Problem 140 - Fundamentals of Physics

Question

Suppose the farthest distance a person can see without visual aid is \(50 \mathrm{~cm} .\) (a) What is the focal length of the corrective lens that will allow the person to see very far away? (b) Is the lens converging or diverging? (c) The power \(P\) of a lens (in diopters) is equal to \(1 / f,\) where \(f\) is in meters. What is \(P\) for the lens?

Step-by-Step Solution

Answer

(a) Focal length is -0.5 m. (b) The lens is diverging. (c) Power is -2 diopters.

Step 1: Identify Focal Point for Infinite Vision

To correct the person's vision such that they can see objects at infinity clearly, we need to set the object-distance as infinity. Using the lens formula \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \(u\) is the object distance (infinity) and \(v\) is the image distance (50 cm or 0.5 m), the equation simplifies to \( \frac{1}{f} = \frac{1}{0.5} - \frac{1}{\infty} \).

Step 2: Calculate Focal Length

Since \( \frac{1}{\infty} \) is 0, the lens formula becomes \( \frac{1}{f} = \frac{1}{0.5} \), which simplifies to \(f = 0.5\) meters. Thus, the focal length of the lens is -0.5 meters (since it is a diverging lens for correcting short sight).

Step 3: Determine Lens Type

The negative sign of the focal length indicates that the lens is diverging. Diverging lenses are used to correct myopia or nearsightedness.

Step 4: Calculate Lens Power

Using the formula for power \( P = \frac{1}{f} \), where \(f\) is in meters, we have \( P = \frac{1}{-0.5} = -2 \) diopters. Therefore, the power of the lens is \(-2\) diopters.

Key Concepts

Focal Length

In the world of optics, the focal length of a lens is a fundamental concept. It defines the distance between the lens and the point where parallel rays of light converge or appear to diverge. In simpler terms, it's the spot where images come into focus. When a person has difficulty seeing objects far away, corrective lenses change this focal point to help bring those distant images into sharper focus.

For example, if someone can only see clearly up to 50 cm, a lens with a specific focal length can adjust how the light enters their eyes. The lens aids in shifting the focus from a near point to a far point, essentially extending their vision to infinity, where distant objects appear sharp.

• Positive focal length: used in converging lenses.
• Negative focal length: used in diverging lenses.

Lens Power

Lens power is a term often encountered when discussing corrective eyewear. It relates directly to the focal length and is expressed in diopters. Calculating lens power is straightforward when you know the focal length:\[ P = \frac{1}{f} \] Here, the focal length \( f \) is always considered in meters.

A lens with a focal length of -0.5 meters would have a power of:
\[ P = \frac{1}{-0.5} = -2 \] which signifies that the lens is designed to correct nearsightedness. Diopters not only indicate the need for lens correction but also provide insights into the strength of the lens.

• Positive diopters: used for farsightedness (hyperopia).
• Negative diopters: used for nearsightedness (myopia).

Diverging Lenses

Diverging lenses, also known as concave lenses, are uniquely significant in correcting myopia (nearsightedness). They work by spreading out light rays before they enter the eye, which helps in focusing the image correctly on the retina.

Key features of diverging lenses include:

• They have a negative focal length.
• The image formed is virtual and upright.
• They spread out light rays, making them appear to come from a point that is closer than the actual object.

In the example provided, a focal length of -0.5 meters signifies a diverging lens, which is appropriate for someone who has difficulty seeing far objects. This type of lens adjusts the visual accommodation of the eye, aiding in clearer vision of distant objects.