Question

The radius of curvature of a convex mirror is 40 cm and the object size is double that of image size. The image distance will be

• A. 60 cm
• B. 40 cm
• C. 30 cm
• D. 20 cm

Hint:

For convex mirrors, the image formed is always virtual, smaller, and located behind the mirror. The image distance is negative in the mirror equation.

Answer 1

The Correct Option is D

Step 1: Understanding the properties of convex mirrors.
For a convex mirror, the image formed is always virtual, erect, and diminished. The magnification \( m \) for a convex mirror is
given by the relation:
\[ m = \frac{\text{Image size}}{\text{Object size}} \] Here, the object size is twice the image size, so \( m = \frac{1}{2} \).
Step 2: Using the mirror equation.
The mirror equation is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. The focal length for a convex mirror is related to the radius of curvature \( R \) by: \[ f = \frac{R}{2} \]
Given that the radius of curvature is 40 cm, we find:
\[ f = \frac{40}{2} = 20 \, \text{cm} \]
Step 3: Finding the image distance.
The magnification \( m \) for a convex mirror is also related to the object and image distances by: \[ m = -\frac{v}{u} \] Substituting \( m = \frac{1}{2} \), we get: \[ \frac{1}{2} = -\frac{v}{u} \quad \Rightarrow \quad v = -\frac{u}{2} \] Substituting this into the mirror equation, we can solve for \( v \), yielding \( v = 20 \, \text{cm} \). Thus, the correct answer is
(D) 20 cm
.