If both object distance 'u' and image distance 'v' from a concave mirror is -20 cm and both object size and image size is 4 cm, then what is the radius of curvature 'R' of the mirror?

Correct option is D

We are given:

• Object distance (u) = −20 cm
• Image distance (v) = −20 cm
• Object size = Image size = 4 cm

The object size being equal to the image size indicates that the magnification (M) is equal to 1.

Step 1: Magnification Formula

$$M= -\frac{v}{u}$$​

Substitute v=−20cm  and u=−20 cm:

​$$M= -\frac{-20}{-20} = 1$$​​

This confirms the magnification is 1, consistent with the image size being equal to the object size.

Step 2: Mirror Formula

The mirror formula is:

​$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$​​

Substitute v=−20 cm and u=−20 cm:

​$$\frac{1}{f} = \frac{1}{-20} - \frac{1}{-20} \\ \ \\\frac{1}{f} = \frac{-1}{20} + \frac{1}{20} = 0$$​​

This means the focal length (f) is at infinity, which is not possible for a concave mirror.

Step 3: Radius of Curvature

For a concave mirror, the radius of curvature (R) and focal length (f) are related as:

R=2f

In this case, u=v implies the object and image are at the center of curvature. Thus, the radius of curvature (R) is equal to the distance of the object or image from the mirror:

R=−20 cm

Key Facts:

•    The radius of curvature is twice the focal length for any spherical mirror.
• When the object and image distances are equal, they lie at the center of curvature.
• A concave mirror forms a real, inverted image at the center of curvature when the object is placed there.

·         For concave mirrors, distances are negative as per the sign convention.