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Chapter 24: Problem 37

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?

Short Answer

Expert verified
Answer: The angular magnification of the microscope is -11.5. Question: What would be the required focal length of the objective lens in order to double the microscope's magnification? Answer: A focal length of approximately 0.61 cm for the objective lens is required to double the magnification of the microscope.

Step by step solution

01

Understand angular magnification formula for a microscope

The angular magnification (M) of a microscope can be found using the formula: \( M = - \dfrac{f_e (L - f_o)}{f_o f_e} \) where \(f_e\) is the focal length of the eyepiece, \(f_o\) is the focal length of the objective lens, and \(L\) is the tube length.
02

Calculate the angular magnification for given values

Now, we plug in the values given in the exercise for \(f_e = 1.25 \mathrm{cm}\), \(f_o = 1.44 \mathrm{cm}\), and \(L = 18.0 \mathrm{cm}\). \( M = - \dfrac{1.25 (18 - 1.44)}{1.44 \cdot 1.25} \) Now, perform the arithmetic operations to get the value of \(M\): \( M = - \dfrac{1.25 (16.56)}{1.44 \cdot 1.25} = - \dfrac{20.7}{1.8} \approx -11.5 \) So, the angular magnification of the microscope is -11.5. (Note that the negative sign indicates that the image is inverted).
03

Calculate the required objective focal length to double the magnification

We're asked to find the objective focal length required to double the magnification, which means we want a magnification of \(-2 \times M = -2 \times(-11.5) = 23\). Let's denote the new objective focal length as \(f'_o\). So, \( 23 \approx - \dfrac{1.25(18 - f'_o)}{f'_o \cdot 1.25} \) Now, we need to solve for \(f'_o\). First, we simplify and multiply both sides by \(-1.25 f'_o\): \( 23 \cdot -1.25 f'_o = -(18 - f'_o) \) Continue solving for \(f'_o\): \( -28.75 f'_o = -18 + f'_o \) \( -29.75 f'_o = -18 \) \( f'_o \approx \dfrac{18}{29.75} = 0.61 \mathrm{cm} \) Therefore, an objective lens focal length of approximately \(0.61 \mathrm{cm}\) would be required to double the magnification of the microscope.

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Most popular questions from this chapter

An object is located \(16.0 \mathrm{cm}\) in front of a converging lens with focal length \(12.0 \mathrm{cm} .\) To the right of the converging lens, separated by a distance of \(20.0 \mathrm{cm},\) is a diverging lens of focal length \(-10.0 \mathrm{cm} .\) Find the location of the final image by ray tracing and verify using the lens equations.
Comprehensive Problems Good lenses used in cameras and other optical devices are actually compound lenses, made of five or more lenses put together to minimize distortions, including chromatic aberration. Suppose a converging lens with a focal length of \(4.00 \mathrm{cm}\) is placed right next to a diverging lens with focal length of \(-20.0 \mathrm{cm} .\) An object is placed \(2.50 \mathrm{m}\) to the left of this combination. (a) Where will the image be located? (b) Is the image real or virtual?
A slide projector has a lens of focal length \(12 \mathrm{cm} .\) Each slide is \(24 \mathrm{mm}\) by \(36 \mathrm{mm}\) (see the figure with Problem 16). The projector is used in a room where the screen is \(5.0 \mathrm{m}\) from the projector. How large must the screen be?
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.
Repeat Problem \(40(\mathrm{c})\) using a different eyepiece that gives an angular magnification of 5.00 for a final image at the viewer's near point \((25.0 \mathrm{cm})\) instead of at infinity.
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