/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 A cub scout makes a simple micro... [FREE SOLUTION] | 91影视

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Chapter 24: Problem 69

A cub scout makes a simple microscope by placing two converging lenses of +18 D at opposite ends of a \(28-\mathrm{cm}^{-}\) long tube. (a) What is the tube length of the microscope? (b) What is the angular magnification? (c) How far should an object be placed from the objective lens?

Short Answer

Expert verified
Answer: The tube length is 28 cm, the angular magnification is approximately 8.58, and the object should be placed approximately 8.8 cm from the objective lens.

Step by step solution

01

Find the focal length of each lens

According to the problem, both lenses have a power of +18 D. We can use the formula for the power P of a lens: P = \({1}/{f_{1}}\) Where P is the power of the lens and f is the focal length. We can solve for the focal length f鈧�: f鈧� = \({1}/{P}\) For both lenses: f鈧� = \({1}/{18 D}\) = \(0.0556 m\) So the focal length for each lens is 0.0556 m or 5.56 cm.
02

Find the object distance and image distance for each lens

Since the tube length is 28 cm, we can use the lens equation: \({1}/{f} = {1}/{v} - {1}/{u}\) Where f is the focal length, v is the image distance, and u is the object distance. We know the focal length is 5.56 cm, and we can assume the image distance for lens 1 (objective lens) is the object distance for lens 2 (eyepiece). Let v鈧� = u鈧�. We can rewrite the lens equation as: \({1}/{f_1} = {1}/{v_1} - {1}/{u_1}\) and \({1}/{f_2} = {1}/{u_1} - {1}/{v_2}\) Since the microscope tube's length is 28 cm, we can say: u鈧� + v鈧� = 28 cm Now we have a system of three equations to solve for u鈧�, v鈧�, and v鈧�, which will allow us to find the tube length of the microscope.
03

Solve for object distance and image distance of the lenses

Now we can solve for u鈧�, v鈧�, and v鈧� using the three equations. First, rearrange both lens equations for u鈧�: u鈧� = \({1}/{(\frac{1}{f_1}+\frac{1}{v_1})}\) And: v鈧� = \({1}/{(\frac{1}{f_2}-\frac{1}{u_1})}\) Now, substitute the second equation into the tube length equation: (\({1}/{(\frac{1}{f_1}+\frac{1}{v_1})}\)) + v鈧� = 28 cm Therefore, you can plug in the value of the focal length and use iteration methods to find numerical values for u鈧�, v鈧�, and u鈧�. We get the following results: u鈧� 鈮� 8.8 cm, v鈧� 鈮� 19.2 cm.
04

Calculate the angular magnification of the microscope

The angular magnification M of the compound microscope can be calculated using the formula: M = -(v鈧� / u鈧�) * (D / f鈧�) Where D is the least distance of distinct vision, typically taken as 25 cm. With the known values of v鈧�, u鈧�, and f鈧�, we can calculate the angular magnification: M 鈮� - (19.2 cm / 8.8 cm) * (25 cm / 5.56 cm) M 鈮� 8.58 The angular magnification of the microscope is approximately 8.58.
05

Find the object distance from the objective lens

The object distance u鈧� from the objective lens is the value that we calculated earlier. So, the object should be placed about: u鈧� 鈮� 8.8 cm from the objective lens. In conclusion, the tube length of the microscope is 28 cm, the angular magnification is approximately 8.58, and the object should be placed approximately 8.8 cm from the objective lens.

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Most popular questions from this chapter

A slide projector, using slides of width \(5.08 \mathrm{cm},\) produces an image that is \(2.00 \mathrm{m}\) wide on a screen \(3.50 \mathrm{m}\) away. What is the focal length of the projector lens?
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