Both a converging lens and a concave mirror can produce virtual images that are larger than the object. Concave mirrors can be used as magnifying makeup or shaving mirrors, but converging lenses cannot be. (a) Draw ray diagrams to explain why not. (b) If a concave mirror has the same focal length as a converging lens, and an object is placed first at a distance of (1/2)ƒ from the lens and then at a distance of (1/2)ƒ from the mirror, how will the magnification of the object compare in the two cases?

Verified step by step guidance

1. Step 1: Understand the problem. Part (a) asks us to explain why converging lenses cannot be used as magnifying mirrors, while concave mirrors can. Part (b) involves comparing the magnification of an object placed at a distance of 1/2ƒ from both a converging lens and a concave mirror with the same focal length.
2. Step 2: For part (a), draw ray diagrams for both the converging lens and the concave mirror. For the converging lens, place the object closer to the lens than the focal point (inside the focal length). Use the lens equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to determine the image distance \( d_i \), and note that the image will be virtual, upright, and magnified. For the concave mirror, place the object closer to the mirror than the focal point. Use the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to determine the image distance \( d_i \), and note that the image will also be virtual, upright, and magnified. However, the key difference is that the concave mirror reflects light, allowing the virtual image to appear in front of the mirror, making it suitable for use as a magnifying mirror. The converging lens refracts light, and the virtual image forms on the same side as the object, making it less practical for this purpose.
3. Step 3: For part (b), use the magnification formula \( M = -\frac{d_i}{d_o} \) for both the lens and the mirror. Start with the lens: Substitute \( d_o = \frac{1}{2}f \) into the lens equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to solve for \( d_i \). Then calculate the magnification \( M \) using the magnification formula. Note that the image will be virtual, upright, and magnified.
4. Step 4: Repeat the process for the concave mirror. Substitute \( d_o = \frac{1}{2}f \) into the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to solve for \( d_i \). Then calculate the magnification \( M \) using the magnification formula. Note that the image will also be virtual, upright, and magnified.
5. Step 5: Compare the magnifications obtained for the lens and the mirror. Since the focal lengths are the same and the object distances are identical, the magnifications will be the same in magnitude. However, the practical usability differs due to the reasons explained in part (a).

Key Concepts

Ray Diagrams

Ray diagrams are graphical representations used to illustrate how light rays interact with optical devices like lenses and mirrors. They help visualize the paths of light as it travels through a lens or reflects off a mirror, showing how images are formed. By tracing specific rays, such as the parallel ray, the focal ray, and the center ray, one can determine the characteristics of the image, including its size and orientation.

Magnification

Magnification is a measure of how much larger an image appears compared to the actual object. It is defined as the ratio of the height of the image to the height of the object and can be calculated using the formula M = h'/h, where h' is the image height and h is the object height. In the context of mirrors and lenses, the magnification also depends on the object's distance from the optical device and its focal length.

Virtual Images

Virtual images are images formed by optical devices where light rays appear to diverge from a point, but do not actually converge there. They cannot be projected onto a screen and are typically upright and larger than the object when formed by concave mirrors or converging lenses under specific conditions. Understanding the conditions under which virtual images are produced is crucial for analyzing the behavior of lenses and mirrors in practical applications.