Problem 58 - Physics for Scientists and Engineers with Modern Physics

Question

(II) A magnifying glass is rated at \(3.0 \times\) for a normal eye that is relaxed. What would be the magnification for a relaxed eye whose near point is (a) \(65 \mathrm{~cm},\) and (b) \(17 \mathrm{~cm} ?\) Explain the differences.

Step-by-Step Solution

Answer

The magnification is 1.15 for a 65 cm near point and 4.41 for a 17 cm near point.

Step 1: Understand the Formula for Magnifying Glass

The magnifying power of a magnifying glass is given by the formula \( M = \frac{25 \text{ cm}}{N} \times M_0 \), where \( M \) is the magnification for an observer with a near point of \( N \), and \( M_0 \) is the rated magnification when the near point is 25 cm, which is 3.0 in this case.

Step 2: Calculate Magnification for a Near Point of 65 cm

Substitute \( N = 65 \text{ cm} \) and \( M_0 = 3.0 \) into the magnification formula: \[ M = \frac{25}{65} \times 3.0 = \frac{75}{65} \approx 1.15. \] Thus, the magnification for a relaxed eye with a near point of 65 cm is approximately 1.15.

Step 3: Calculate Magnification for a Near Point of 17 cm

Substitute \( N = 17 \text{ cm} \) and \( M_0 = 3.0 \) into the magnification formula: \[ M = \frac{25}{17} \times 3.0 = \frac{75}{17} \approx 4.41. \] Thus, the magnification for a relaxed eye with a near point of 17 cm is approximately 4.41.

Step 4: Compare and Explain the Differences in Magnification

The magnification is higher for someone with a near point of 17 cm as opposed to 65 cm. A closer near point allows the eye to accommodate more, effectively increasing the magnifying power because the relaxed state eye can focus on objects placed closer to it.

Key Concepts

Magnifying Glass

A magnifying glass is a simple optical device designed to make small objects appear larger. It consists of a convex lens that bends light rays to converge at a focal point. By bringing an object closer than the normal viewing distance, the lens can make it look larger to the human eye.

Here’s how it works:

• Light passes through the convex lens and is bent inward.
• This causes the light rays to converge, creating a magnified image.
• The image is formed at a point where the lens is able to focus the light rays effectively.

The effectiveness of a magnifying glass is often described in terms of magnifying power, which refers to how many times larger the image appears compared to the actual object. This tool is commonly used by people with normal vision to see finer details or small text, and is ideal when eyesight is not perfectly acute. However, differences in eye capacity, like varying near points, can impact how much magnification is achieved in practice.

Near Point

The near point is the closest distance at which the eye can focus comfortably. For most people, this distance changes with age and conditions, such as originality of the eye lens and overall physical well-being. A typical near point is about 25 cm for a healthy adult.

Distances closer than this make it difficult for the eye to focus without visual strain.

When using a magnifying glass, the near point plays a significant role:

• A shorter near point allows viewing objects at a closer distance, creating a potential for greater magnification naturally by the eye itself.
• A longer near point tends to reduce the effective magnification because the eye cannot focus as close, meaning the object must be further away.

In the context of optical magnification, knowing the near point helps tailor the magnifying glass's effectiveness. A magnifying glass rated for general use assumes a near point of 25 cm, but variations in this can affect how much larger the object is perceived.

Magnifying Power

Magnifying power is a key concept in understanding how much larger an image appears when viewed through a magnifying glass. It is determined by the formula \( M = \frac{25 \text{ cm}}{N} \times M_0 \), where:

• \( M \) is the actual magnification achieved for the observer.
• \( N \) is the observer's near point, measured in centimeters.
• \( M_0 \) is the rated magnification when the standard near point of 25 cm is used.

This formula showcases how the perceived magnification changes based on individual variations in near point. For those with a shorter near point, the magnifying power increases as the eye can naturally focus closer than someone with a longer near point.

Practical use means:

• The longer the near point, the lower the effective magnification, as seen in our exercise where 65 cm provides less power than 17 cm.
• The closer the near point, the stronger the magnification due to increased ability to focus on closer objects.

Thus, understanding the magnifying power is essential for tailoring the use of optical aids like magnifying glasses to individual vision needs.