When a convex lens forms a real, inverted image equal in size to the object, the object is placed at 2F (twice the focal length). Since the image distance (v) is 50 cm, the object distance (u) is also 50 cm. The lens formula 1/f = 1/v + 1/u gives 1/f = 1/50 + 1/50, so f = 25 cm. The power (P) is P = 1/f in meters, so P = 1/0.25 = 4 dioptres. The needle is 50 cm from the lens, and the lens power is 4 dioptres.

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## Introduction to Image Formation by Lenses

*Optical Principles of Lenses*: In optics, lenses are used to form images of objects by refracting light. The nature of the image formed – whether real or virtual, inverted or erect – depends on the object’s distance from the lens and the lens’s focal length.

### Convex Lens Characteristics

*Converging Properties*: A convex lens is a converging lens, meaning it brings light rays together. It can form real, inverted images of objects placed at various distances from the lens.

#### Given Scenario

*Equal Size Image Formation*: In the given scenario, a convex lens forms a real and inverted image of a needle, which is equal in size to the needle itself. This specific condition occurs when the object is placed at twice the focal length (2F) of the lens.

##### Determining Object Distance

*Application of Lens Formula*: Using the lens formula 1/f = 1/v + 1/u, where f is the focal length, v is the image distance, and u is the object distance, we can determine the position of the needle. Given that the image distance v is 50 cm, and knowing that u=v in this case, we find that the object is also 50 cm from the lens.

###### Calculating the Lens Power

*Finding Focal Length and Power*: The focal length f can be calculated using the same lens formula. Substituting v = 50 cm and u = 50 cm, we find f = 25 cm. The power P of a lens is given by P = 1/ f in meters, which in this case is 1/0.25 = = 4 dioptres.

*Summary of Findings*: Therefore, in this scenario, the needle is placed 50 cm in front of the convex lens, and the power of the lens is 4 dioptres. This setup allows for the formation of a real, inverted image of the needle that is equal in size to the needle itself.

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