Problem 141 - Fundamentals of Physics

Question

A simple magnifier of focal length \(f\) is placed near the eye of someone whose near point \(P_{n}\) is \(25 \mathrm{~cm} .\) An object is positioned so that its image in the magnifier appears at \(P_{n}\). (a) What is the angular magnification of the magnifier? (b) What is the angular magnification if the object is moved so that its image appears at infinity? For \(f=10 \mathrm{~cm},\) evaluate the angular magnifications of (c) the situation in (a) and (d) the situation in (b). (Viewing an image at \(P_{n}\) requires effort by muscles in the eye, whereas viewing an image at infinity requires no such effort for many people.)

Step-by-Step Solution

Answer

(a) 3.5, (b) 2.5, (c) 3.5, (d) 2.5.

Step 1: Understanding Angular Magnification at the Near Point

To find the angular magnification when the image is at the near point \(P_n\), we use the formula \( M = 1 + \frac{D}{f} \), where \(D = 25\text{ cm}\) is the near point. This formula accounts for image formation at \(P_n\). According to this formula, the angular magnification \(M\) can be expressed as \(M = 1 + \frac{25}{f} \).

Step 2: Understanding Angular Magnification at Infinity

When the image is at infinity, the angular magnification is given by \( M = \frac{D}{f} \). This formula simplifies the understanding that at infinity, the object distance is such that the lens creates a parallel beam for relaxed viewing by the eye.

Step 3: Calculate Angular Magnification for Part (a)

For \(f = 10\text{ cm}\), substitute into the formula from Step 1. Thus, \( M = 1 + \frac{25}{10} \). Simplifying this expression gives \( M = 1 + 2.5 = 3.5 \).

Step 4: Calculate Angular Magnification for Part (b)

Substitute \(f = 10\text{ cm}\) into the formula from Step 2. Thus, \( M = \frac{25}{10} \). Simplifying this expression gives \( M = 2.5 \).

Step 5: Evaluating Part (c)

Part (c) refers to the setup where the image appears at the near point. From our calculation in Step 3, the angular magnification is \(3.5\).

Step 6: Evaluating Part (d)

Part (d) refers to the setup where the image appears at infinity. From our calculation in Step 4, the angular magnification is \(2.5\).

Key Concepts

Angular Magnification

Angular magnification is a measure of how much larger an image appears through a lens compared to viewing it with the naked eye without any optical aid. This concept is important in optics because it allows us to quantify how "powerful" a magnifying glass or lens is.
In this context, angular magnification is given by:

• For an object at the near point: \[ M = 1 + \frac{D}{f} \]where \(D\) is the near point (typically 25 cm), and \(f\) is the focal length of the lens.
• For an image at infinity:\[ M = \frac{D}{f} \]This setup is often more comfortable as it requires less effort from the eye muscles.

Understanding these formulas helps to distinguish between the different visual experiences offered by variations in object positioning relative to the lens.

Focal Length

The focal length, usually denoted as \(f\), is a crucial property of lenses and mirrors in optics. It describes the distance from the center of a lens to its focal point, where light rays converge. In the exercise provided, the focal length of the magnifying lens is given as 10 cm.

• A shorter focal length means a stronger lens that bends light rays more sharply, focusing them more quickly.
• The focal length determines how the lens magnifies objects and affects the angular magnification directly.

Understanding focal length is important as it influences how images are formed by lenses as well as the magnification provided, making it a central concept in learning about optical devices.

Near Point

The near point is the closest distance at which the eye can focus comfortably on an object. For a "normal" human eye, this distance is typically around 25 cm. In optics problems, it's often used to calculate the maximum allowable magnification that still results in a clear view for the observer.
The exercise uses the near point, \(D\), as a reference point for calculating angular magnification. When an image is at the near point:

• The formula used incorporates \[ M = 1 + \frac{D}{f} \], leading to greater magnification compared to an image at infinity.
• Visualizing at the near point involves eye muscle effort, hence is more strenuous compared to viewing an object at infinity.

By understanding the concept of the near point, students can better comprehend why certain magnification scenarios require more focus adjustment.

Image Formation

Image formation is the process by which lenses and mirrors create visible representations of objects. With a magnifier, the image formation principle is about manipulating light paths to magnify an image for better viewing.
In the exercise:

• When the image appears at the near point, the lens causes rays to converge such that the image is comfortably observed at 25 cm.
• To form an image at infinity, the light rays are made parallel, easing the viewing effort for the eye as the image appears far away.

Understanding how images are formed by lenses helps explain why different set-ups lead to varying angular magnifications, adjusting the image's apparent size perceived by the viewer.