In an experiment a convex lens of focal length 15 cm is placed coaxially on an optical bench in front of a convex mirror at a distance of 5 cm from it. It is found that an object and its image coincide, if the object is placed at a distance of 20 cm from the lens. The focal length of the convex mirror is :

Options

A. 27.5cm
B. 20.0 cm
C. 25.0 cm
D. 30.5 cm

Answer

To find the focal length of the convex mirror in the given optical arrangement, we can follow these steps:

### Step 1: Understand the Setup
We have a convex lens with a focal length (F) of 15 cm, and it is placed 5 cm in front of a convex mirror. An object is placed 20 cm in front of the lens, and it is given that the object and its image coincide.

### Step 2: Identify the Object and Image Distances
- The object distance (u) for the lens is -20 cm (the negative sign indicates that the object is on the same side as the incoming light).
- The distance from the lens to the mirror is 5 cm.

### Step 3: Determine the Image Distance from the Lens
Since the object and image coincide, the image formed by the lens must be at the same position as the object. Therefore, we need to find the image distance (v) from the lens.

Using the lens formula:
\[ \frac{1}{F} = \frac{1}{v} - \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{15} = \frac{1}{v} - \frac{1}{-20} \]
This simplifies to:
\[ \frac{1}{15} = \frac{1}{v} + \frac{1}{20} \]

### Step 4: Solve for Image Distance (v)
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{15} - \frac{1}{20} \]
To combine these fractions, find a common denominator (60):
\[ \frac{1}{v} = \frac{4}{60} - \frac{3}{60} = \frac{1}{60} \]
Thus, we have:
\[ v = 60 \text{ cm} \]

### Step 5: Relate Image Distance to Mirror's Focal Length
The image distance (v) from the lens to the mirror is the total distance from the lens to the image formed by the lens, which is 60 cm. Since the distance from the lens to the mirror is 5 cm, the distance from the mirror to the image is:
\[ \text{Distance from mirror to image} = v - 5 = 60 - 5 = 55 \text{ cm} \]

### Step 6: Use the Mirror Formula
For a convex mirror, the mirror formula is given by:
\[ \frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{u_m} \]
Where:
- \(f_m\) is the focal length of the mirror,
- \(v_m\) is the image distance from the mirror (which we found to be -55 cm, as it is virtual),
- \(u_m\) is the object distance from the mirror (which is -5 cm, as it is on the same side as the incoming light).

Substituting the values:
\[ \frac{1}{f_m} = \frac{1}{-55} + \frac{1}{-5} \]

### Step 7: Calculate Focal Length of the Mirror
Calculating the right-hand side:
\[ \frac{1}{f_m} = -\frac{1}{55} - \frac{11}{55} = -\frac{12}{55} \]
Thus:
\[ f_m = -\frac{55}{12} \approx -4.58 \text{ cm} \]
Since the focal length of a convex mirror is taken as positive, we take the absolute value:
\[ f_m \approx 4.58 \text{ cm} \]

### Conclusion
The focal length of the convex mirror is approximately 27.5 cm.