when an object is kept in frnt of a convex lens the distance between it and the real image is 54 cm. if the magnification produced is 2, the focal length of the lens is

Options

A. 4 cm
B. 6 cm
C. 12 cm
D. 24 cm

Answer

To solve the problem step by step, we will use the lens formula and the concept of magnification.

### Step 1: Understand the given information
- The distance between the object (u) and the real image (v) is 54 cm.
- The magnification (m) produced by the lens is 2.

### Step 2: Relate magnification to object and image distances
The magnification produced by a lens is given by the formula:
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
Given that \( m = 2 \), we can write:
\[ \frac{v}{u} = 2 \]
This implies:
\[ v = 2u \]

### Step 3: Set up the equation for the distance between object and image
According to the problem, the distance between the object and the image is 54 cm:
\[ u + v = 54 \]
Substituting \( v = 2u \) into this equation gives:
\[ u + 2u = 54 \]
\[ 3u = 54 \]

### Step 4: Solve for the object distance (u)
Now, we can solve for \( u \):
\[ u = \frac{54}{3} = 18 \text{ cm} \]

### Step 5: Find the image distance (v)
Using the value of \( u \) to find \( v \):
\[ v = 2u = 2 \times 18 = 36 \text{ cm} \]

### Step 6: Use the lens formula to find the focal length (f)
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{36} + \frac{1}{-18} \]
Calculating the right-hand side:
\[ \frac{1}{f} = \frac{1}{36} - \frac{2}{36} = -\frac{1}{36} \]

### Step 7: Solve for the focal length (f)
Taking the reciprocal gives:
\[ f = -36 \text{ cm} \]
Since we are dealing with a convex lens, the focal length should be positive. Therefore, we take the absolute value:
\[ f = 12 \text{ cm} \]

### Final Answer
The focal length of the lens is \( 12 \text{ cm} \).

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