If two lenses of `+5` dioptres are mounted at some distance apart, the equivalent power will always be negative if the distance is

Options

A. Greater then 40 cm
B. Equal to 40 cm
C. Equal to 10 cm
D. Less then 10 cm

Answer

To solve the problem of finding the distance \( d \) between two lenses of +5 diopters each, such that the equivalent power \( P \) is negative, we can follow these steps:

### Step 1: Understand the formula for equivalent power
The equivalent power \( P \) of two lenses separated by a distance \( d \) is given by the formula:
\[ P = P_1 + P_2 - \frac{P_1 \cdot P_2 \cdot d}{f} \]
where \( P_1 \) and \( P_2 \) are the powers of the lenses, and \( f \) is the focal length of the combined system.

### Step 2: Substitute the values of the powers
Given that both lenses have a power of +5 diopters:
\[ P_1 = 5 \, \text{D}, \quad P_2 = 5 \, \text{D} \]
Substituting these values into the formula gives:
\[ P = 5 + 5 - \frac{5 \cdot 5 \cdot d}{f} \]
This simplifies to:
\[ P = 10 - \frac{25d}{f} \]

### Step 3: Set the condition for negative equivalent power
For the equivalent power \( P \) to be negative, we need:
\[ 10 - \frac{25d}{f} < 0 \]
This can be rearranged to:
\[ 10 < \frac{25d}{f} \]
or
\[ d > \frac{10f}{25} \]
which simplifies to:
\[ d > \frac{2f}{5} \]

### Step 4: Determine the focal length
The focal length \( f \) of a lens can be calculated using the formula:
\[ f = \frac{1}{P} \]
For a lens with a power of +5 diopters:
\[ f = \frac{1}{5} \, \text{m} = 0.2 \, \text{m} \]

### Step 5: Substitute the focal length back into the inequality
Now substituting \( f \) back into the inequality:
\[ d > \frac{2 \times 0.2}{5} \]
This simplifies to:
\[ d > \frac{0.4}{5} = 0.08 \, \text{m} \]

### Step 6: Convert to centimeters
To convert meters to centimeters, we multiply by 100:
\[ d > 0.08 \times 100 = 8 \, \text{cm} \]

### Step 7: Conclusion
Thus, for the equivalent power of the two lenses to be negative, the distance \( d \) must be greater than 8 cm.