The real image which is exactly equal to the size of an object is to be obtained on a screen with the help of a convex glass of focal length 15 cm. For this, what must be in the distance between the object and the screen?

Options

A. 15 cm
B. 30 cm
C. 45 cm
D. 60 cm

Explanation

To solve the problem of finding the distance between the object and the screen when a real image equal to the size of the object is formed by a convex lens with a focal length of 15 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the distance between the object and the screen (which is the distance between the object and the image) when the image formed is real and of the same size as the object.

2. **Use the Lens Formula**: The lens formula for a convex lens is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) = focal length of the lens (15 cm)
- \( v \) = image distance (positive for real images)
- \( u \) = object distance (negative for real objects)

3. **Size of the Image Equals Size of the Object**: Since the size of the image is equal to the size of the object, we can use the magnification formula:
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
Here, \( h_i \) is the height of the image, and \( h_o \) is the height of the object. Since \( h_i = h_o \), we have:
\[ m = 1 \Rightarrow \frac{v}{u} = 1 \Rightarrow v = -u \]

4. **Substituting in the Lens Formula**: Since \( v = -u \), we can substitute \( v \) in the lens formula:
\[ \frac{1}{15} = \frac{1}{-u} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{15} = -\frac{1}{u} - \frac{1}{u} = -\frac{2}{u} \]

5. **Solving for \( u \)**: Rearranging gives:
\[ \frac{2}{u} = -\frac{1}{15} \Rightarrow u = -30 \text{ cm} \]
Since \( u \) is negative, the object is placed 30 cm in front of the lens.

6. **Finding the Image Distance \( v \)**: Since \( v = -u \):
\[ v = 30 \text{ cm} \]

7. **Calculating the Distance Between Object and Screen**: The distance between the object and the screen is given by:
\[ \text{Distance} = |u| + |v| = 30 \text{ cm} + 30 \text{ cm} = 60 \text{ cm} \]

### Final Answer:
The distance between the object and the screen is **60 cm**.

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