A convex lens forms a real image of an object for its two different positions on a screen. If height of the image in both the cases be 8 cm and 2 cm , then height of the object is

Options

A. 16cm
B. 8cm
C. 4cm
D. 2cm

Answer

To solve the problem, we will use the concept of magnification in optics. The magnification (m) produced by a lens is given by the formula:

\[ m = \frac{h_i}{h_o} \]

where:
- \( h_i \) is the height of the image,
- \( h_o \) is the height of the object.

In this case, we have two different positions of the object, leading to two different heights of the image. Let's denote:
- For the first position:
- Height of image \( h_{i1} = 8 \, \text{cm} \)
- For the second position:
- Height of image \( h_{i2} = 2 \, \text{cm} \)

Let the height of the object be \( h_o \).

### Step 1: Write the magnification equations for both positions
For the first position:
\[ m_1 = \frac{h_{i1}}{h_o} = \frac{8}{h_o} \]

For the second position:
\[ m_2 = \frac{h_{i2}}{h_o} = \frac{2}{h_o} \]

### Step 2: Use the property of magnification
The product of the magnifications for the two positions of the object is equal to 1 (since both images are real and formed on the same screen):
\[ m_1 \times m_2 = 1 \]

Substituting the magnification equations:
\[ \left(\frac{8}{h_o}\right) \times \left(\frac{2}{h_o}\right) = 1 \]

### Step 3: Simplify the equation
This simplifies to:
\[ \frac{16}{h_o^2} = 1 \]

### Step 4: Solve for \( h_o^2 \)
Multiplying both sides by \( h_o^2 \):
\[ 16 = h_o^2 \]

### Step 5: Take the square root to find \( h_o \)
Taking the square root of both sides:
\[ h_o = \sqrt{16} = 4 \, \text{cm} \]

Thus, the height of the object is \( 4 \, \text{cm} \).

### Final Answer
The height of the object is \( 4 \, \text{cm} \).

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