In the displacement method , a convex lens is placed in between an object and a screen . If one of them magnification is 3 and the displacement of the lens between the two positions is 24 cm , then the focal length of the lens is :-

Options

A. 10 cm
B. 9 cm
C. 6 cm
D. 16/3cm

Explanation

To solve the problem, we will use the displacement method for a convex lens and apply the relevant formulas.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a convex lens placed between an object and a screen. The magnification (M) is given as 3, and the displacement (d) of the lens between two positions is 24 cm. We need to find the focal length (F) of the lens.

2. **Using the Magnification Formula**:
The magnification (M) is defined as:
\[ M = \frac{v}{u} \]
Where \(v\) is the image distance and \(u\) is the object distance. Given that \(M = 3\), we can express this as:
\[ v = 3u \]

3. **Relating Object Distances**:
When the lens is displaced, the new object distance becomes \(u_2\) and the old object distance is \(u_1\). The relationship can be established as:
\[ u_2 = 3u_1 \]
The displacement of the lens is given as:
\[ d = u_2 - u_1 \]
Substituting \(u_2\):
\[ d = 3u_1 - u_1 = 2u_1 \]
Given that \(d = 24\) cm, we can set up the equation:
\[ 2u_1 = 24 \implies u_1 = 12 \text{ cm} \]

4. **Finding the Value of \(u_2\)**:
Now substituting back to find \(u_2\):
\[ u_2 = 3u_1 = 3 \times 12 = 36 \text{ cm} \]

5. **Using the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
We can use the average of the two object distances and their corresponding image distances. The average distance \(d\) is:
\[ d = u_2 + u_1 = 36 + 12 = 48 \text{ cm} \]
Now substituting \(u_1\) and \(u_2\) into the lens formula:
\[ \frac{1}{f} = \frac{1}{v_1} - \frac{1}{u_1} \]
Where \(v_1 = 3u_1 = 36\) cm.

6. **Calculating the Focal Length**:
Substitute the values into the lens formula:
\[ \frac{1}{f} = \frac{1}{36} - \frac{1}{12} \]
Finding a common denominator (36):
\[ \frac{1}{f} = \frac{1}{36} - \frac{3}{36} = -\frac{2}{36} = -\frac{1}{18} \]
Thus, the focal length \(f\) is:
\[ f = 18 \text{ cm} \]

### Final Answer:
The focal length of the lens is **9 cm**.