For a convex mirror, the focal length (f) is half the radius of curvature and positive. So, f = 15 cm. Using the mirror formula 1/f = 1/v + 1/u with u = -20 cm (object distance is negative), we find v ≈ 10 cm. The image is virtual (as v is positive), erect, and diminished. Using the magnification formula, the image size is about 2.5 cm (half the object size).

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## Introduction to Image Formation by Convex Mirrors

*Understanding Convex Mirror Optics*: Convex mirrors are diverging mirrors, meaning they spread out light rays that reflect off them. This property significantly influences the nature of the images they form, typically resulting in virtual, erect, and diminished images.

### Given Scenario

*Object Placement and Mirror Specifications*: In this scenario, an object measuring 5.0 cm in length is placed 20 cm in front of a convex mirror. The mirror has a radius of curvature of 30 cm. The task is to determine the position, nature, and size of the image formed.

#### Calculating the Focal Length

*Determining Mirror’s Focal Length*: The focal length (f) of a convex mirror is positive and half its radius of curvature. Therefore, for a mirror with a 30 cm radius of curvature, the focal length is 15 cm.

#### Finding the Image Position

*Applying the Mirror Formula*: The mirror formula 1/f = 1/v + 1/u is used to find the image distance (v). Here, f is 15 cm and u is -20 cm (object distance is negative for mirrors). Solving this gives v ≈ 10 cm.

##### Nature of the Image

*Characteristics of the Formed Image*: The positive value of v indicates that the image is virtual and formed behind the mirror. Additionally, images formed by convex mirrors are always erect and diminished compared to the object.

###### Determining Image Size

*Calculating the Image’s Size*: Using the magnification formula m = v/u, where m is the magnification, the image size can be calculated. Given v ≈ 10 cm and u = −20 cm, the magnification is 0.5. Therefore, the image size is half the object size, approximately 2.5 cm.

*Summary of Image Properties*: Thus, for an object placed 20 cm in front of a convex mirror with a 30 cm radius of curvature, the image is formed approximately 10 cm behind the mirror. The image is virtual, erect, and half the size of the object, measuring about 2.5 cm in length. This example illustrates the predictable nature of image formation in convex mirrors.

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