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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 f₁: f₁ = \({1}/{P}\) For both lenses: f₁ = \({1}/{18 D}\) = \(0.0556 m\) 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 v₂ = u₁. 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: u₁ + v₁ = 28 cm Now we have a system of three equations to solve for u₁, v₁, and v₂, 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 u₁, v₁, and v₂ using the three equations. First, rearrange both lens equations for u₁: u₁ = \({1}/{(\frac{1}{f_1}+\frac{1}{v_1})}\) And: v₂ = \({1}/{(\frac{1}{f_2}-\frac{1}{u_1})}\) Now, substitute the second equation into the tube length equation: (\({1}/{(\frac{1}{f_1}+\frac{1}{v_1})}\)) + v₁ = 28 cm Therefore, you can plug in the value of the focal length and use iteration methods to find numerical values for u₁, v₁, and u₂. We get the following results: u₁ ≈ 8.8 cm, v₁ ≈ 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 = -(v₁ / u₁) * (D / f₂) Where D is the least distance of distinct vision, typically taken as 25 cm. With the known values of v₁, u₁, and f₂, we can calculate the angular magnification: M ≈ - (19.2 cm / 8.8 cm) * (25 cm / 5.56 cm) M ≈ 8.58 The angular magnification of the microscope is approximately 8.58.
05

Find the object distance from the objective lens

The object distance u₁ from the objective lens is the value that we calculated earlier. So, the object should be placed about: u₁ ≈ 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

The area occupied by one frame on 35 -mm film is \(24 \mathrm{mm}\) by $36 \mathrm{mm}-\( see the figure with Problem \)16 .$ The focal length of the camera lens is \(50.0 \mathrm{mm}\). A picture is taken of a person $182 \mathrm{cm}$ tall. What is the minimum distance from the camera for the person to stand so that the image fits on the film? Give two answers; one for each orientation of the camera.
You have two lenses of focal length \(25.0 \mathrm{cm}\) (lens 1 ) and $5.0 \mathrm{cm}$ (lens 2 ). (a) To build an astronomical telescope that gives an angular magnification of \(5.0,\) how should you use the lenses (which for objective and which for eyepiece)? Explain. (b) How far apart should they be?
Callum is examining a square stamp of side \(3.00 \mathrm{cm}\) with a magnifying glass of refractive power \(+40.0 \mathrm{D}\). The magnifier forms an image of the stamp at a distance of \(25.0 \mathrm{cm} .\) Assume that Callum's eye is close to the magnifying glass. (a) What is the distance between the stamp and the magnifier? (b) What is the angular magnification? (c) How large is the image formed by the magnifier?
A convex lens of power +12 D is used as a magnifier to examine a wildflower. What is the angular magnification if the final image is at (a) infinity or (b) the near point of \(25 \mathrm{cm} ?\)
Telescopes (a) If you were stranded on an island with only a pair of 3.5 -D reading glasses, could you make a telescope? If so, what would be the length of the telescope and what would be the best possible angular magnification? (b) Answer the same questions if you also had a pair of 1.3 -D reading glasses.
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