An object is placed at a distance of `1.5 m` from a screen and a convex lens is interposed between them. The magnification produced is `4`. What is the focal length of the lens ?

Options

A. `1m`
B. `0.5 m`
C. `0.24 m`
D. `2 m`

Explanation

To solve the problem step by step, we will use the given information about the object distance, image distance, and magnification to find the focal length of the convex lens.

### Step 1: Understand the given data
- The distance between the object and the screen is \(1.5 \, \text{m}\).
- The magnification \(m\) produced by the lens is \(4\).

### Step 2: Set up the equations
We know that the total distance between the object and the image (screen) is given by:
\[ |u| + |v| = 1.5 \, \text{m} \]
where \(u\) is the object distance (negative in lens formula convention) and \(v\) is the image distance (positive).

The magnification \(m\) is given by the formula:
\[ m = -\frac{v}{u} \]
Given that \(m = 4\), we can rewrite this as:
\[ 4 = -\frac{v}{u} \implies u = -\frac{v}{4} \]

### Step 3: Substitute \(u\) in the distance equation
Substituting \(u\) in the total distance equation:
\[ |u| + |v| = 1.5 \]
Substituting \(u = -\frac{v}{4}\):
\[ \left| -\frac{v}{4} \right| + |v| = 1.5 \]
This simplifies to:
\[ \frac{v}{4} + v = 1.5 \]
Combining the terms:
\[ \frac{v + 4v}{4} = 1.5 \implies \frac{5v}{4} = 1.5 \]

### Step 4: Solve for \(v\)
Multiplying both sides by \(4\):
\[ 5v = 6 \implies v = \frac{6}{5} = 1.2 \, \text{m} \]

### Step 5: Find \(u\)
Now substituting \(v\) back to find \(u\):
\[ u = -\frac{v}{4} = -\frac{1.2}{4} = -0.3 \, \text{m} \]

### Step 6: Use the lens formula to find the focal length
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting the values of \(u\) and \(v\):
\[ \frac{1}{f} = \frac{1}{1.2} - \frac{1}{-0.3} \]
Calculating each term:
\[ \frac{1}{f} = \frac{1}{1.2} + \frac{1}{0.3} \]
Finding a common denominator (which is \(1.2 \times 0.3 = 0.36\)):
\[ \frac{1}{f} = \frac{0.3}{1.2 \times 0.3} + \frac{1.2}{1.2 \times 0.3} = \frac{0.3 + 1.2}{0.36} = \frac{1.5}{0.36} \]
Calculating:
\[ f = \frac{0.36}{1.5} = 0.24 \, \text{m} \]

### Final Answer
The focal length of the lens is \(0.24 \, \text{m}\).

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