An object is 60 cm from a screen. What are the radii of a symmetric converging plastic lens (i.e., two equally curved surfaces) that will form an image on the screen twice the height of the object?

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Determine the focal length of the lens using the magnification formula. Since the image is twice the height of the object, the magnification \( M \) is \( M = -2 \) (negative because the image is inverted). Use the relationship \( M = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance and \( d_o \) is the object distance. Solve for \( d_i \).

Use the lens equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to calculate the focal length \( f \). Substitute \( d_o = 60 \, \text{cm} \) and the value of \( d_i \) obtained from the previous step.

Apply the lensmaker's equation for a symmetric lens: \( \frac{1}{f} = (n - 1) \left( \frac{2}{R} \right) \), where \( n \) is the refractive index of the plastic, \( R \) is the radius of curvature of each surface, and \( f \) is the focal length. Rearrange the equation to solve for \( R \).

Substitute the known values into the lensmaker's equation. Use the refractive index of the plastic (typically around \( n = 1.5 \), unless otherwise specified) and the focal length \( f \) calculated earlier to find \( R \).

Verify the solution by checking that the calculated radius of curvature \( R \) satisfies the lensmaker's equation and produces the desired magnification and image distance.

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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 given by the equation 1/f = 1/v - 1/u. Understanding this formula is crucial for determining the focal length needed to form an image at a specific distance.

Magnification

Magnification is the ratio of the height of the image (h') to the height of the object (h), expressed as h'/h = v/u. In this case, the problem states that the image height is twice that of the object, indicating a magnification of 2. This relationship helps in calculating the necessary distances for the lens.

Radius of Curvature

The radius of curvature (R) of a lens is the radius of the sphere from which the lens is made. For a symmetric converging lens, the focal length (f) is related to the radii of curvature by the lensmaker's equation: 1/f = (n - 1)(1/R1 - 1/R2). Since the lens is symmetric, R1 = R2, simplifying the calculations for the required lens shape.