Illustration of image formation by a convex lens for various object positions.

The image you provided outlines the behavior of a convex lens based on the object distance relative to the focal length. This is a fundamental concept in optics, particularly when dealing with lenses and their imaging properties. Below, I will explain each case in detail and provide a solution or interpretation for the problem.

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Key Concepts

1. Convex Lens: A lens that converges light rays.
2. Focal Length (f): The distance from the lens to the point where parallel light rays converge (focus).
3. Object Distance (u): The distance between the object and the lens.
4. Image Distance (v): The distance between the lens and the image formed.
5. Lens Formula:
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\]
6. Magnification (m):
\[
m = \frac{\text{Height of Image}}{\text{Height of Object}} = -\frac{v}{u}
\]

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Cases Explained

#### Case 1: Convex lens - Object distance is greater than two focal lengths (\( u > 2f \))
- Description: The object is placed beyond \( 2f \) (twice the focal length).
- Image Characteristics:
- The image is real, inverted, and diminished (smaller than the object).
- The image forms between \( f \) and \( 2f \) on the other side of the lens.
- Example Calculation:
If \( u = 3f \):
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{3f} \implies \frac{1}{v} = \frac{1}{f} - \frac{1}{3f} = \frac{2}{3f} \implies v = \frac{3f}{2}
\]
The image is real, inverted, and smaller.

#### Case 2: Object distance is equal to two focal lengths (\( u = 2f \))
- Description: The object is placed exactly at \( 2f \).
- Image Characteristics:
- The image is real, inverted, and same size as the object.
- The image forms at \( 2f \) on the other side of the lens.
- Example Calculation:
If \( u = 2f \):
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{2f} \implies \frac{1}{v} = \frac{1}{f} - \frac{1}{2f} = \frac{1}{2f} \implies v = 2f
\]
The image is real, inverted, and the same size as the object.

#### Case 3: Object distance is between one and two focal lengths (\( f < u < 2f \))
- Description: The object is placed between \( f \) and \( 2f \).
- Image Characteristics:
- The image is real, inverted, and enlarged (larger than the object).
- The image forms beyond \( 2f \) on the other side of the lens.
- Example Calculation:
If \( u = 1.5f \):
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{1.5f} \implies \frac{1}{v} = \frac{1}{f} - \frac{1}{1.5f} = \frac{0.5}{1.5f} = \frac{1}{3f} \implies v = 3f
\]
The image is real, inverted, and enlarged.

#### Case 4: Object distance is equal to one focal length (\( u = f \))
- Description: The object is placed exactly at \( f \).
- Image Characteristics:
- No image is formed because the rays become parallel after passing through the lens.
- This is a special case where the lens fails to form a real image.
- Explanation:
If \( u = f \):
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{f} \implies \frac{1}{v} = 0 \implies v \to \infty
\]
The rays diverge and do not converge to form an image.

#### Case 5: Object distance is less than one focal length (\( u < f \))
- Description: The object is placed closer than \( f \).
- Image Characteristics:
- The image is virtual, erect, and enlarged.
- The image forms on the same side as the object, behind the lens.
- Example Calculation:
If \( u = 0.5f \):
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{0.5f} \implies \frac{1}{v} = \frac{1}{f} - \frac{2}{f} = -\frac{1}{f} \implies v = -f
\]
The negative sign indicates a virtual image. The image is erect, virtual, and enlarged.

#### Case 6: Concave lens - The object can be located anywhere
- Description: For a concave lens, the object can be placed at any distance.
- Image Characteristics:
- The image is always virtual, erect, and diminished.
- The image forms on the same side as the object, between the lens and the object.
- Explanation:
Concave lenses always produce virtual images because they diverge light rays.

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Summary of Key Points

1. Real Images: Formed when \( u > f \). These images can be projected onto a screen.
2. Virtual Images: Formed when \( u < f \). These images cannot be projected onto a screen.
3. Inverted Images: Formed when \( u > f \).
4. Erect Images: Formed when \( u < f \) (convex lens) or always (concave lens).

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Final Answer

The problem involves understanding how the position of the object relative to the focal length affects the nature, size, and location of the image formed by a convex lens. Each case provides specific conditions and resulting image characteristics. The key takeaway is to use the lens formula and magnification formula to analyze and predict the behavior of the lens under different scenarios.

\[
\boxed{\text{The solution involves analyzing the object distance relative to the focal length and applying the lens formula to determine the image properties.}}
\]