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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

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?
A microscope has an eyepicce that gives an angular magnification of 5.00 for a final image at infinity and an objective lens of focal length $15.0 \mathrm{mm}\(. The tube length of the microscope is \)16.0 \mathrm{cm} .$ (a) What is the transverse magnification due to the objective lens alone? (b) What is the angular magnification due to the microscope? (c) How far from the objective should the object be placed?
Kim says that she was less than 10 ft away from the president when she took a picture of him with her \(50-\mathrm{mm}\) focal length camera lens. The picture shows the upper half of the president's body (or \(3.0 \mathrm{ft}\) of his total height). On the negative of the film, this part of his body is $18 \mathrm{mm}$ high. How close was Kim to the president when she took the picture?
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?
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