In an experiment for the determination of focal length of the convex mirror a convex lens of focal length 20cm is placed on the optical bench and an object pin is placed at a distance 30cm from the lens. When a convex mirror is introduced in between the lens and the real and inverted image of the object, the final image of object O is formed at O itself. If the distance between the lens and the mirror is 10cm, then the focal length of the mirror is

Options

A. `10 cm`
B. `20 cm`
C. `24 cm`
D. `50 cm`

Answer

To determine the focal length of the convex mirror in the given experiment, we can follow these steps:

### Step 1: Understand the Setup
We have a convex lens with a focal length (f) of 20 cm and an object placed at a distance (u) of 30 cm from the lens. A convex mirror is placed 10 cm away from the lens.

### Step 2: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( f \) = focal length of the lens
- \( v \) = image distance from the lens
- \( u \) = object distance from the lens (taken as negative in lens formula)

Substituting the known values:
- \( f = 20 \) cm (positive for a convex lens)
- \( u = -30 \) cm (since the object is on the same side as the incoming light)

Rearranging the formula to find \( v \):
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
\[ \frac{1}{v} = \frac{1}{20} - \frac{1}{30} \]

### Step 3: Calculate \( v \)
Finding a common denominator (60):
\[ \frac{1}{v} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \]
Thus,
\[ v = 60 \text{ cm} \]

### Step 4: Determine Object Distance for the Mirror
The distance from the lens to the mirror is given as 10 cm. Therefore, the distance from the image formed by the lens to the mirror is:
\[ \text{Distance from lens to mirror} = 60 \text{ cm} - 10 \text{ cm} = 50 \text{ cm} \]
This distance (50 cm) is the object distance for the convex mirror, which we will denote as \( u_m \).

### Step 5: Use the Mirror Formula
The mirror formula is:
\[ \frac{1}{f_m} = \frac{1}{v_m} + \frac{1}{u_m} \]
Where:
- \( f_m \) = focal length of the mirror
- \( v_m \) = image distance from the mirror (which is negative for virtual images)
- \( u_m \) = object distance from the mirror (positive for real objects)

From the setup, the image formed by the mirror is at the same position as the object (O), which means:
\[ u_m = 50 \text{ cm} \quad (\text{positive}) \]
And since the final image is at the object position:
\[ v_m = -10 \text{ cm} \quad (\text{negative, as it is virtual}) \]

### Step 6: Substitute into the Mirror Formula
Substituting the values into the mirror formula:
\[ \frac{1}{f_m} = \frac{1}{-10} + \frac{1}{50} \]
Finding a common denominator (50):
\[ \frac{1}{f_m} = -\frac{5}{50} + \frac{1}{50} = -\frac{4}{50} = -\frac{2}{25} \]
Thus,
\[ f_m = -\frac{25}{2} = -12.5 \text{ cm} \]

### Step 7: Conclusion
The focal length of the convex mirror is approximately 12.5 cm. However, since the options provided were 10 cm, 20 cm, 24 cm, and 50 cm, we can conclude that the closest answer is 10 cm.