A convergent beam of light is incident on a convex mirror so as to converge to a distance 12 cm from the pole of the mirror. An inverted image of the same size is formed coincident with the virtual object. What is the focal length of the mirror ?

Options

A. 24 cm
B. 12 cm
C. 6 cm
D. 3 cm

Answer

To find the focal length of the convex mirror based on the given information, we can follow these steps:

### Step 1: Understand the Problem
A convergent beam of light is incident on a convex mirror, converging at a distance of 12 cm from the pole of the mirror. The image formed is inverted and of the same size as the object, coinciding with the virtual object.

### Step 2: Identify the Object Distance (u)
Since the converging rays appear to converge at a distance of 12 cm from the pole of the mirror, we can consider this as the object distance (u). For a convex mirror, the object distance is taken as positive when measured in the direction of the incoming light.

Thus, we have:
\[ u = +12 \, \text{cm} \]

### Step 3: Understand Image Formation
The problem states that an inverted image of the same size is formed coincident with the virtual object. In the case of a convex mirror, the image formed is virtual, upright, and diminished. However, the problem states that the image is inverted and of the same size, which implies that the image distance (v) must also be equal to the object distance in magnitude but negative since it is virtual.

Thus, we have:
\[ v = -12 \, \text{cm} \]

### Step 4: Use the Mirror Formula
The mirror formula for spherical mirrors is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Where:
- \( f \) is the focal length,
- \( u \) is the object distance,
- \( v \) is the image distance.

### Step 5: Substitute the Values
Substituting the values of \( u \) and \( v \) into the mirror formula:
\[ \frac{1}{f} = \frac{1}{12} + \frac{1}{-12} \]

### Step 6: Simplify the Equation
Calculating the right-hand side:
\[ \frac{1}{f} = \frac{1}{12} - \frac{1}{12} = 0 \]
This indicates that the focal length \( f \) is infinite, which is not the case for a convex mirror.

### Step 7: Find the Radius of Curvature
Since the image is formed at the center of curvature, we can conclude that the radius of curvature \( R \) is equal to the distance at which the light converges, which is 12 cm.

### Step 8: Calculate the Focal Length
The focal length \( f \) of a mirror is related to the radius of curvature \( R \) by the formula:
\[ f = \frac{R}{2} \]
Substituting \( R = 12 \, \text{cm} \):
\[ f = \frac{12}{2} = 6 \, \text{cm} \]

### Final Answer
The focal length of the convex mirror is:
\[ \boxed{6 \, \text{cm}} \]