The distance between an object and the screen is 100cm. A lens produces an image on the screen when the lens is placed at either of the positions 40cm apart. The power of the lens is nearly

Options

A. 3 diopter
B. 5 diopter
C. 2 diopter
D. 9 diopter

Explanation

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

### Step 1: Understand the setup
We have an object and a screen that are 100 cm apart. The lens can be placed in two positions, which are 40 cm apart, and both positions produce an image on the screen.

### Step 2: Define the variables
Let:
- \( D \) = distance between the object and the screen = 100 cm
- \( d_1 \) = distance from the object to the lens in the first position
- \( d_2 \) = distance from the object to the lens in the second position

Since the two positions of the lens are 40 cm apart, we can express this as:
\[ d_2 = d_1 - 40 \]

### Step 3: Write the lens formula
For a lens, the lens formula is given by:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
Where:
- \( f \) = focal length of the lens
- \( d_o \) = object distance from the lens
- \( d_i \) = image distance from the lens

In our case, since the image is formed on the screen, the image distance \( d_i \) can be expressed as:
\[ d_i = D - d_o \]

### Step 4: Set up the equations
For the first position of the lens:
\[ \frac{1}{f} = \frac{1}{d_1} + \frac{1}{100 - d_1} \]

For the second position of the lens:
\[ \frac{1}{f} = \frac{1}{d_2} + \frac{1}{100 - d_2} \]
Substituting \( d_2 = d_1 - 40 \):
\[ \frac{1}{f} = \frac{1}{d_1 - 40} + \frac{1}{100 - (d_1 - 40)} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{d_1 - 40} + \frac{1}{140 - d_1} \]

### Step 5: Set the two equations equal
Since both expressions equal \( \frac{1}{f} \), we can set them equal to each other:
\[ \frac{1}{d_1} + \frac{1}{100 - d_1} = \frac{1}{d_1 - 40} + \frac{1}{140 - d_1} \]

### Step 6: Solve for \( d_1 \)
Cross-multiply and simplify the equation to find \( d_1 \). This will involve some algebraic manipulation.

### Step 7: Calculate the focal length
Once we have \( d_1 \), we can substitute back into one of the lens equations to find \( f \).

### Step 8: Calculate the power of the lens
The power \( P \) of the lens is given by:
\[ P = \frac{1}{f} \quad \text{(in meters)} \]
Convert \( f \) from centimeters to meters if necessary.

### Step 9: Final calculation
After calculating \( f \), compute \( P \) to find the power of the lens.