A magnifier has a magnification of 5x. How far from the lens should an object be held so that its is seen at the near-point distance of 25 cm? Assume that your eye is immediately behind the lens.

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Understand the relationship between magnification (M), the focal length (f), and the object distance (d_o). For a magnifier, the magnification is given by:

M = \frac{25}{f}

, where 25 cm is the near-point distance.

Rearrange the magnification formula to solve for the focal length (f):

f = \frac{25}{M}

. Substitute M = 5 into the equation to find the focal length.

Use the lens formula to relate the focal length (f), object distance (d_o), and image distance (d_i):

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

. Here, the image distance (d_i) is 25 cm because the image is formed at the near-point distance.

Rearrange the lens formula to solve for the object distance (d_o):

\frac{1}{d_o} = \frac{1}{f} - \frac{1}{d_i}

. Substitute the values of f (calculated in step 2) and d_i = 25 cm into the equation.

Take the reciprocal of the result from step 4 to find the object distance (d_o):

d_o = \frac{1}{(}

. This gives the distance at which the object should be held from the lens.

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주요 개념

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

Magnification

Magnification is the ratio of the size of the image produced by a lens to the size of the object. In this case, a magnification of 5x means that the image appears five times larger than the actual object. This concept is crucial for determining the relationship between the object distance and the image distance in lens systems.

Lens Formula

The lens formula relates the object distance (u), image distance (v), and the focal length (f) of a lens. It is expressed as 1/f = 1/v - 1/u. Understanding this formula is essential for calculating the position of the object when given the magnification and the near-point distance, which serves as the image distance in this scenario.

Near Point Distance

The near point distance is the closest distance at which the eye can comfortably focus on an object, typically around 25 cm for a normal human eye. In this problem, the near point distance is used as the image distance, allowing us to determine how far the object should be placed from the lens to achieve the desired magnification.