A lightbulb is 3.0 m from a wall. What are the focal length and the position (measured from the bulb) of a lens that will form an on the wall that is twice the size of the lightbulb?

검증된 단계별 안내

Identify the given information: The object distance \( d_o \) is 3.0 m, the image distance \( d_i \) is unknown, and the magnification \( M \) is given as 2 (since the image is twice the size of the object).

Use the magnification formula \( M = -\frac{d_i}{d_o} \) to relate the image distance \( d_i \) to the object distance \( d_o \). Substitute \( M = 2 \) and \( d_o = 3.0 \) m into the equation to solve for \( d_i \).

Once \( d_i \) is determined, use the lens equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to calculate the focal length \( f \). Substitute the values of \( d_o \) and \( d_i \) into the equation.

Rearrange the lens equation to isolate \( f \): \( f = \frac{1}{\frac{1}{d_o} + \frac{1}{d_i}} \). Perform the necessary algebraic manipulations to simplify the expression for \( f \).

Verify the results by checking that the calculated focal length and image distance satisfy both the lens equation and the magnification condition. Ensure the units are consistent throughout the calculations.

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주요 개념

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

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens through the equation 1/f = 1/v - 1/u. This formula is essential for determining how a lens will manipulate light to form images at specific distances, which is crucial for solving the problem of image formation on the wall.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), expressed as M = h'/h. In this scenario, the problem states that the image formed on the wall is twice the size of the lightbulb, indicating a magnification of 2. Understanding magnification helps in determining the relationship between the object and image distances.

Sign Convention in Optics

In optics, the sign convention dictates how distances are measured in relation to the lens. Typically, distances measured in the direction of the incoming light are negative, while those in the direction of outgoing light are positive. This convention is crucial for correctly applying the lens formula and calculating the focal length and image position.