Problem 54 - University Physics with Modern Physics

Question

Contact Lenses. Contact lenses are placed right on the eyeball, so the distance from the eye to an object (or image) is the same as the distance from the lens to that object (or image). A certain person can see distant objects well, but his near point is 45.0 \(\mathrm{cm}\) from his eyes instead of the usual 25.0 \(\mathrm{cm} .(\mathrm{a})\) Is this person nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed to correct his vision? (c) If the correcting lenses will be contact lenses, what focal length lens is needed and what is its power in diopters?

Step-by-Step Solution

Answer

The person is farsighted and needs a converging lens with a focal length of 16.07 cm and a power of 6.22 diopters.

Step 1: Define Nearsightedness and Farsightedness

Nearsighted individuals can see close objects clearly but have difficulty seeing distant objects. Farsighted individuals can see distant objects clearly but struggle to see close ones. This person can see distant objects well but not close ones (closer than 45 cm). Thus, this person is farsighted.

Step 2: Determine the Type of Lens Needed

Farsightedness is corrected using converging lenses, which help in focusing light on the retina for closer objects. Therefore, this person requires a converging lens for correction.

Step 3: Calculate Focal Length Using Lens Formula

To find the required focal length, we use the lens formula: \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\). Here, the desired near point \(v\) is 25.0 cm, and the actual near point \(u\) is 45.0 cm. Since contact lenses are worn directly on the eye, \(v = 25\) cm (positive for a real image), \(u = -45\) cm (negative for the virtual object). Substituting these values, \(\frac{1}{f} = \frac{1}{25} - \frac{1}{-45} = \frac{1}{25} + \frac{1}{45}\).

Step 4: Solve for Focal Length

Let's calculate the focal length from the previous equation: \[\frac{1}{f} = \frac{45 + 25}{1125} = \frac{70}{1125}\].Hence, \(f = \frac{1125}{70} \approx 16.07\) cm.

Step 5: Calculate Lens Power in Diopters

The power \(P\) of a lens is given by the formula \(P = \frac{1}{f}\), where \(f\) is the focal length in meters. Convert 16.07 cm to meters: \(f = 0.1607\) m.So, \(P = \frac{1}{0.1607} \approx 6.22\) diopters.

Key Concepts

Farsightedness

Farsightedness, also known as hyperopia, is a common vision condition where distant objects are seen clearly, but close objects appear blurry. It occurs when the light entering the eye is focused behind the retina rather than directly on it. This can be due to an eyeball that is too short or a lens that is not curved enough.

If you're farsighted, your eye struggles to bring near objects into focus, causing you to experience strain and fatigue when reading or focusing on anything up close. Ordinary tasks like reading a book or using a smartphone can become challenging.

Some key facts about farsightedness include:

• It often runs in families and can be detected early in life.
• Many children are slightly farsighted but outgrow it as their eyes grow longer and the lens can focus light more effectively.
• Farsightedness is corrected by using convex lenses which help to converge the light rays before they enter the eye.

Understanding farsightedness is the first step towards finding the right corrective measures.

Converging Lens

A converging lens, commonly referred to as a convex lens, is thicker in the middle than at the edges. Its primary function is to bend light rays towards each other, or converge them, to focus on a single point. This is particularly useful for farsighted individuals, as these lenses help focus light rays on the retina instead of behind it, providing a clearer image of close objects.

Here’s how a converging lens works:

• Light rays that pass through the lens are bent towards the center, making them converge.
• The point where the rays converge is called the focal point, and the distance from the lens to this point is the focal length.
• By adjusting the focal length, these lenses can adapt to correct various levels of farsightedness.

Converging lenses are integral to eyeglasses and contact lenses designed to rectify hyperopia, helping individuals see both near and far objects more clearly.

Lens Power in Diopters

The power of a lens is measured in diopters, a unit that quantifies the ability of the lens to bend light. A diopter describes the refractive power of the lens, calculated as the inverse of its focal length (in meters). This means that a lens with a shorter focal length has a higher diopter value and thus greater light-bending ability.

To understand lens power better, consider the formula:
\[ P = \frac{1}{f} \]
Where `P` is the lens power in diopters and `f` is the focal length in meters. For example, if a lens has a focal length of 0.1607 meters, its power would be approximately 6.22 diopters.

Here are some key points about diopters:

• Positive diopter values indicate converging (convex) lenses, ideal for treating farsightedness.
• Negative diopter values are used for diverging (concave) lenses, which correct nearsightedness.
• Optometrists use diopters to prescribe the appropriate lens strength for correcting vision issues.

By understanding how diopters work, individuals can better comprehend the specifications of their corrective lenses.