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 66

Chapter 34, Problem 66

A 25-cm-long rod lies along the optical axis of a converging lens, perpendicular to the lens plane. The lens has a 30 cm focal length. The rod's real , along the optical axis on the other side of the lens, is also 25 cm long. What is the distance from the lens to the nearest end of the rod?

Verified step by step guidance

Understand the problem: A rod is placed along the optical axis of a converging lens, and its real image is formed on the other side of the lens. The goal is to find the distance from the lens to the nearest end of the rod. This involves using the lens formula and understanding the geometry of the image formation.

Recall the lens formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the lens, \( d_o \) is the object distance (distance from the lens to the object), and \( d_i \) is the image distance (distance from the lens to the image).

Let the nearest end of the rod be at a distance \( d_{o1} \) from the lens. The farthest end of the rod will then be at a distance \( d_{o2} = d_{o1} + 25 \ \text{cm} \). The corresponding image distances for these ends will be \( d_{i1} \) and \( d_{i2} \), respectively.

Use the lens formula for the nearest end of the rod: \( \frac{1}{f} = \frac{1}{d_{o1}} + \frac{1}{d_{i1}} \). Rearrange to find \( d_{i1} \): \( d_{i1} = \frac{1}{\frac{1}{f} - \frac{1}{d_{o1}}} \).

For the farthest end of the rod, use the lens formula again: \( \frac{1}{f} = \frac{1}{d_{o2}} + \frac{1}{d_{i2}} \). Since the image length is given as 25 cm, the difference between \( d_{i2} \) and \( d_{i1} \) must equal 25 cm. Solve these equations simultaneously to find \( d_{o1} \), the distance from the lens to the nearest end of the rod.

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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. 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 to the height of the object, and it can also be expressed as the negative ratio of the image distance to the object distance (M = -v/u). Understanding magnification helps in analyzing how the size of the image relates to the size of the object, which is crucial in this problem.

Real and Virtual Images

In optics, a real image is formed when light rays converge and can be projected onto a screen, while a virtual image occurs when light rays appear to diverge from a point. In this scenario, the rod's real image is formed on the opposite side of the lens, which is important for determining the distances involved.

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