A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a 3.0 x upright image. What kind of mirror must be used and what must its radius of curvature be?

검증된 단계별 안내

Determine the type of mirror: Since the image is upright and magnified, the mirror must be a concave mirror. Concave mirrors can produce upright, magnified images when the object is within the focal length of the mirror.

Use the magnification formula to relate the image distance \(d_i\) and object distance \(d_o\): \(M = \frac{d_i}{d_o}\), where \(M\) is the magnification. Substitute \(M = 3.0\) and \(d_o = 2.00\,\text{cm}\) to solve for \(d_i\).

Apply the mirror equation to find the focal length \(f\): \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\). Use the values of \(d_o\) and \(d_i\) from the previous step to calculate \(f\).

Relate the focal length \(f\) to the radius of curvature \(R\): \(R = 2f\). Use the focal length obtained in the previous step to find the radius of curvature.

Summarize the results: The mirror is a concave mirror, and its radius of curvature can be determined using the steps above.

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주요 개념

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

Types of Mirrors

There are two main types of mirrors: concave and convex. Concave mirrors can produce real and inverted images or virtual and upright images, depending on the object's distance from the mirror. In contrast, convex mirrors always produce virtual, upright images. For the scenario described, a concave mirror is required to achieve the specified upright image at a given distance.

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror: 1/f = 1/v + 1/u. This formula is essential for determining the characteristics of the image produced by the mirror. In this case, knowing the object distance and the magnification allows us to find the image distance and subsequently the focal length.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed in terms of distances: M = -v/u. A positive magnification indicates an upright image, while a negative value indicates an inverted image. In this problem, the magnification of 3.0 signifies that the image is upright and larger than the object, which is a key factor in determining the type of mirror and its properties.