A convex lens forms a real image 16 mm long on a screen. If the lens is shifted to a new position without disturbing the object or the screen then again a real image of length 81 mm is formed. The length of the object must be

Options

A. 48.5 mm
B. 36 mm
C. 6 mm
D. 72 mm

Answer

To solve the problem step by step, we will use the concept of magnification in optics.

### Step 1: Understand the problem
We have a convex lens that forms two real images of different lengths on a screen when the lens is shifted. The lengths of the images are given as:
- Length of the first image (l1) = 16 mm
- Length of the second image (l2) = 81 mm

We need to find the length of the object (L).

### Step 2: Write the magnification formulas
The magnification (m) produced by a lens is given by the formula:
\[ m = \frac{\text{Height of Image}}{\text{Height of Object}} \]

For the first image:
\[ m_1 = \frac{l_1}{L} = \frac{16}{L} \]

For the second image:
\[ m_2 = \frac{l_2}{L} = \frac{81}{L} \]

### Step 3: Use the displacement law
According to the displacement law of lenses, when the lens is shifted, the product of the magnifications of the two images formed remains equal to 1:
\[ m_1 \times m_2 = 1 \]

Substituting the expressions for \( m_1 \) and \( m_2 \):
\[ \left(\frac{16}{L}\right) \times \left(\frac{81}{L}\right) = 1 \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{16 \times 81}{L^2} = 1 \]

### Step 5: Solve for L
Now, we can solve for \( L^2 \):
\[ L^2 = 16 \times 81 \]

Calculating \( 16 \times 81 \):
\[ 16 \times 81 = 1296 \]

Thus,
\[ L^2 = 1296 \]

Taking the square root of both sides:
\[ L = \sqrt{1296} = 36 \, \text{mm} \]

### Final Answer
The length of the object is \( L = 36 \, \text{mm} \).

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