d_i \): \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \).

Substitute the given values into the lens formula. Here, \( f = 235 \ \text{mm} \) and \( d_i = 373 \ \text{mm} \): \( \frac{1}{235} = \frac{1}{d_o} + \frac{1}{373} \).

Rearrange the equation to isolate \( \frac{1}{d_o} \): \( \frac{1}{d_o} = \frac{1}{235} - \frac{1}{373} \).

Simplify the right-hand side by finding a common denominator for the fractions and performing the subtraction: \( \frac{1}{d_o} = \frac{373 - 235}{235 \cdot 373} \).

Finally, take the reciprocal of \( \frac{1}{d_o} \) to find \( d_o \), the object distance: \( d_o = \frac{235 \cdot 373}{373 - 235} \).

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주요 개념

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Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is expressed as 1/f = 1/v + 1/u. This equation is fundamental for solving problems involving lenses, as it allows us to find one of the distances when the other two are known.

Sign Convention for Lenses

In optics, the sign convention is crucial for correctly applying the lens formula. For a converging lens, the focal length (f) is positive, the object distance (u) is negative if the object is on the same side as the incoming light, and the image distance (v) is positive if the image is on the opposite side of the lens. Understanding this convention helps avoid errors in calculations.

Converging Lens

A converging lens, or convex lens, is thicker at the center than at the edges and focuses parallel rays of light to a point known as the focal point. The focal length is the distance from the lens to this focal point. Converging lenses are commonly used in applications such as cameras and magnifying glasses, making them essential in optics.