An object 5.0 cm in length is placed at a distance of 20 cm in front of a convex mirror of radius of curvature 30 cm. Find the position of the image, its nature and size.
Position of image: 8.57 cm behind the mirror Nature: Virtual and erect Size: 2.14 cm
Explanation
This numerical problem requires applying the mirror formula and magnification formula for a convex mirror. The key steps are: (1) finding focal length from radius of curvature, (2) applying the mirror formula with proper sign conventions, and (3) calculating magnification to find image size. For convex mirrors, focal length is positive and images are always virtual, erect, and diminished.
Solution Steps
Step 1: Given values Object-size, h = +5.0 cm Object-distance, u = -20 cm Radius of curvature, R = +30 cm
Step 2: Find focal length Focal length, f = R/2 = +30/2 = +15 cm (as the principal focus of a convex mirror is behind the mirror)
Step 3: Apply mirror formula 1/v + 1/u = 1/f 1/v = 1/f - 1/u = 1/(+15) - 1/(-20) = 1/15 + 1/20 = (20 + 15)/300 = 35/300 v = +300/35 = +8.57 cm The image is 8.57 cm at the back of the mirror.
Step 4: Calculate magnification m = h'/h = -v/u = -(+8.57)/(-20) = +0.43
Step 5: Find image size h' = m × h = + = +2.14 cm The image is virtual, erect and smaller in size by a factor of 0.43.