Problem 93 - Fundamentals of Physics

Question

Someone with a near point \(P_{n}\) of \(25 \mathrm{~cm}\) views a thimble through a simple magnifying lens of focal length \(10 \mathrm{~cm}\) by placing the lens near his eye. What is the angular magnification of the thimble if it is positioned so that its image appears at (a) \(P_{n}\) and (b) infinity?

Step-by-Step Solution

Answer

(a) 1.5 and (b) 2.5

Step 1: Understanding Angular Magnification

Angular magnification, denoted as \( M \), is defined by the formula \( M = \frac{d_i}{f} + 1 \), where \( d_i \) is the image distance and \( f \) is the focal length of the lens. Angular magnification represents how much larger the object appears compared to when it's viewed without the lens.

Step 2: Case (a) Image at Near Point \(P_n = 25 \mathrm{~cm}\)

For this case, the image is formed at the near point \(P_n = 25 \mathrm{~cm}\). The lens formula is \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \( v \) is the image distance, and \( u \) is the object distance. Substituting \( f = 10 \mathrm{~cm} \) in, we have: \( \frac{1}{10} = \frac{1}{25} - \frac{1}{u} \). Solving for \( u \), we find that \( u \approx 16.67 \mathrm{~cm} \). The angular magnification \( M \) is then \( M = \frac{25}{u} \approx \frac{25}{16.67} = 1.5 \).

Step 3: Case (b) Image at Infinity

For an image formed at infinity, the object is placed at the focal point of the lens, \( u = f = 10 \mathrm{~cm}\). The angular magnification is then given by \( M = \frac{P_n}{f} \). Substituting \( P_n = 25 \mathrm{~cm} \) and \( f = 10 \mathrm{~cm} \), the magnification is \( M = \frac{25}{10} = 2.5 \).

Key Concepts

Focal Length

The focal length (\(f\)) of a lens is an essential concept in optics. It refers to the distance from the center of the lens to the point where the light rays converge after passing through the lens. This point is known as the focal point. The focal length helps determine how the lens bends the light and whether it will produce a magnification or a reduction in the image size.

• For converging lenses (like the magnifying glass in the exercise), shorter focal lengths mean stronger bending of light rays, resulting in larger magnification.
• For diverging lenses, the opposite is true, where light rays appear to emanate from a focal point behind the lens.

Understanding focal length helps in positioning objects to achieve desired image properties.

Lens Formula

The lens formula is a crucial tool in understanding image formation in optics. It is expressed mathematically as:\[\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\]where:

• \(f\) is the focal length of the lens,
• \(v\) is the image distance from the lens,
• \(u\) is the object distance from the lens.

This formula relates the focal length to the distances of the object and image. It helps us determine where the image will form for a particular lens and object placement.By rearranging the formula, you can solve for any unknown variable if the other two are known. It's a fundamental equation for tasks involving lens systems like microscopes or telescopes.

Angular Magnification

Angular magnification (\(M\)) describes how much larger an object appears when viewed through a lens compared to the naked eye. It is especially relevant for optical instruments like magnifying glasses.The formula for angular magnification is:\[M = \frac{d_i}{f} + 1\]where \(d_i\) is the image distance and \(f\) is the focal length. In simple terms, it combines the effect of the lens's focal properties with the viewing distance to show the increase in apparent size.For high magnification, a balance between focal length and image distance is key, allowing users to see fine details clearly. Angular magnification is crucial for applications ranging from reading aids to complex microscopy.

Image Distance

Image distance (\(v\)) is the distance from the lens to the point where the image forms. This concept is fundamental for understanding how lenses project images onto a surface.The image distance can vary based on:

• The position of the object relative to the lens.
• The focal length of the lens.

When the image distance aligns with the focal length, unique behaviors occur, such as forming an image at infinity.This concept is crucial in applications like camera focusing and lens-based projectors, where precise image placement is often needed for clear visuals.

Object Distance

Object distance (\(u\)) is defined as the distance from the object to the lens. It plays a vital role in determining how an image is projected by a lens.In the lens formula (\(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)), the object distance is one of the parameters you can adjust to control image formation. Depending on its value:

• If the object is at the focal point, the image forms at infinity, as in case (b) of the original exercise.
• If the object is within twice the focal length, a real, magnified image can be observed.

In optical devices, managing object distance ensures users see crisp images, such as in focusing a camera or through a microscope.