An object is placed at a distance of 15cm from a convex lenx of focal length 10cm. On the other side of the lens, a convex mirror is placed at its focus such that the image formed by the combination coincides with the object itself. The focal length of the convex mirror is

Options

A. 20 cm
B. 10 cm
C. 15 cm
D. 30 cm

Answer

To solve the problem step by step, we will use the lens formula and the properties of mirrors.

### Step 1: Identify the given values
- Object distance from the convex lens (U) = -15 cm (negative because it is on the same side as the incoming light)
- Focal length of the convex lens (F) = +10 cm (positive for a convex lens)

### Step 2: Use the lens formula to find the image distance (V)
The lens formula is given by:
\[ \frac{1}{V} - \frac{1}{U} = \frac{1}{F} \]
Rearranging the formula gives:
\[ \frac{1}{V} = \frac{1}{F} + \frac{1}{U} \]
Substituting the known values:
\[ \frac{1}{V} = \frac{1}{10} + \frac{1}{-15} \]
Finding a common denominator (30):
\[ \frac{1}{V} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \]
Thus,
\[ V = 30 \text{ cm} \]

### Step 3: Determine the position of the image formed by the lens
The image formed by the lens is at a distance of 30 cm on the opposite side of the lens.

### Step 4: Analyze the convex mirror
The convex mirror is placed at the focus of the lens, which is 10 cm from the lens. Since the image formed by the lens is at 30 cm, the distance from the lens to the mirror is:
\[ 30 \text{ cm} - 10 \text{ cm} = 20 \text{ cm} \]
This means that the image formed by the lens is 20 cm away from the convex mirror.

### Step 5: Use the mirror formula to find the focal length of the convex mirror
For a convex mirror, the focal length (f) is related to the radius of curvature (R) by:
\[ f = \frac{R}{2} \]
Since the image coincides with the object, the distance from the convex mirror to the image is equal to the radius of curvature. Therefore, the radius of curvature (R) is 20 cm.

### Step 6: Calculate the focal length of the convex mirror
Substituting R into the focal length formula:
\[ f = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \]

### Conclusion
The focal length of the convex mirror is **10 cm**.