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Chapter 2: Problem 126

What will be the formula for the angular magnification of a convex lens of focal length \(f\) if the eye is very close to the lens and the near point is located a distance \(D\) from the eye?

Short Answer

Expert verified
The formula for the angular magnification of a convex lens of focal length \(f\) when the eye is very close to the lens and the near point is located a distance \(D\) from the eye is \(M = \frac{D}{f}\).

Step by step solution

01

Use the lens formula

The lens formula is defined as \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\), where \(f\) is the focal length of the lens, \(u\) is the object distance, and \(v\) is the image distance. If \(u\) is large compared to the focal length, i.e., \(u >> f\), then \(v \approx f\), as the term \(\frac{1}{u}\) is negligible compared to \(\frac{1}{f}\). In this case, the near point is located far away enough so \(D >> f\). Therefore, we can say that \(v \approx f\).
02

Calculate the angular magnification

The angular magnification of a lens is given by the ratio of the angle subtended by the image to the angle subtended by the object. According to the formula, for an unaided eye, the angular magnification is \(M = \frac{D}{f}\). Therefore, the angular magnification is inversely proportional to the focal length.
03

Final Result

Formulating the result from Step 2, it can be concluded that the angular magnification \(M\) of a convex lens with a focal length \(f\), when the eye is very close to the lens and the near point is located a distance \(D\) away, is given by \(M = \frac{D}{f}\).

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Most popular questions from this chapter

The far point of a myopic administrator is \(50.0 \mathrm{cm}\). (a) What is the relaxed power of his eyes? (b) If he has the normal \(8.00 \%\) ability to accommodate, what is the closest object he can see clearly?

Can an image be larger than the object even though its magnification is negative? Explain.

What is the angular size of the Moon if viewed from a binocular that has a focal length of \(1.2 \mathrm{cm}\) for the eyepiece and a focal length of \(8 \mathrm{cm}\) for the objective? Use the radius of the moon \(1.74 \times 10^{6} \mathrm{m}\) and the distance of the moon from the observer to be \(3.8 \times 10^{8} \mathrm{m}\)

Derive the spherical interface equation for refraction at a concave surface. (Hint: Follow the derivation in the text for the convex surface.)

Use a ruler and a protractor to draw rays to find images in the following cases. (a) A point object located on the axis of a concave mirror located at a point within the focal length from the vertex. (b) A point object located on the axis of a concave mirror located at a point farther than the focal length from the vertex. (c) A point object located on the axis of a convex mirror located at a point within the focal length from the vertex. (d) A point object located on the axis of a convex mirror located at a point farther than the focal length from the vertex. (e) Repeat (a)-(d) for a point object off the axis.
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