Problem 75 - Physics

Question

Bill is farsighted and has a near point located \(125 \mathrm{~cm}\) from his eyes. Anne is also farsighted, but her near point is \(75.0 \mathrm{~cm}\) from her eyes. Both have glasses that correct their vision to a normal near point \((25.0 \mathrm{~cm}\) from the eyes), and both wear the glasses 2.0 \(\mathrm{cm}\) from the eyes. Relative to the eyes, what is the closest object that can be seen clearly (a) by Anne when she wears Bill's glasses and (b) by Bill when he wears Anne's glasses?

Step-by-Step Solution

Answer

(a) Anne can see clearly at about 99.65 cm with Bill's glasses. (b) Bill can see clearly at about 568.5 cm with Anne's glasses.

Step 1: Understand the Problem

The problem involves determining how farsighted people's corrective lenses affect their ability to see objects at a normal near point of 25 cm. We'll consider the difference in near points between Bill and Anne, and how their glasses correct the images formed.

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 of the lens, \( v \) is the image distance, and \( u \) is the object distance. Both individuals' lenses form images 25 cm (an image distance of 23 cm relative to the lens) from the lens.

Step 3: Calculate Bill's Focal Length

For Bill's glasses, the image distance \( v_b = 23 \text{ cm} \). His near point when corrected is 25 cm, uncorrected 125 cm, thus \( u_b = -125 \text{ cm} \). Using the lens formula: \[ \frac{1}{f_b} = \frac{1}{23} - \left( -\frac{1}{125} \right) \Rightarrow f_b \approx 27.31 \text{ cm} \]

Step 4: Calculate Anne's Focal Length

For Anne's glasses, the image distance \( v_a = 23 \text{ cm} \). Her near point when corrected is 25 cm, uncorrected 75 cm, thus \( u_a = -75 \text{ cm} \). Using the lens formula: \[ \frac{1}{f_a} = \frac{1}{23} - \left( -\frac{1}{75} \right) \Rightarrow f_a \approx 22.66 \text{ cm} \]

Step 5: Calculate Object Distance for Anne using Bill's Glasses

Using Bill's focal length, \( f_b = 27.31 \text{ cm} \), find the new object distance with Anne's near point (v_a) 23 cm: \[ \frac{1}{27.31} = \frac{1}{23} - \frac{1}{u} \Rightarrow u \approx -97.65 \qquad \Rightarrow \text{Relative to the eyes: 97.65 cm + 2 cm = 99.65 cm} \]

Step 6: Calculate Object Distance for Bill using Anne's Glasses

Using Anne's focal length, \( f_a = 22.66 \text{ cm} \), find the new object distance with Bill's near point (v_b) 23 cm: \[ \frac{1}{22.66} = \frac{1}{23} - \frac{1}{u} \Rightarrow u \approx -566.5 \qquad \Rightarrow \text{Relative to the eyes: 566.5 cm + 2 cm = 568.5 cm} \]

Key Concepts

Farsightedness

Farsightedness, also known as hyperopia, is a common vision condition where distant objects can be seen more clearly than close ones. This happens because the light entering the eye focuses behind the retina, rather than on it. The eye shape and corneal curvature often cause this issue.

People who are farsighted typically have trouble with tasks like reading or sewing, which require seeing details at close range. To diagnose farsightedness, an eye examination is necessary. The test typically involves reading text or identifying objects at various distances.

Common symptoms of farsightedness include blurred vision when focusing on close objects, eye strain, headaches, and fatigue after near work. In children, it might not be evident because their eyes can compensate for the condition. However, in adults, this capacity diminishes with age, leading to more pronounced symptoms.

Lens Formula

The lens formula is a fundamental equation in optics, used to calculate the relationship between the object distance (\( u \)), the image distance (\( v \)), and the focal length (\( f \)) of a lens. The formula is expressed as \[\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\]This equation helps to determine how lenses converge or diverge light rays to form images.

For corrective lenses, knowing the focal length allows us to understand how a particular lens adjusts light to help focus images on the retina. In the context of the original problem, we calculated the focal lengths required for both Bill's and Anne's glasses to correct their near points.

The lens formula is handy in ophthalmic optics, as it provides the means to design lenses that cater to specific vision problems. By adjusting the focal length, optometrists can prescribe the right glasses to bring near or distant objects into clear view.

Corrective Lenses

Corrective lenses are specially designed to adjust the way light enters the eye, aiding in proper focus directly onto the retina. This type of lens can be either concave or convex, depending on the vision impairment it aims to correct. For example, farsighted individuals need convex lenses to help focus light appropriately.

In the original exercise, both Bill and Anne use corrective lenses to adjust their vision to a normal near point of 25 cm. When calculating the closest object distance for each wearing the other's glasses, the compensating power of the lenses thus plays a crucial role. Bill and Anne's respective glasses have differing focal lengths designed to meet their unique needs, which significantly affects the calculation for the clear viewing distances when exchanged.

Corrective lenses not only adjust clear vision but also improve other symptoms related to vision strain. They can be used in glasses or contact lenses and offer critical support for everyday tasks involving both near and far objects.