A 1.0-cm-tall candle flame is 60 cm from a lens with a focal length of 20 cm. What are the distance and the height of the flame's image?

검증된 단계별 안내

Identify the given values: The object height \( h_o \) is 1.0 cm, the object distance \( d_o \) is 60 cm, and the focal length \( f \) is 20 cm. We need to find the image distance \( d_i \) and the image height \( h_i \).

Use the lens formula to find the image distance \( d_i \): \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Rearrange this equation to solve for \( \frac{1}{d_i} \): \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). Substitute the given values for \( f \) and \( d_o \).

After calculating \( \frac{1}{d_i} \), take the reciprocal to find \( d_i \), the image distance. This will tell you where the image is located relative to the lens.

To find the image height \( h_i \), use the magnification formula: \( M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \). Rearrange this to solve for \( h_i \): \( h_i = M \cdot h_o \). Substitute the values for \( M \), \( h_o \), and \( d_i \).

Interpret the results: The sign of \( d_i \) will indicate whether the image is real or virtual (positive for real, negative for virtual), and the sign of \( h_i \) will indicate whether the image is upright or inverted (positive for upright, negative for inverted).

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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. It is expressed as 1/f = 1/v - 1/u. This formula is essential for determining the position of the image formed by the lens based on the position of the object.

Magnification

Magnification is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed as the negative ratio of the image distance (v) to the object distance (u). It is given by the formula M = h'/h = -v/u. Understanding magnification helps in determining how the size of the image compares to the size of the object.

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 considered negative, while those in the direction of outgoing light are positive. This convention is crucial for correctly applying the lens formula and calculating image properties.