Problem 140 - Fundamentals of Physics

Question

Suppose the farthest distance a person can see without visual aid is \(50 \mathrm{~cm} .\) (a) What is the focal length of the corrective lens that will allow the person to see very far away? (b) Is the lens converging or diverging? (c) The power \(P\) of a lens (in diopters) is equal to \(1 / f\), where \(f\) is in meters. What is \(P\) for the lens?

Step-by-Step Solution

Answer

The focal length is -0.5 m, the lens is diverging, and the power is -2 D.

Step 1: Identify the Problem

We have a person who can see up to a maximum distance of 50 cm. The goal is to find the focal length of a corrective lens that allows this person to see distant objects clearly.

Step 2: Convert Distance to Meters

Convert the maximum distance the person can see without visual aid from centimeters to meters: \[ 50 \text{ cm} = 0.5 \text{ m} \]

Step 3: Apply Lens Formula for Distant Vision

For a person to see distant objects clearly, the image formed by the corrective lens must be at the farthest point the person can see, which is 0.5 m (negative since it is a virtual image), and the object is at infinity. The lens formula is:\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Where:- \(u\) is the object distance (\(u = \infty\))- \(v\) is the image distance (\(v = -0.5 \text{ m}\))Substitute the values to find the focal length \(f\).\[ \frac{1}{f} = \frac{1}{-0.5} - 0 \]Solving that equation gives:\[ \frac{1}{f} = -2 \rightarrow f = -0.5 \text{ m} \]

Step 4: Determine the Type of Lens

If the focal length \(f\) is negative, the lens is diverging. Hence, the corrective lens required is a diverging lens.

Step 5: Calculate Power of Lens

The power \(P\) of a lens in diopters is given by the formula:\[ P = \frac{1}{f} \]Substituting the focal length obtained:\[ P = \frac{1}{-0.5} = -2 \text{ D} \]

Step 6: Conclusion

The focal length of the corrective lens is -0.5 meters, the lens is diverging, and its power is -2 diopters.

Key Concepts

Lens Formula

The lens formula is a fundamental concept in optics used to find the focal length of lenses in optical devices such as glasses, cameras, and microscopes.
It is expressed as:\[\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\]In the formula:

• \(f\) represents the focal length of the lens.
• \(v\) indicates the image distance from the lens.
• \(u\) is the object distance from the lens.

When dealing with lenses, the sign conventions are important. For a virtual image, which our eyesight correction often involves, the image distance \(v\) is considered negative.
The objective is to place the corrective lens so that the image of an object at a far distance (essentially at infinity) is formed at the near point of the person’s vision. This way, they can see clearly without strain, even if they have myopia (nearsightedness). By knowing these conventions and applying them correctly, we can calculate the necessary focal length for any lens.

Focal Length

Focal length is a critical parameter in lenses that measures how strongly the lens converges or diverges light.
It is the distance from the lens to the point where parallel rays of light converge or appear to diverge. In the context of corrective lenses, determining the correct focal length means providing clarity for distant or near objects, depending on the type of vision correction required. For instance, if someone is nearsighted, they need a lens that diverges light rays before reaching their eyes, to make distant objects appear clear.
Here, the focal length is negative, indicating that a diverging lens is used. Conversely, farsighted individuals require positive focal lengths for converging lenses to see nearby objects clearly. Thus, the focal length provides insight into the type and effectiveness of a lens needed to correct one's vision defects.

Diopters

Diopters are the unit of measurement for the optical power of a lens, which directly ties into the focal length. The diopter value tells us how strong a lens is, with higher diopters indicating more pronounced bending of light.
This power is crucial because it allows us not only to describe lenses succinctly but also to ensure the right corrective measure for specific eyesight needs.The formula that links diopters \(P\) with focal length \(f\) is:\[P = \frac{1}{f}\]In this formula, the focal length is in meters.
Since the solved focal length was -0.5 meters, the corresponding power is -2 diopters. Negative diopters reflect diverging lenses, commonly used in glasses for people with nearsightedness. The diopter system is an invaluable tool for opticians to prescribe and make lenses that precisely match the required correction strength for each individual.