Problem 67 - Physics

Question

Interactive Solution 26.67 at illustrates one approach to solving problems such as this one. A farsighted person has a near point that is \(67.0 \mathrm{~cm}\) from her eyes. She wears eyeglasses that are designed to enable her to read a newspaper held at a distance of 25.0 \(\mathrm{cm}\) from her eyes. Find the focal length of the eyeglasses, assuming that they are worn (a) \(2.2 \mathrm{~cm}\) from the eyes and (b) \(3.3 \mathrm{~cm}\) from the eyes.

Step-by-Step Solution

Answer

(a) The focal length is approximately 32.4 cm. (b) The focal length is approximately 32.1 cm.

Step 1: Understand the problem

We need to find the focal length of eyeglasses for a farsighted person whose near point is initially 67 cm. The glasses should make it possible to read at 25 cm with (a) eyeglasses worn 2.2 cm from the eyes and (b) eyeglasses worn 3.3 cm from the eyes.

Step 2: Use the Lens Formula

The lens formula is given by \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), where \(f\) is the focal length, \(v\) is the image distance (the near point), and \(u\) is the object distance (the reading distance).

Step 3: Account for Eyeglass Distance (a)

For part (a), eyeglasses are worn at 2.2 cm from the eyes. So the object distance is \(u = 25 \text{ cm} - 2.2 \text{ cm} = 22.8 \text{ cm}\) and the image distance is \(v = 67 \text{ cm} - 2.2 \text{ cm} = 64.8 \text{ cm}\).

Step 4: Calculate Focal Length (a)

Using the lens formula: \[\frac{1}{f} = \frac{1}{64.8} - \frac{1}{22.8}\] Calculate \(f\) to find the focal length for part (a).

Step 5: Account for Eyeglass Distance (b)

For part (b), eyeglasses are worn at 3.3 cm from the eyes. So the object distance is \(u = 25 \text{ cm} - 3.3 \text{ cm} = 21.7 \text{ cm}\) and the image distance is \(v = 67 \text{ cm} - 3.3 \text{ cm} = 63.7 \text{ cm}\).

Step 6: Calculate Focal Length (b)

Using the lens formula: \[\frac{1}{f} = \frac{1}{63.7} - \frac{1}{21.7}\] Calculate \(f\) to find the focal length for part (b).

Key Concepts

Lens Formula

The lens formula is essential when solving problems related to optics, particularly those involving lenses, such as eyeglasses. It is expressed as \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), where:

• \(f\) represents the focal length of the lens.
• \(v\) is the image distance, meaning the distance from the lens to the image.
• \(u\) is the object distance, which is the distance from the lens to the object being viewed.

The lens formula helps to determine the focal length when the distances are known. It rearranges the relationship between these variables in such a way that by knowing any two, the third can be found. This formula is particularly useful in situations like eyeglasses calculations, where understanding focal length adjustments can significantly improve vision.

Farsightedness

Farsightedness, also known as hyperopia, is a common vision condition where distant objects are seen more clearly than close ones. This occurs because light entering the eye is focused behind the retina rather than directly on it. Therefore, people with farsightedness might struggle with tasks such as reading or seeing objects up close.

Corrective lenses, such as eyeglasses or contact lenses, are used to help adjust the way light enters the eyes, thus aiding in focusing light correctly onto the retina. By using lenses with the correct focal length, it is possible to bring closer objects into sharper focus. This adjustment allows people with hyperopia to engage in activities that require clear near vision, without experiencing the blurred vision commonly associated with this condition.

Optics

Optics is the branch of physics dealing with the behavior and properties of light, including its interactions with objects. Fundamental concepts in optics include reflection, refraction, diffraction, and interference. Lenses are a significant component within the field of optics, as they manipulate light to form images.

In eyeglasses, lenses are used to correct refractive errors, like farsightedness, by changing the way light is bent when entering the eye. Concave lenses cause light to diverge, helping in near-sighted conditions, while convex lenses converge light rays to aid farsighted individuals. Understanding these optic principles allows lenses to be designed so they correct vision effectively by ensuring the focus falls precisely on the retina, providing clear vision.

Eyeglasses Calculation

The calculation for eyeglasses involves determining the focal length of the lenses necessary to correct a refractive error, such as farsightedness. This is done using the lens formula, by taking into account the specific needs of an individual's vision.

• The required object distance (e.g., the distance at which the person wants to read clearly) is first determined.
• Eyeglass distance, which is the distance of the lens from the eye, is considered in the calculations to ensure accurate correction.
• Both the near point (current point of clear vision) and desired reading distance are used to calculate the accurate focal length needed for correction.

Adjustments such as those noted for eyeglasses worn at different distances allow for the personalization of lenses, ensuring that individuals experience the best possible clarity in different visual tasks.