Problem 67 - College Physics

Question

An optometrist prescribes glasses with a power of \(-4.0 \mathrm{D}\) for a nearsighted student. What is the focal length of the glass lenses?

Step-by-Step Solution

Answer

The focal length of the glasses is \(-0.25\) meters.

Step 1: Understanding the Relationship between Power and Focal Length

The power of a lens is connected to its focal length using the formula \( P = \frac{1}{f} \), where \( P \) is the power in diopters and \( f \) is the focal length in meters. Our goal is to solve for \( f \).

Step 2: Rearrange the Formula to Solve for Focal Length

We need to rearrange the formula \( P = \frac{1}{f} \) to solve for the focal length \( f \). By taking the reciprocal, the formula becomes \( f = \frac{1}{P} \).

Step 3: Substitute the Given Power into the Formula

Substitute the given power of the lens, \( -4.0 \, \text{D} \), into the formula: \( f = \frac{1}{-4.0} \).

Step 4: Calculate the Focal Length

Perform the calculation \( f = \frac{1}{-4.0} \) which results in \( f = -0.25 \, \text{m} \). This value indicates that the lens has a focal length of \(-0.25\) meters, or \(-25\) centimeters.

Key Concepts

Lens Power

Lens power is a fundamental concept in optics and helps determine how effective a lens is at bending light. This power is measured in diopters (D) and tells us how strong or weak a lens is. When a lens has a high positive power, it converges light rays more sharply, while a high negative power means it diverges light rays more.
The formula for lens power is given by:

• \( P = \frac{1}{f} \)

where \( P \) is the power in diopters, and \( f \) is the focal length in meters. A negative power, like \(-4.0 \text{ D}\), implies that the lens is concave and is used for correcting vision problems such as nearsightedness. In our case, the lens power helps us find the focal length of the glasses prescribed by the optometrist.

Focal Length Calculation

Focal length is an essential parameter that describes how strongly the lens converges or diverges light. It is the distance over which initially collimated rays are brought to a focus. For our calculation, knowing the lens power allows us to find the focal length using the rearranged formula:

• \( f = \frac{1}{P} \)

This step is straightforward. We substitute the given lens power into the formula. So, for a lens power of \(-4.0 \text{ D}\), the focal length is:

• \( f = \frac{1}{-4.0} \)

This calculation results in a focal length, \( f = -0.25 \text{ m} \). The negative sign indicates that it is a divergent lens, appropriate for nearsightedness correction.

Nearsightedness Correction

Nearsightedness, also known as myopia, is a common vision condition where close objects are seen clearly, but distant objects appear blurry. This occurs because the eye focuses images in front of the retina instead of on it. To correct nearsightedness, lenses are used to adjust the focal point onto the retina.
Concave lenses, which have negative power, spread out light rays before they enter the eye. This helps shift the focus back onto the retina. The glasses with a power of \(-4.0 \text{ D}\) form an example of a concave lens used for this purpose. The prescribed lens helps the eye to project distant objects clearly, ensuring the wearer can see them as sharply as nearby objects.
Corrective lenses are personalized based on the degree of myopia, often calculated by an optometrist through standard eye tests.