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Sagot :
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.
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.
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