(III) A bright object is placed on one side of a converging lens of focal length f, and a white screen for viewing the image is on the opposite side. The distance d_T = d_i + d_o between the object and the screen is kept fixed, but the lens can be moved. Determine a formula for the distance between the two lens positions in part (a), and the ratio of the image sizes.

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Step 1: Begin by understanding the lens equation, which relates the object distance (dₒ), image distance (dᵢ), and focal length (f) of a converging lens:

\frac{1}{dₒ} + \frac{1}{dᵢ} = \frac{1}{f} .

This equation will be central to solving the problem.

Step 2: Recognize that the total distance between the object and the screen is fixed, denoted as d_T = dₒ + dᵢ. Use this relationship to express one variable in terms of the other. For example,

dₒ = d_T - dᵢ .

Step 3: Substitute

dₒ = d_T - dᵢ

into the lens equation. This gives:

\frac{1}{dₒ} = \frac{1}{(} .

Combine this with

\frac{1}{dᵢ} = \frac{1}{f} - \frac{1}{(} .

Step 4: Solve the resulting equation for dᵢ. This will yield two possible solutions for dᵢ, corresponding to the two positions of the lens where the image is formed. The distance between these two lens positions can be determined by finding the difference between the two values of dᵢ.

Step 5: To determine the ratio of the image sizes, recall that the magnification (M) of a lens is given by

M = \frac{dᵢ}{dₒ} .

Use the two values of dᵢ and their corresponding dₒ values to calculate the magnifications for each lens position. The ratio of the image sizes will be the ratio of these magnifications.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

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

Lens Formula

The lens formula relates the object distance (dₒ), the image distance (dᵢ), and the focal length (f) of a lens. It is expressed as 1/f = 1/dₒ + 1/dᵢ. This formula is essential for understanding how the position of the object and the image changes when the lens is moved, allowing for the calculation of distances and image characteristics.

Magnification

Magnification (M) is the ratio of the height of the image (hᵢ) to the height of the object (hₒ), and it can also be expressed as M = -dᵢ/dₒ. This concept is crucial for determining how the size of the image changes relative to the object when the lens is adjusted, which is necessary for solving the problem regarding the ratio of image sizes.

Converging Lens Behavior

A converging lens focuses parallel rays of light to a point known as the focal point. The behavior of light through a converging lens is fundamental to understanding image formation, including the characteristics of real and virtual images, their positions, and sizes, which are all influenced by the lens's focal length and the distances of the object and image.