/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 A nearsighted woman has a far po... [FREE SOLUTION] | 91Ó°ÊÓ

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A nearsighted woman has a far point of \(300 \mathrm{cm}\). What kind of lens, converging or diverging, should be prescribed for her to see distant objects more clearly? What refractive power should the lens have?

Short Answer

Expert verified
The woman needs a diverging lens with power of \(-0.33\) diopters.

Step by step solution

01

Selecting the Correct Lens

To correct nearsightedness, a diverging lens is used. This type of lens spreads out the incoming light so it can focus directly on the retina.
02

Lens Focal Length Identification

Given that the far point of the woman is 300 cm, this distance acts as the focal length of the corrective lens. However, it must be converted to meters before calculation of the lens power. Therefore, the focal length \(f\) is \(300 \, cm = 3 \, m\).
03

Calculating Lens Power

The power \(P\) of a lens is the reciprocal of its focal length (in meters). In this case, to calculate the power of the lens, the formula \(P = 1/f\) is used. Substituting the given focal length, \(P = 1/3 = 0.33\) diopters. As this is a diverging lens used to correct nearsightedness, this should be represented as a negative power. Therefore, the power of the required lens is \(-0.33\) diopters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Optical Lenses
Optical lenses play a crucial role in vision correction and work by redirecting light to achieve better focus. These lenses are pieces of transparent material, typically glass or plastic, that have at least one curved surface. There are two main types of lenses:
  • Converging lenses: Also known as convex lenses. They bring light rays together and are used to correct farsightedness.
  • Diverging lenses: Also known as concave lenses. They spread light rays apart and are used to correct nearsightedness.
In the case of nearsightedness, the light focuses in front of the retina. A diverging lens helps by spreading the light rays out so that they focus correctly on the retina. This improves clarity when looking at distant objects.
Refractive Power Calculation
The refractive power of a lens is an essential parameter and is a measure of how strongly the lens can bend light. It is measured in diopters. The formula for calculating the refractive power (P) of a lens is given by:\[P = \frac{1}{f}\]Where \(f\) is the focal length in meters. Calculating refractive power helps in determining the corrective strength needed for glasses. For instance, if a nearsighted person has a far point of 300 cm, equivalent to 3 meters, then the power is calculated as:\[P = \frac{1}{3} = 0.33 \, \text{diopters}\]However, because the lens is diverging, the power is represented with a negative sign, turning it into \(-0.33\) diopters. Always remember, diverging lenses have negative power values, while converging lenses have positive power values.
Diverging Lens
A diverging lens, or concave lens, is specifically shaped to help scatter light rays. This lens features at least one surface that curves inward, resembling the shape of a cave. It is thinner in the center and thicker at the edges. When parallel light rays pass through, a diverging lens spreads them out.

This spreading causes artificial convergence toward a focal point before reaching the eye, refocusing light onto the retina. As a result, it aids nearsighted people by allowing them to see distant objects clearly.

By affecting the path of light, diverging lenses correct the issue of light focusing too soon. The extent of refraction depends on the lens's curvature and material. The larger the negative refractive power, the stronger the divergence.

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Most popular questions from this chapter

Rank the following people from the most nearsighted to the most farsighted, indicating any ties: A. Bernie has a prescription of \(+2.0 \mathrm{D}\) B. Carol needs diverging lenses with a focal length of \(-0.35 \mathrm{m}\) C. Maria Elena wears converging lenses with a focal length of \(0.50 \mathrm{m}\) D. Janet has a prescription of +2.5 D. E. Warren's prescription is \(-3.2 \mathrm{D}\)

You are using a microscope with a \(10 \times\) eyepiece. What focal length of the objective lens will give a total magnification of \(200 \times ?\) Assume a length \(L=160 \mathrm{mm}\).

A student has built a 20 -cm-long pinhole camera for a science fair project. She wants to photograph the Washington Monument, which is \(167 \mathrm{m}(550 \mathrm{ft})\) tall, and to have the image on the detector be \(5.0 \mathrm{cm}\) high. How far should she stand from the Washington Monument?

Measure your near point by bringing this page up to the closest distance at which the image is still crisp \(-\) not the closest at which you can still read the letters, but the closest at which they remain sharp. (If you wear glasses or contacts, keep them on.) Measure the distance to determine your near point. What is your range of accommodation?

The Hubble Space Telescope has a mirror diameter of \(2.4 \mathrm{m}\). Suppose the telescope was used to photograph the surface of the moon from a distance of \(3.8 \times 10^{8} \mathrm{m} .\) What is the distance between two objects that the telescope can barely resolve? Assume the wavelength of light is \(600 \mathrm{nm}\).

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