Certainly! Let's examine the behavior of a convex lens and the formation of images by it.
When an object is placed at twice the focal length (2F) of a convex lens, it is a specific case that follows the lens formula and principles of optics. Here's the detailed solution for understanding why a particular type of image is produced:
### Step-by-Step Solution:
1. Lens Formula and Image Formation:
- The lens formula is given by:
[tex]\[\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\][/tex]
where [tex]\(f\)[/tex] is the focal length, [tex]\(v\)[/tex] is the image distance, and [tex]\(u\)[/tex] is the object distance.
- For a convex lens, if an object is placed at twice the focal length (2F), the object distance [tex]\(u\)[/tex] is [tex]\(-2f\)[/tex] (taking the convention that distances measured against the direction of incoming light are negative).
2. Image Characteristics for an Object at 2F:
- According to the lens formula, when the object is at [tex]\(-2f\)[/tex], the image distance [tex]\(v\)[/tex] will also be [tex]\(2f\)[/tex]. This can be derived using:
[tex]\[\frac{1}{f} = \frac{1}{v} - \frac{1}{-2f} \Rightarrow \frac{1}{f} = \frac{1}{v} + \frac{1}{2f}\][/tex]
Solving this yields [tex]\(v = 2f\)[/tex].
3. Nature of the Image:
- Position: The image is formed at a distance of twice the focal length (2F) on the opposite side of the lens.
- Orientation: The image is inverted because in a convex lens, an image formed at 2F (on the other side) is always inverted.
- Size: The image is of the same size as the object. This is a unique property when the object is placed at 2F.
- Type: The image is real because it is formed on the opposite side of the lens and can be projected on a screen.
Summarizing all these properties:
- The image is real, because it is formed by actual convergence of light rays.
- The image is inverted, as typical for a convex lens when the object is placed at 2F.
- The image is of the same size as the object under this specific condition.
### Conclusion:
Given this analysis, the correct answer is:
- OC. real, inverted, and of the same size
This explains why the correct choice matches the detailed ray diagram and theoretical analysis of the convex lens behavior.
