Ch 34: Ray Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 34: Ray Optics

Problem 71

Chapter 34, Problem 71

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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