Two identical thin converging lenses brought in contact so that their axes coincides are placed `12.5cm` form an object. What is the optical power of the system and each lens, if the real image formed by the system of lenses if four times as large as the object?

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the given data
- Distance of the object (U) = -12.5 cm (the negative sign indicates that the object is on the same side as the incoming light).
- Magnification (M) = 4 (the image is four times larger than the object).
- Since the image is real, the magnification is given by the formula:
\[ M = \frac{V}{U} \]
where V is the image distance.

### Step 2: Calculate the image distance (V)
Using the magnification formula:
\[ M = \frac{V}{U} \]
Substituting the values:
\[ 4 = \frac{V}{-12.5} \]

Rearranging gives:
\[ V = 4 \times (-12.5) = -50 \text{ cm} \]

### Step 3: Use the lens formula to find the focal length (F)
The lens formula is given by:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]

Substituting the values of U and V:
\[ \frac{1}{F} = \frac{1}{-50} - \frac{1}{-12.5} \]

Calculating the right side:
\[ \frac{1}{F} = -\frac{1}{50} + \frac{1}{12.5} \]

Finding a common denominator (50):
\[ \frac{1}{F} = -\frac{1}{50} + \frac{4}{50} = \frac{3}{50} \]

Thus:
\[ F = \frac{50}{3} \text{ cm} \approx 16.67 \text{ cm} \]

### Step 4: Calculate the optical power of the system
The optical power (P) is given by:
\[ P = \frac{1}{F} \]
Where F must be in meters:
\[ F = \frac{50}{3} \text{ cm} = \frac{50}{300} \text{ m} = \frac{1}{6} \text{ m} \]

Thus:
\[ P = \frac{1}{\frac{1}{6}} = 6 \text{ diopters} \]

### Step 5: Calculate the power of each lens
Since the two lenses are identical and in contact, the total power of the system is the sum of the powers of the individual lenses:
\[ P_{\text{total}} = P_1 + P_2 \]
Since \( P_1 = P_2 = P_0 \):
\[ P_{\text{total}} = 2P_0 \]
Thus:
\[ 6 = 2P_0 \]
So:
\[ P_0 = 3 \text{ diopters} \]

### Summary of Results
- The optical power of the system is \( 6 \text{ diopters} \).
- The power of each lens is \( 3 \text{ diopters} \).