Question

Use the mirror formula to deduce that a convex mirror always produces a virtual image of an object kept in front of it.

Hint:

In convex mirrors, the image is always virtual, upright, and diminished. The image is formed behind the mirror, and the object distance is negative. The sign conventions are crucial when applying the mirror formula.

Answer 1

The mirror formula relates the focal length \( f \), object distance \( u \), and image distance \( v \) of a mirror: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] For a convex mirror, the focal length \( f \) is positive, and the image formed is always virtual. 1. Object Distance: In the case of a convex mirror, the object is placed in front of the mirror, so the object distance \( u \) is negative according to the sign convention. 2. Image Distance: For a convex mirror, the image distance \( v \) is always positive and virtual. The image is formed behind the mirror. 3. Derivation: For a convex mirror, the image formed is always virtual, upright, and diminished. Since \( v \) is always positive (virtual image) and \( u \) is negative (real object), the image distance always satisfies the mirror formula for a virtual image. Thus, a convex mirror always produces a virtual image, irrespective of the object distance.