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Chapter 24: Problem 38

Jordan is building a compound microscope using an eyepiece with a focal length of \(7.50 \mathrm{cm}\) and an objective with a focal length of $1.500 \mathrm{cm} .\( He will place the specimen a distance of \)1.600 \mathrm{cm}$ from the objective. (a) How far apart should Jordan place the lenses? (b) What will be the angular magnification of this microscope?

Short Answer

Expert verified
Answer: Jordan should place the lenses 4.5 cm apart, and the approximate angular magnification of the microscope is 9.75 times.

Step by step solution

01

Note down the given values

We have the following values: Focal length of the eyepiece: \(f_e = 7.5 \mathrm{cm}\) Focal length of the objective: \(f_o = 1.5 \mathrm{cm}\) Distance from the specimen to the objective: \(s_o = 1.6 \mathrm{cm}\)
02

Find the image distance for the objective lens

We can use the thin lens equation: \(\frac{1}{f_o} = \frac{1}{s_o} + \frac{1}{s_i^o}\) We can solve this equation for the image distance \(s_i^o\): \(s_i^o = \frac{1}{\frac{1}{f_o} - \frac{1}{s_o}}\) Now, plug in the values of \(f_o\) and \(s_o\): \(s_i^o = \frac{1}{\frac{1}{1.5} - \frac{1}{1.6}} \approx 12.0 \mathrm{cm}\)
03

Calculate the magnification for the objective lens

We can use the magnification formula for a single lens: \(M_o = \frac{s_i^o}{s_o}\) Now, plug in the values of \(s_i^o\) and \(s_o\): \(M_o = \frac{12.0}{1.6} = 7.5\)
04

Determine the location for the final image using the eyepiece lens

The final image will be at the focus of the eyepiece, so the image distance for the eyepiece lens is equal to its focal length: \(s_i^e = f_e = 7.5 \mathrm{cm}\)
05

Determine the distance between the lenses (objective and eyepiece)

To find the distance between the objective and eyepiece, we'll calculate the object distance for the eyepiece lens: \(s_o^e = s_i^o - s_i^e\) Now, plug in the values of \(s_i^o\) and \(s_i^e\): \(s_o^e = 12.0 - 7.5 = 4.5 \mathrm{cm}\) So, Jordan should place the lenses \(4.5 \mathrm{cm}\) apart.
06

Calculate the magnification for the eyepiece lens

Since the object distance for the eyepiece lens is placed at the focal point, the angular magnification of the eyepiece lens is happening at infinity. Thus, we can use the following formula for the angular magnification of the eyepiece lens: \(M_e = 1 + \frac{f_e}{D}\), where \(D\) is the nearest clear vision distance, which is typically assumed to be \(25 \mathrm{cm}\). Now, plug in the value of \(f_e\): \(M_e = 1 + \frac{7.5}{25} = 1.3\)
07

Calculate the total angular magnification of the microscope

The total angular magnification of the microscope is the product of the magnification of the objective lens and the angular magnification of the eyepiece lens: \(M_{total} = M_o \times M_e\) Now, plug in the values of \(M_o\) and \(M_e\): \(M_{total} = 7.5 \times 1.3 \approx 9.75\) The angular magnification of the microscope will be approximately \(9.75\) times.

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Most popular questions from this chapter

Keesha is looking at a beetle with a magnifying glass. She wants to see an upright, enlarged image at a distance of \(25 \mathrm{cm} .\) The focal length of the magnifying glass is \(+5.0 \mathrm{cm} .\) Assume that Keesha's eye is close to the magnifying glass. (a) What should be the distance between the magnifying glass and the beetle? (b) What is the angular magnification? (tutorial: magnifying glass II).
A microscope has an eyepiece of focal length \(2.00 \mathrm{cm}\) and an objective of focal length \(3.00 \mathrm{cm} .\) The eyepiece produces a virtual image at the viewer's near point \((25.0 \mathrm{cm}\) from the eye). (a) How far from the eyepiece is the image formed by the objective? (b) If the lenses are \(20.0 \mathrm{cm}\) apart, what is the distance from the objective lens to the object being viewed? (c) What is the angular magnification?
A statue is \(6.6 \mathrm{m}\) from the opening of a pinhole camera, and the screen is \(2.8 \mathrm{m}\) from the pinhole. (a) Is the image erect or inverted? (b) What is the magnification of the image? (c) To get a brighter image, we enlarge the pinhole to let more light through, but then the image looks blurry. Why? (d) To admit more light and still have a sharp image, we replace the pinhole with a lens. Should it be a converging or diverging lens? Why? (e) What should the focal length of the lens be?
The eyepiece of a microscope has a focal length of \(1.25 \mathrm{cm}\) and the objective lens focal length is \(1.44 \mathrm{cm} .\) (a) If the tube length is \(18.0 \mathrm{cm},\) what is the angular magnification of the microscope? (b) What objective focal length would be required to double this magnification?
Two converging lenses, separated by a distance of \(50.0 \mathrm{cm},\) are used in combination. The first lens, located to the left, has a focal length of \(15.0 \mathrm{cm} .\) The second lens, located to the right, has a focal length of \(12.0 \mathrm{cm} .\) An object, \(3.00 \mathrm{cm}\) high, is placed at a distance of \(20.0 \mathrm{cm}\) in front of the first lens. (a) Find the intermediate and final image distances relative to the corresponding lenses. (b) What is the total magnification? (c) What is the height of the final image?
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