Lens Magnification Factor Problems: A Comprehensive Guide

Lens magnification factor problems involve calculating the magnification of a lens based on the object distance, image distance, and focal length. This comprehensive guide will provide you with a deep understanding of the underlying principles, formulas, and practical examples to help you master this topic.

Understanding Lens Magnification

The magnification of a lens is the ratio of the size of the image to the size of the object. It is a dimensionless quantity that represents the degree of enlargement or reduction of the object. The magnification of a lens can be expressed as:

$m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$

Where:
– $m$ is the magnification factor
– $h_i$ is the height of the image
– $h_o$ is the height of the object
– $d_i$ is the distance between the image and the lens
– $d_o$ is the distance between the object and the lens

The negative sign in the second equation indicates that the image is inverted compared to the object.

Lens Equation and Magnification

The relationship between the object distance, image distance, and focal length of a lens is described by the lens equation:

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

Where $f$ is the focal length of the lens.

By rearranging the lens equation, we can express the magnification in terms of the object distance and focal length:

$m = \frac{d_i}{d_o} = -\frac{f}{d_o}$

This equation allows us to calculate the magnification of a lens given the object distance and focal length.

Example 1: Calculating Magnification Using Object and Image Distances

Problem Statement: An object is placed 20 cm in front of a convex lens with a focal length of 10 cm. Determine the magnification of the lens.

Solution:
1. Gather the given information:
– Object distance, $d_o = -20$ cm (negative because the object is on the left side of the lens)
– Focal length, $f = 10$ cm
2. Substitute the values into the magnification equation:
$m = -\frac{f}{d_o} = -\frac{10\ cm}{-20\ cm} = 0.5$

Therefore, the magnification of the lens is 0.5, indicating a reduced and upright image.

Example 2: Calculating Magnification Using Object and Image Heights

Problem Statement: An object with a height of 4 cm is placed in front of a lens, and the image has a height of 8 cm. Calculate the magnification of the lens.

Solution:
1. Gather the given information:
– Object height, $h_o = 4$ cm
– Image height, $h_i = 8$ cm
2. Substitute the values into the magnification equation:
$m = \frac{h_i}{h_o} = \frac{8\ cm}{4\ cm} = 2$

Therefore, the magnification of the lens is 2, indicating an enlarged and upright image.

Practical Applications of Lens Magnification

Lens magnification is crucial in various applications, such as:

1. Microscopes: Microscopes use a series of lenses to magnify small objects, allowing us to observe them in greater detail.
2. Telescopes: Telescopes use lenses or mirrors to magnify distant objects, enabling us to study celestial bodies in more detail.
3. Magnifying glasses: Magnifying glasses are simple lenses that provide a magnified view of small objects, making them useful for reading, inspecting, or performing detailed work.
4. Camera lenses: Camera lenses use the principles of lens magnification to capture images at different scales, from wide-angle to telephoto.

Numerical Problems and Exercises

1. Problem: A 5 cm tall object is placed 20 cm in front of a convex lens with a focal length of 10 cm. Calculate the image distance and the magnification of the lens.

Solution:
– Using the lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
– Substituting the values: $\frac{1}{10\ cm} = \frac{1}{-20\ cm} + \frac{1}{d_i}$
– Solving for $d_i$: $d_i = 20\ cm$
– Using the magnification equation: $m = -\frac{d_i}{d_o} = -\frac{20\ cm}{-20\ cm} = 1$

Therefore, the image distance is 20 cm, and the magnification of the lens is 1, indicating a real and upright image.

1. Problem: A concave lens has a focal length of -15 cm. An object is placed 30 cm in front of the lens. Determine the image distance and the magnification of the lens.

Solution:
– Using the lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
– Substituting the values: $\frac{1}{-15\ cm} = \frac{1}{30\ cm} + \frac{1}{d_i}$
– Solving for $d_i$: $d_i = -10\ cm$
– Using the magnification equation: $m = -\frac{d_i}{d_o} = -\frac{-10\ cm}{30\ cm} = \frac{1}{3}$

Therefore, the image distance is -10 cm, and the magnification of the lens is 1/3, indicating a reduced and inverted image.

1. Problem: A 2 cm tall object is placed 40 cm in front of a convex lens with a focal length of 20 cm. Determine the image distance, image height, and magnification of the lens.

Solution:
– Using the lens equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
– Substituting the values: $\frac{1}{20\ cm} = \frac{1}{40\ cm} + \frac{1}{d_i}$
– Solving for $d_i$: $d_i = 40\ cm$
– Using the magnification equation: $m = -\frac{d_i}{d_o} = -\frac{40\ cm}{40\ cm} = -1$
– Calculating the image height: $h_i = mh_o = -1 \cdot 2\ cm = -2\ cm$

Therefore, the image distance is 40 cm, the image height is -2 cm, and the magnification of the lens is -1, indicating a real and inverted image.

Measuring Magnification Factor Using a Water Drop Lens

One interesting way to explore lens magnification is by using a water drop as a simple magnifying glass. Here’s how you can do it:

1. Obtain a transparent surface, such as a glass slide or a piece of clear plastic.
2. Place a small drop of water on the surface, creating a water drop lens.
3. Position the water drop lens over a printed text or a small object.
4. Observe the magnified image through the water drop lens.
5. Measure the height of the magnified text or object and compare it to the original size.
6. Calculate the magnification factor by dividing the magnified height by the original height.

By experimenting with different water drop sizes and positions, you can observe how the magnification factor changes. This hands-on activity can help you develop a better understanding of the principles of lens magnification.

Conclusion

Lens magnification factor problems are a fundamental concept in optics and have numerous practical applications. By understanding the underlying principles, formulas, and examples, you can confidently solve a wide range of lens magnification problems. This comprehensive guide has provided you with the necessary tools and knowledge to master this topic and apply it in various scientific and technological contexts.

References

1. Hecht, E. (2016). Optics (5th ed.). Pearson.
2. Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers with Modern Physics (10th ed.). Cengage Learning.
3. Giancoli, D. C. (2018). Physics for Scientists and Engineers with Modern Physics (4th ed.). Pearson.
4. Young, H. D., & Freedman, R. A. (2016). University Physics with Modern Physics (14th ed.). Pearson.