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Magazine Chapter 26: Problem 121
At age forty, a man requires contact lenses \((f=65.0 \mathrm{cm})\) to read a book held 25.0 \(\mathrm{cm}\) his eyes. At age forty-five, while wearing these contacts he must now hold a book 29.0 \(\mathrm{cm}\) from his eyes. (a) By what distance has his near point changed? (b) What focal-length lenses does he require at age forty-five to read a book at 25.0 \(\mathrm{cm} ?\)
Short Answer
Expert verified
(a) Near point changed by 11.7 cm. (b) New focal length required is 16.9 cm.
Step by step solution
01
Understand the Problem
The problem involves determining changes in the near point of a man's eyesight using contact lenses at two different ages. At age forty, with a focal length of 65.0 cm, the man reads at 25.0 cm. At age forty-five, holding the book at 29.0 cm with the same lenses indicates a change in his near point. We need to calculate the new near point and adjust the focal length lenses required to read at 25.0 cm at age forty-five.
02
Analyze Initial Condition (Age 40)
At age forty, the man needs contact lenses with a focal length of 65.0 cm to read at 25.0 cm. This implies his unaided near point can be found using the lens formula: \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \), where \( u = 25.0 \) cm and \( f = 65.0 \) cm. Rearrange to find \( v \), the near point without lenses.
03
Calculate Near Point Without Lenses (Age 40)
Using the lens formula: \[ \frac{1}{65} = \frac{1}{25} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{65} - \frac{1}{25} \] \[ \frac{1}{v} = \frac{25 - 65}{1625} = \frac{-40}{1625} \] \[ v = -40.625 \text{ cm} \] Since near point is positive, the physiologic near point (distance from the lens) is 40.625 cm.
04
Analyze Condition at Age 45
At age forty-five, the man holds the book at 29.0 cm while still using 65.0 cm lenses, indicating a new near point. Use the lens formula again, substituting \( u = 29.0 \) cm.
05
Calculate Near Point Without Lenses (Age 45)
Apply the lens formula: \[ \frac{1}{65} = \frac{1}{29} + \frac{1}{v} \] \[ \frac{1}{v} = \frac{1}{65} - \frac{1}{29} \] \[ \frac{1}{v} = \frac{29 - 65}{1885} = \frac{-36}{1885} \] \[ v = -52.36 \text{ cm} \] Converting to positive gives 52.36 cm as the new near point.
06
Calculate Distance Change in Near Point
Compare the near point at age forty and forty-five without lenses: \[ 52.36 - 40.625 = 11.735 \text{ cm} \]. The distance change in near point is approximately 11.7 cm.
07
Determine Focal Length Required (Age 45)
Now adjust the focal length such that the man can read at 25.0 cm at age forty-five. Use: \( u = 25.0 \) cm and \( v = 52.36 \) cm, from the new near point: \[ \frac{1}{f} = \frac{1}{25} + \frac{1}{52.36} \] \[ \frac{1}{f} = \frac{52.36 + 25}{1309} = \frac{77.36}{1309} \] Evaluate to find \( f \).
08
Calculate New Focal Length
Continuing from previous: \[ f = \frac{1309}{77.36} \approx 16.9 \text{ cm} \]. The required focal length for reading at 25.0 cm at age forty-five is approximately 16.9 cm.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Contact Lenses
Contact lenses are thin, curved lenses placed directly on the eye's surface to correct vision problems. They work by bending light rays to focus them onto the retina, thus improving vision. Unlike eyeglasses, contact lenses move with your eye and provide a wider field of view without obstructions. This can be particularly beneficial for tasks like reading or using a computer.
- Daily wear lenses are removed nightly for cleaning.
- Extended wear lenses can be worn overnight.
- Toric lenses are designed to correct astigmatism.
- Multifocal lenses help with presbyopia, allowing for vision correction at different distances.
Near Point
The near point is the closest distance at which the eye can focus on an object comfortably. As we age, the elasticity of the eye's lens decreases, making it difficult to see items up close clearly. This condition, known as presbyopia, typically begins to affect individuals in their forties.
- In a young adult, the near point is usually around 25 cm.
- Presbyopia causes the near point to recede.
- Reading glasses or contact lenses modify the near point for better near vision.
Focal Length
Focal length is the distance between the center of a lens and its focus, where light is most sharply concentrated. In optics, it is crucial for determining how strongly the lens converges or diverges light.
- It is commonly measured in centimeters or meters.
- A lens with a short focal length bends light more sharply than one with a longer focal length.
- Positive focal lengths indicate converging lenses, while negative focal lengths indicate diverging lenses.
Lens Formula
The lens formula is an equation that describes the relationship between the focal length ( (f)), the object distance ( (u)), and the image distance ( (v)). It is expressed as: \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \). This mathematical relationship helps in calculating the variables needed to correct vision or focus light via lenses.
- It can be rearranged to find unknowns when two values are known.
- Understanding this formula is essential for designing eyeglasses and contact lenses.
- Construction of optical instruments like cameras and microscopes also utilizes the lens formula.
