Problem 47 - College Physics

Question

An object is placed in front of a concave lens whose focal length is \(-18 \mathrm{~cm}\). Where is the image located and what are its characteristics, if the object distance is (a) \(10 \mathrm{~cm}\) and (b) \(25 \mathrm{~cm}\) ? Sketch ray diagrams for each case.

Step-by-Step Solution

Answer

For case (a), image at 22.5 cm, virtual, upright, smaller. For case (b), image at ~-64.29 cm, virtual, upright, smaller.

Step 1: Understanding the Lens Formula

To find the image location using a lens, we use the lens formula: \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \). Here, \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. We are given a concave lens with \( f = -18 \) cm.

Step 2: Calculating Image Distance for Case (a)

For case (a), the object distance \( u = -10 \) cm (negative sign because object is on the same side as the light source for a lens). Using the lens formula: \( \frac{1}{-18} = \frac{1}{v} - \frac{1}{-10} \). Simplifying gives: \( \frac{1}{v} = \frac{1}{-18} + \frac{1}{10} = \frac{-10 + 18}{180} = \frac{8}{180} = \frac{4}{90} \). Thus, \( v = \frac{90}{4} = 22.5 \) cm.

Step 3: Analyzing Image Characteristics for Case (a)

The positive image distance (\( v = 22.5 \) cm) indicates the image is virtual and formed on the same side as the object. As it is virtual, it is upright and smaller compared to the object.

Step 4: Calculating Image Distance for Case (b)

For case (b), the object distance \( u = -25 \) cm. Using the lens formula: \( \frac{1}{-18} = \frac{1}{v} - \frac{1}{-25} \). Solving gives: \( \frac{1}{v} = \frac{1}{-18} + \frac{1}{25} = \frac{-25 + 18}{450} = \frac{-7}{450} \). Thus, \( v = \frac{450}{-7} \approx -64.29 \) cm.

Step 5: Analyzing Image Characteristics for Case (b)

The negative image distance (\( v \approx -64.29 \) cm) indicates the image is virtual and on the opposite side of the lens from the object. The image is also upright and smaller than the object.

Step 6: Summarizing Image Characteristics

For both cases, the image formed is virtual, upright, and smaller than the object. In case (a), the image appears \( 22.5 \) cm from the lens on the same side as the object. In case (b), the image appears approximately \( 64.29 \) cm from the lens but on the opposite side.

Key Concepts

Lens Formula

Understanding how images are formed by lenses is crucial in optics. The lens formula is a mathematical representation that helps us calculate where an image will form and its characteristics. The formula is written as:

• \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \)

Here, each variable represents an aspect of the optical system:

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

For concave lenses, the focal length \( f \) is always negative because concave lenses diverge light rays; unlike convex lenses which converge them.
The sign conventions in optics dictate this negative sign, as it keeps calculations standard, especially when determining the nature and position of images formed.

Virtual Image

In the study of optics, an image is said to be "virtual" when it cannot be projected onto a screen. It appears to be located in a position where light does not actually come from. Virtual images arise when diverging lenses, like concave lenses, are used.
For a concave lens, the virtual image is usually:

• Upright.
• Smaller than the object.
• On the same side as the object with respect to the lens.

The characteristics of a virtual image can be found using the lens formula. In the example problem, we calculated this image distance \( v \), which was positive or negative depending on the situation. When \( v \) is positive, as in case (a), the image appears on the same side as the object. When \( v \) is negative, it implies the image location is interpreted as being on the opposite side from the object. But keep in mind, physically with respect to concave lenses, it's on the same side because it's virtual.

Optics

Optics is a fascinating branch of physics that deals with the behavior of light and how it interacts with different mediums. It is often divided into several branches, such as geometrical optics (which primarily deals with the laws of reflection and refraction) and physical optics (which considers the wave nature of light).
Concave lenses are vital components in optics due to their ability to diverge light. They are used in various applications like in eyeglasses for people with myopia (nearsightedness) to correct their vision. Understanding how light behaves when it passes through a lens helps in designing optical devices effectively.
In our problem, applying the lens formula and understanding virtual images has shown us how complex optical phenomena can be simplified using consistent mathematical rules. This underlines the universal applicability of optics' principles, not just for study, but for practical technology and everyday tools. By mastering these concepts, one can unlock numerous technological marvels involving light!