Ch 35: Optical Instruments

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 35: Optical Instruments

Problem 16

Chapter 35, Problem 16

A microscope has a 160 mm tube length. What focal-length objective will give total magnification ≈ 500x when used with an eyepiece having a focal length of 5.0 cm?

Verified step by step guidance

Step 1: Understand the relationship between total magnification, the magnification of the objective lens, and the magnification of the eyepiece. Total magnification is the product of the magnification of the objective lens and the magnification of the eyepiece: \( M_{total} = M_{objective} \times M_{eyepiece} \).

Step 2: Calculate the magnification of the eyepiece using its focal length. The magnification of the eyepiece is given by \( M_{eyepiece} = \frac{25 \text{ cm}}{f_{eyepiece}} \), where 25 cm is the near point of the human eye. Substitute \( f_{eyepiece} = 5.0 \text{ cm} \) into the formula.

Step 3: Rearrange the total magnification formula to solve for the magnification of the objective lens: \( M_{objective} = \frac{M_{total}}{M_{eyepiece}} \). Substitute \( M_{total} = 500 \) and the value of \( M_{eyepiece} \) calculated in Step 2.

Step 4: Use the formula for the magnification of the objective lens: \( M_{objective} = \frac{L}{f_{objective}} \), where \( L \) is the tube length (160 mm or 16.0 cm) and \( f_{objective} \) is the focal length of the objective lens. Rearrange this formula to solve for \( f_{objective} \): \( f_{objective} = \frac{L}{M_{objective}} \).

Step 5: Substitute \( L = 16.0 \text{ cm} \) and the value of \( M_{objective} \) from Step 3 into the formula for \( f_{objective} \). This will give the focal length of the objective lens required to achieve the desired total magnification.

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Key Concepts

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

Total Magnification

Total magnification in a microscope is the product of the magnification of the objective lens and the eyepiece. It indicates how much larger an object appears compared to its actual size. For example, if the objective lens has a magnification of 40x and the eyepiece has a magnification of 10x, the total magnification would be 400x. Understanding this relationship is crucial for determining the required specifications of the lenses.

Focal Length

Focal length is the distance from the lens to the point where parallel rays of light converge to a single point. In microscopes, the focal length of the objective lens affects the magnification and the working distance. A shorter focal length typically results in higher magnification, which is essential for observing small details in specimens. Knowing how to calculate and apply focal lengths is vital for achieving the desired total magnification.

Lens Formula

The lens formula relates the object distance, image distance, and focal length of a lens. It is expressed as 1/f = 1/v - 1/u, where f is the focal length, v is the image distance, and u is the object distance. This formula is fundamental in optics for determining how lenses will behave in a system, especially in microscopes where precise calculations are necessary to achieve specific magnifications.

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