Problem 99 - Physics for Scientists and Engineers with Modern Physics

Question

What is the magnifying power of a +4.0 -D lens used as a magnifier? Assume a relaxed normal eye.

Step-by-Step Solution

Answer

The magnifying power of the lens is 2.

Step 1: Understanding Magnifying Power

Magnifying power, or magnification, of a lens is given by the formula for a relaxed eye: \( M = 1 + \frac{D}{F} \), where \( D \) is the least distance of distinct vision and \( F \) is the focal length of the lens.

Step 2: Identifying Provided Values and Assumptions

The least distance of distinct vision \( D \) is generally taken as 25 cm for a normal eye. The power of the lens \( P \) is given as +4.0 diopters, which is the reciprocal of the focal length \( F \) when expressed in meters. Hence, \( F = \frac{1}{P} = \frac{1}{4.0} \) meters.

Step 3: Calculating Focal Length

Convert the focal length from meters to centimeters for compatibility with \( D \). Since \( F = 0.25 \) meters, \( F = 25 \) cm.

Step 4: Using the Magnifying Power Formula

Substitute the values for \( D \) and \( F \) into the magnifying power formula: \[ M = 1 + \frac{25}{25} = 1 + 1 = 2. \]

Step 5: Conclusion of Magnifying Power

The magnifying power of the lens is 2, meaning it magnifies objects to twice their size when viewed with a relaxed eye.

Key Concepts

Magnifying Power

When using lenses, the magnifying power is a measure of how much larger a lens can make an object appear. This value is essential for understanding how effective a lens is in magnifying objects. In mathematical terms, for a relaxed eye, it is expressed as:\[ M = 1 + \frac{D}{F} \]where:

• \( M \) is the magnifying power.
• \( D \) represents the least distance of distinct vision, commonly 25 cm for normal vision.
• \( F \) is the focal length of the lens.

A positive magnifying power indicates that the lens is converging and enlarging the object, rather than scattering the light.

Diopters

Diopters measure the optical power of a lens and are significant in dictating how a lens affects vision. A lens with a positive diopter value, like +4.0 D, indicates a converging lens which brings parallel rays of light to a focus. The formula that relates diopters to focal length is:\[ P = \frac{1}{F} \]where:

• \( P \) is the power in diopters.
• \( F \) is the focal length in meters.

Therefore, a lens with +4.0 D has a focal length of 0.25 meters (or 25 cm), which is crucial for determining the magnifying power.

Focal Length

The focal length of a lens is the distance over which parallel rays of light are brought to a focus. It is a core property determining a lens's convergence or divergence capability. For our given lens with a +4.0 diopter value:- The focal length is calculated by taking the reciprocal of the diopter value.- Thus, \( F = \frac{1}{P} = \frac{1}{4.0} \), which equals 0.25 meters.This indicates that the lens focuses light 25 cm from itself, making it essential to use this focal length in magnification calculations.

Least Distance of Distinct Vision

The least distance of distinct vision is the minimum distance at which the eye can comfortably focus on an object. For most people with normal vision, this distance is about 25 cm. This standard measurement helps define metrics like magnifying power efficiently. Utilizing a lens changes the effective least distance, improving how well we can see small details at a shorter distance, effectively doubling the size of these details when the magnifying power is 2, as was calculated in our solution. This concept ensures that when we use magnifying lenses, the reduced strain and increased clarity enhance everyday tasks and scientific observations.