Question

A converging lens will be used as a magnifying glass. In order for this to work, the object must be placed at a distance a) \(d_{\mathrm{o}}>f\) c) \(d_{\mathrm{o}}

Short answer

Question: To function properly as a magnifying glass, the distance of the object from a converging lens with focal length \(f\) should be: a) \(d_{\mathrm{o}}>f\) b) \(d_{\mathrm{o}}=f\) c) \(d_{\mathrm{o}}<f\) d) None of the above Answer: c) \(d_{\mathrm{o}}<f\)

Step-by-step solution

Understanding a converging lens and magnifying glass

A converging lens is a lens that bends light rays towards a central focus point when they pass through it. The distance between the lens and the focus point is called the focal length (denoted by \(f\)). Magnifying glasses, on the other hand, creates a larger, magnified virtual image of an object held close to the lens.

Establishing the lens formula and magnification

To determine the required distance of the object from the lens, we use the lens formula: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\] where \(d_o\) is the object distance, \(d_i\) is the image distance, and \(f\) is the focal length of the lens. Moreover, the magnification \(m\) of a lens is given by the ratio of the image distance to the object distance: \[m = \frac{d_i}{d_o}\]

Analyzing the magnifying glass's requirements

To function properly as a magnifying glass, the image formed must be virtual, upright, and larger than the object. In case of a converging lens, a virtual image is formed when the object is placed within the focal length of the lens. Thus, to determine the object distance (\(d_o\)), we can evaluate the given options and find the one that will result in a virtual, magnified image.

Evaluating the given options

Let's evaluate each option: a) \(d_{\mathrm{o}}>f\) If the object distance is greater than the focal length, the image formed by a converging lens will be real and inverted. This does not meet the requirements of a magnifying glass. b) \(d_{\mathrm{o}}=f\) When the object is placed at the focal length of the lens, the rays of light will never converge to form an image. This also does not meet the requirements of a magnifying glass. c) \(d_{\mathrm{o}}<f\) If the object distance is less than the focal length, a virtual, upright, and magnified image will be formed, meeting the requirements of a magnifying glass. d) None of the above This option can be disregarded since we have found a correct option in (c). Based on the analysis, the correct answer is: c) \(d_{\mathrm{o}}<f\)

Key concepts

Lens Formula

The lens formula is a fundamental equation that relates the focal length (f) of a lens to the distances of the object (d_o) and the image (d_i). It is often represented as:

\[\frac{1}{f} = \frac{1}{d_{o}} + \frac{1}{d_{i}}\]
This formula helps us understand how light bends when passing through a lens and enables us to calculate where an image will be formed relative to the lens and the object. Converging lenses, which are thicker at the center than at the edges, bring parallel rays of light to a focus point. To use this formula properly, it's important to note that distances measured in the direction of the incoming light are considered positive, while those measured against the incoming light are negative.

In practice, once you know two of these values, you can find the third. This is incredibly useful in a variety of applications, from designing optical devices to solving textbook problems related to lenses.

Focal Length

Focal length, denoted by (f), is a key concept in the study of optics. It is defined as the distance between the center of a lens and the focus point where it converges light rays or appears to diverge them.

For converging lenses, the focal point is a real point where light rays meet. However, for diverging lenses, it's a virtual point from which light appears to spread out. The focal length determines how strongly the lens converges or diverges light, influencing the size and position of the image formed. Typically, shorter focal lengths result in greater levels of magnification because the lens is stronger and bends light rays more sharply. Understanding focal length is essential when determining how to create certain types of images with a lens, such as magnified images in magnifying glasses.

Virtual Image

A virtual image is a type of image formed by lenses and mirrors where the light rays appear to converge but do not actually do so. Unlike a real image, a virtual image cannot be projected on a screen because it doesn't come from actual light convergence. Instead, it seems to be behind the lens or mirror.

In the case of a magnifying glass (a converging lens), when an object is placed within the focal length of the lens, a virtual image is formed. This virtual image is upright and larger than the object, making it crucial for tasks that require enlarged visuals such as reading small print. A virtual image's size and position are also dictated by the lens formula, but it is important to remember that for virtual images, the value for the image distance (d_i) will be negative because it is on the same side of the lens as the object.

Magnifying Glass

A magnifying glass is a converging lens designed to produce a magnified virtual image of an object. To achieve this, the object must be placed at a distance less than the focal length of the lens. This is because, according to the lens formula, when the object distance (d_o) is less than the focal length (f), the result is a negative image distance (d_i), indicating a virtual image.

Furthermore, the magnification (m) achieved by a magnifying glass is the ratio of the image distance to the object distance:

\[m = \frac{d_{i}}{d_{o}}\]
Here, magnification will be positive and greater than one, meaning the image is upright and enlarged. The amount of magnification depends on how close the object is to the focal point; the closer it is, the larger the magnification. Magnifying glasses are quintessential tools in various fields, including biology for observing small specimens, and philately for examining stamp details.