Problem 107 - Physics

Question

A nearsighted person cannot read a sign that is more than \(5.2 \mathrm{~m}\) from his eyes. To deal with this problem, he wears contact lenses that do not correct his vision completely, but do allow him to read signs located up to distances of \(12.0 \mathrm{~m}\) from his eyes. What is the focal length of the contacts?

Step-by-Step Solution

Answer

Focal length of the contacts is approximately 8.73 m.

Step 1: Understand the Problem

A nearsighted person has a far point of 5.2 meters without correction. With contact lenses, this person can see up to 12.0 meters. We are to find the focal length of these lenses.

Step 2: Use the Lens Formula

The lens formula relates the object distance \(d_o\), image distance \(d_i\), and the focal length \(f\) of a lens: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\] In this situation, the far point without lenses (5.2 meters) serves as the image distance \(d_i\), and the farthest point visible with lenses (12.0 meters) is the object distance \(d_o\).

Step 3: Assign Known Values

Assign the known distances to the formula: \(d_o = 12.0\) m and \(d_i = -5.2\) m. The image distance \(d_i\) is negative because the image forms on the same side as the object for a nearsighted person.

Step 4: Substitute Values into the Formula

Substitute the known values into the lens formula: \[\frac{1}{f} = \frac{1}{12.0} + \frac{1}{-5.2}\]

Step 5: Calculate \(\frac{1}{f}\)

Calculate the right side of the equation: \[\frac{1}{f} = \frac{1}{12.0} - \frac{1}{5.2}\]Find common denominators and perform the subtraction.

Step 6: Solve for \(f\)

After calculating the right side, invert the result to find \(f\): \(f = 1 / \text{(result from previous step)}\).

Step 7: Conclusion

Compute the value of \(f\) and confirm the units in meters. The focal length of the contacts is the value you have calculated.

Key Concepts

Lens Formula

The lens formula is a fundamental concept in optics. It helps us relate the focal length of a lens to the distances of an object and its image. The formula is given by:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]where:

• \(f\) is the focal length of the lens.
• \(d_o\) is the distance from the lens to the object (object distance).
• \(d_i\) is the distance from the lens to the image (image distance).

The key thing about the lens formula is that it uses reciprocal values. You find the reciprocal of the focal length by adding the reciprocals of the object distance and the image distance. This equation is vital for designing optical systems, like glasses and cameras.
In our exercise, understanding the relationship between these distances and the focal length enables us to determine the power needed in lenses to correct vision.

Nearsightedness

Nearsightedness, or myopia, is a common vision condition where distant objects appear blurry while close objects can be seen clearly. This happens because the eye is elongated or the cornea is too curved, which causes light to focus in front of the retina rather than directly on it.
To address nearsightedness:

• Corrective lenses such as glasses or contact lenses are used to diverge light rays and move the focus back onto the retina.
• The far point (farthest distance an object can be seen clearly without correction) is closer than normal.

In the context of the exercise, the far point without lenses is 5.2 meters. This distance tells us how much vision correction is needed for the person in question.

Focal Length

Focal length is a critical term used when discussing lenses. It represents the distance from the lens to its focus, where parallel rays of light meet. A lens with shorter focal length bends light more than one with a longer focal length.
Key considerations for focal length include:

• Lenses for nearsighted individuals will have a negative focal length.
• Shorter focal length lenses provide stronger correction due to greater bending of light.

In our problem, the focal length helps determine how much correction the lens needs to provide to extend the far point from 5.2 meters to 12.0 meters. Once we calculate the focal length using the lens formula, we know exactly how much the contact lenses adjust the focus.

Vision Correction

Vision correction aims to adjust the eye's focusing ability to see clearly at various distances. For nearsightedness, lenses need to diverge light before it reaches the eye.
Methods of vision correction include:

• Eyeglasses which can easily be adjusted for different lighting needs and distances.
• Contact lenses which sit right on the eye for convenience and comfort.

Using the focal length found through calculations, we adjust the strength of these lenses. In the given exercise, contact lenses correct the person’s sight enough to see distant objects previously indistinct. The corrected vision now extends to 12.0 meters, indicating improved visual capacity.