Problem 101 - Physics

Question

A nearsighted person has a far point that is located only \(220 \mathrm{~cm}\) from his eyes. Determine the focal length of contact lenses that will enable him to see distant objects clearly.

Step-by-Step Solution

Answer

The focal length required is 220 cm.

Step 1: Understanding Nearsightedness

A nearsighted person can see nearby objects clearly but struggles with distant objects. This means their eyes focus images of distant objects in front of the retina instead of on it.

Step 2: Defining the Problem

The person has a far point at 220 cm, which is where they can see distant objects clearly without assistance. We want to find a lens that allows them clear vision beyond this range.

Step 3: Using 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.

Step 4: Assigning Values

For distant objects, \( u = \infty \) (object distance), and for the person, \( v = -220 \) cm (image distance is negative as it forms on the same side as the object).

Step 5: Solving the Lens Formula

With \( u = \infty \), \( \frac{1}{u} = 0 \). Substituting into the lens formula gives \( \frac{1}{f} = 0 - \frac{1}{-220} \), simplifying to \( \frac{1}{f} = \frac{1}{220} \).

Step 6: Calculating Focal Length

Solving for \( f \) gives \( f = 220 \) cm. This is the focal length of the contact lenses needed.

Key Concepts

Physics of Optics and Vision

Optics is a fascinating branch of physics that explores how light behaves as it travels through different mediums. Understanding optics is crucial for solving vision-related problems, such as nearsightedness or farsightedness. The fundamental principle of optics involves the interaction of light waves with lenses and mirrors.
To address vision issues, like nearsightedness, we use lenses to correct the way light is focused in the eye. Light that enters the eye is refracted by the cornea and lens so that it converges on the retina, creating a clear image. If the light cannot focus directly on the retina, vision issues occur.
Lenses alter the path of light, allowing us to re-focus it correctly on the retina, improving clarity and focus of objects at various distances, whether close up or far away.

Understanding the Lens Formula

The lens formula is a valuable tool in optics, especially for resolving vision problems. It provides a mathematical relationship between the object distance (\( u \)), image distance (\( v \)), and the focal length of a lens (\( f \)).
This formula is expressed as:

• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

The lens formula helps us determine the correct power needed for eyeglasses or contact lenses. By substituting known values into the formula, we can either solve for the focal length or the position of the object or image.
Knowing how to manipulate and apply this formula is key in designing lenses that correct specific types of vision problems, such as myopia (nearsightedness).

Nearsightedness: Myopia Explained

Nearsightedness, or myopia, is a common vision issue where distant objects appear blurry. This occurs when the eyeball is too long or the cornea is too curved, causing the light to focus in front of the retina instead of directly on it.
For a person with myopia, the farthest point they can see clearly without correction is called their far point. In our example, a nearsighted person has a far point of 220 cm, meaning beyond this distance, everything becomes blurry.
Contact lenses or glasses can help refocus light onto the retina. These lenses are often concave, spreading light rays slightly outward before they reach the eye, which helps them focus properly on the retina.

How to Calculate Focal Length for Lenses

Determining the focal length of a corrective lens is essential for solving nearsightedness. The focal length (\( f \)) is the distance from the lens where parallel rays of light converge. It's crucial in determining how much correction is needed to adjust a person's focus from their far point to infinity (distant vision).
In the solution process, for a distant object, the object distance (\( u \)) is considered infinite (\( \infty \)), while the image distance (\( v \)) is given by the person's far point (negative in sign convention since it forms on the same side).

• Using the formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)
• Given \( v = -220 \text{ cm} \) and \( u = \infty \), we find \( \frac{1}{f} = \frac{1}{220} \)
• Resulting in \( f = 220 \text{ cm} \) for the focal length required.

This calculation ensures that light will focus directly on the retina, resolving the blurriness of distant objects.