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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 refracting telescope is used to view the Moon. The focal lengths of the objective and eyepiece are \(+2.40 \mathrm{m}\) and \(+16.0 \mathrm{cm},\) respectively. (a) What should be the distance between the lenses? (b) What is the diameter of the image produced by the objective? (c) What is the angular magnification?
A nearsighted man cannot clearly see objects more than \(2.0 \mathrm{m}\) away. The distance from the lens of the eye to the retina is \(2.0 \mathrm{cm},\) and the eye's power of accommodation is \(4.0 \mathrm{D}\) (the focal length of the cornea-lens system increases by a maximum of \(4.0 \mathrm{D}\) over its relaxed focal length when accommodating for nearby objects). (a) As an amateur optometrist, what corrective eyeglass lenses would you prescribe to enable him to clearly see distant objects? Assume the corrective lenses are $2.0 \mathrm{cm}$ from the eyes. (b) Find the nearest object he can see clearly with and without his glasses.
The distance from the lens system (cornea + lens) of a particular eye to the retina is \(1.75 \mathrm{cm} .\) What is the focal length of the lens system when the eye produces a clear image of an object \(25.0 \mathrm{cm}\) away?
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?
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).
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