Question

A concave mirror produces an image of an object such that the distance between the object and image is 20 cm. If the magnification of the image is \( -3 \), then the magnitude of the radius of curvature of the mirror is:

• A. 7.5 cm
• B. 30 cm
• C. 15 cm
• D. 3.75 cm

Hint:

For concave mirrors, use the mirror equation \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), and the magnification to solve for unknown distances or focal lengths.

Answer 1

The Correct Option is C

The magnification \( m \) is given by: \[ m = -\frac{v}{u} \] Where \( v \) is the image distance and \( u \) is the object distance. Also, the mirror equation is: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Using the given magnification and the relation between focal length \( f \) and radius of curvature \( R \): \[ f = \frac{R}{2} \] By solving these equations, we find that the radius of curvature \( R = 15 \, \text{cm} \).

Answer 2

Step 1: Given data.
Magnification of the image, \( m = -3 \)
Distance between object and image, \( |u - v| = 20 \, \text{cm} \)
We need to find the magnitude of the radius of curvature \( R \) of the concave mirror.

Step 2: Relation between object distance, image distance, and magnification.
For mirrors, the magnification \( m \) is given by:
\[ m = -\frac{v}{u}. \] Substitute \( m = -3 \):
\[ -3 = -\frac{v}{u} \Rightarrow v = 3u. \] For a concave mirror, both \( u \) and \( v \) are taken as positive in magnitude here since we are dealing with absolute distances.

Step 3: Use the distance between object and image.
Given that the distance between object and image is \( |v - u| = 20 \):
\[ 3u - u = 20 \Rightarrow 2u = 20 \Rightarrow u = 10 \, \text{cm}. \] Then, \[ v = 3u = 30 \, \text{cm}. \]

Step 4: Apply the mirror formula.
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v}. \] Substitute the values of \( u \) and \( v \):
\[ \frac{1}{f} = \frac{1}{10} + \frac{1}{30} = \frac{3 + 1}{30} = \frac{4}{30} = \frac{2}{15}. \] Thus, \[ f = \frac{15}{2} = 7.5 \, \text{cm}. \]

Step 5: Find the radius of curvature.
For a mirror, \[ R = 2f = 2 \times 7.5 = 15 \, \text{cm}. \]

Final Answer:
\[ \boxed{R = 15 \, \text{cm}} \]