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 follow the information given in the question and apply the lens formula.

### Step 1: Understand the setup
The distance between the object and the screen is given as 100 cm. We denote this distance as \( d = 100 \, \text{cm} \). The lens can be placed in two different positions that are 40 cm apart.

### Step 2: Define the variables
Let:
- \( d \) = distance between the object and the screen = 100 cm
- \( d' \) = distance between the two positions of the lens = 40 cm

### Step 3: Use the lens formula
We can use the lens formula to relate the focal length \( f \) of the lens to the distances involved. The formula for the focal length when the lens is moved between two positions is given by:

\[ f = \frac{d^2 - d'^2}{4d} \]

### Step 4: Substitute the values
Now, substituting the values into the formula:

\[ f = \frac{(100)^2 - (40)^2}{4 \times 100} \]

Calculating \( (100)^2 \) and \( (40)^2 \):

\[ f = \frac{10000 - 1600}{400} \]

\[ f = \frac{8400}{400} \]

\[ f = 21 \, \text{cm} \]

### Step 5: Convert focal length to meters
To find the power of the lens, we need to convert the focal length from centimeters to meters:

\[ f = 21 \, \text{cm} = 0.21 \, \text{m} \]

### Step 6: Calculate the power of the lens
The power \( P \) of a lens is given by the formula:

\[ P = \frac{1}{f} \, (\text{in meters}) \]

Substituting the value of \( f \):

\[ P = \frac{1}{0.21} \approx 4.76 \, \text{diopters} \]

### Step 7: Round the result
Rounding this value gives us:

\[ P \approx 5 \, \text{diopters} \]

### Final Answer
The power of the lens is nearly 5 diopters.

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