It is desired to magnify reading material by a factor of 3.0 x when a book is placed 9.0 cm behind a lens.
(a) Draw a ray diagram and describe the type of image this would be.
(b) What type of lens is needed?
(c) What is the power of the lens in diopters?

검증된 단계별 안내

To solve this problem, we first need to understand the relationship between the magnification (M), the object distance (d_o), and the image distance (d_i). The magnification is given by the formula:

M = - d

_id_o. Here, M = 3.0, and d_o = 9.0 cm.

Rearrange the magnification formula to solve for the image distance (d_i):

d

_i = -M × d_o. Substitute the given values to find d_i.

Next, use the lens equation to find the focal length (f) of the lens. The lens equation is:

\frac{1}{f} = \frac{1}{d}

_o + 1d_i. Substitute the values of d_o and d_i into this equation to calculate f.

Determine the type of lens needed. Since the magnification is positive and the image is virtual (as indicated by the negative sign in d_i), the lens must be a converging lens (convex lens).

Finally, calculate the power of the lens in diopters using the formula:

P = \frac{100}{f}

, where f is in centimeters. Convert f to meters before substituting into the formula to find the power in diopters.

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주요 개념

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

Magnification

Magnification is the ratio of the height of the image to the height of the object. In optics, it also relates to the distance of the object and the image from the lens. A magnification factor of 3.0 means the image will be three times larger than the object. Understanding this concept is crucial for determining the characteristics of the image formed by the lens.

Lens Types

Lenses can be classified as converging (convex) or diverging (concave). A converging lens can produce a magnified image when the object is placed within its focal length, while a diverging lens always produces a virtual, smaller image. Identifying the correct type of lens is essential for achieving the desired magnification and image characteristics.

Lens Power

The power of a lens, measured in diopters, is the reciprocal of its focal length in meters. A positive power indicates a converging lens, while a negative power indicates a diverging lens. Calculating the power is important for understanding how strong the lens needs to be to achieve the desired magnification and image distance.