The lateral magnification of a convex mirror is +0.75 for objects 3.2 m from the mirror. What is the focal length of this mirror?

검증된 단계별 안내

Start by recalling the formula for lateral magnification \( M \) in a mirror: \( M = -\frac{q}{p} \), where \( q \) is the image distance and \( p \) is the object distance. Here, \( M = +0.75 \) and \( p = 3.2 \ \text{m} \).

Rearrange the magnification formula to solve for \( q \): \( q = -M \cdot p \). Substitute the given values of \( M \) and \( p \) into the equation to find \( q \).

Next, use the mirror equation: \( \frac{1}{f} = \frac{1}{p} + \frac{1}{q} \), where \( f \) is the focal length, \( p \) is the object distance, and \( q \) is the image distance. Substitute the known values of \( p \) and \( q \) into this equation.

Simplify the terms \( \frac{1}{p} \) and \( \frac{1}{q} \) to calculate \( \frac{1}{f} \).

Finally, take the reciprocal of \( \frac{1}{f} \) to find the focal length \( f \) of the convex mirror.

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주요 개념

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

Lateral Magnification

Lateral magnification (m) is the ratio of the height of the image (h') to the height of the object (h). For mirrors, it can also be expressed in terms of the object distance (u) and the image distance (v) as m = -v/u. A positive magnification indicates that the image is upright relative to the object, which is typical for virtual images produced by convex mirrors.

Convex Mirror Properties

Convex mirrors are curved outward and always produce virtual, upright, and reduced images. The focal length (f) of a convex mirror is considered positive, and it is located behind the mirror. The mirror formula, 1/f = 1/v + 1/u, relates the focal length to the object and image distances, allowing for calculations involving image formation.

Mirror Formula

The mirror formula is a fundamental equation in optics that relates the object distance (u), image distance (v), and focal length (f) of a mirror. It is expressed as 1/f = 1/v + 1/u. This formula is essential for solving problems involving mirrors, as it allows for the determination of unknown distances when the other parameters are known.