Problem 54 - University Physics with Modern Physics

Question

A person can see clearly up close but cannot focus on objects beyond 75.0 cm. She opts for contact lenses to correct her vision. (a) Is she nearsighted or farsighted? (b) What type of lens (converging or diverging) is needed to correct her vision? (c) What focal length contact lens is needed, and what is its power in diopters?

Step-by-Step Solution

Answer

(a) She's nearsighted. (b) A diverging lens is needed. (c) The focal length is -75.0 cm, and the power is -1.33 diopters.

Step 1: Determine the Type of Vision Problem

Since the person can see clearly only objects that are close, specifically up to 75.0 cm, she is nearsighted (myopic). This means she struggles to see distant objects clearly, which is characteristic of being nearsighted.

Step 2: Identify the Needed Lens Type

To correct nearsightedness, a diverging lens is needed. Diverging lenses help to spread out light rays more so that they focus further back on the retina, allowing distant objects to be seen more clearly.

Step 3: Use the Lens Formula to Find Focal Length

To correct her vision to normal, which means seeing distant objects clearly (assume infinity), we'll use the formula \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \( f \) is the focal length, \( v \) is the image distance (for distant objects, \( v \approx 0 \)), and \( u = -75.0 \) cm (image formed at the near point initially). This simplifies to \( \frac{1}{f} = -\frac{1}{75.0} \).

Step 4: Calculate Focal Length

Solving \( \frac{1}{f} = -\frac{1}{75.0} \), the focal length \( f = -75.0 \) cm. The negative sign indicates a diverging lens is needed, consistent with correcting nearsightedness.

Step 5: Convert to Diopters

The power of a lens in diopters \( P \) is given by \( P = \frac{1}{f} \) with \( f \) in meters. Convert \( f \): \( -75.0 \) cm = \( -0.75 \) m. So, \( P = \frac{1}{-0.75} \approx -1.33 \) diopters.

Key Concepts

Diverging Lens

A diverging lens is a type of lens that spreads out light rays that are initially traveling parallel to each other. This type of lens is thinner at the center than it is at the edges. When light rays pass through a diverging lens, they are bent away from the lens. This process helps to extend the focus point backwards, thereby correcting vision for individuals who are nearsighted. The diverging lens effectively decreases the power of the eye's natural lens, making it possible for distant objects to be focusable onto the retina.

In practical terms, diverging lenses are identified by their negative focal length—a characteristic that is vital for correcting myopia, or nearsightedness, which involves focusing too soon in front of the retina.

Lens Formula

The lens formula is a critical equation in optics that relates the focal length of a lens to the distance of the object from the lens and the distance of the image formed. The formula is typically written as:

\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

where:

• \( f \) is the focal length
• \( v \) is the image distance (the distance between the image and the lens)
• \( u \) is the object distance (the distance between the object and the lens)

This formula is extremely useful: solving it allows us to determine the focal length required for a lens to correct vision problems. In the case of nearsightedness, the near point (the maximum distance at which the person can see clearly) is used as the object distance \( u \).

Understanding how light bends through lenses using this formula is key to designing solutions for many vision problems.

Focal Length

The focal length of a lens is the distance between the lens and the point where light rays converge or appear to diverge. For a diverging lens, the focal length is negative, which indicates that the focal point is virtual and on the same side as the object.

In the context of vision correction, the focal length determines how much refraction (bending of light) a lens must provide to bring images into sharp focus on the retina. For a nearsighted person who can only see clearly up to 75 cm, a specific focal length lens is needed to adjust the eye's converging power and enable clear sight of distant objects. Using the lens formula, we can calculate that the required focal length must be \(-75\) cm, fulfilling the requirement for a diverging lens.

Diopter Calculation

The power of a lens is measured in diopters, which is a unit that describes how strongly the lens can refocus light. Diopter calculation is crucial in optometry to determine the correct prescription needed for glasses or contact lenses.

The formula to compute lens power in diopters is:

\[ P = \frac{1}{f} \]

Here, \( P \) is the power in diopters and \( f \) is the focal length in meters. For the nearsighted person who needs a focal length of \(-75\) cm, converting this measurement to meters gives \(-0.75\) m. Thus, the lens power is:

• \( P = \frac{1}{-0.75} = -1.33 \) diopters

The negative sign here is significant as it confirms the presence of a diverging lens, suitable for correcting the individual’s myopia.