The ratio of powers of a thin convex and thin concave lens is 3/2 and equivalent focal length of their combination is 30 cm. Then their focal lengths respectively are

Options

A. 75, -50
B. 75, 50
C. 10, -15
D. `-75, 50`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between power and focal length
The power \( P \) of a lens is related to its focal length \( f \) by the formula:
\[ P = \frac{1}{f} \]
where \( f \) is in meters. For a convex lens, the power is positive, and for a concave lens, the power is negative.

### Step 2: Set up the ratio of powers
Given that the ratio of the powers of the thin convex lens \( P_c \) to the thin concave lens \( P_n \) is:
\[ \frac{P_c}{P_n} = \frac{3}{2} \]
Let’s denote the power of the concave lens as \( P_n = -2p \) (negative because it is concave) and the power of the convex lens as \( P_c = 3p \) (positive).

### Step 3: Write the equation for equivalent focal length
The equivalent power \( P \) of the combination of the two lenses is given by:
\[ P = P_c + P_n = 3p - 2p = p \]
We also know that the equivalent focal length \( f \) of the combination is 30 cm, which can be converted to meters as \( 0.3 \) m. Thus, the power of the combination is:
\[ P = \frac{1}{f} = \frac{1}{0.3} = \frac{10}{3} \text{ D} \]

### Step 4: Set up the equation
From the previous steps, we have:
\[ p = \frac{10}{3} \]

### Step 5: Find the focal lengths
Now we can find the individual focal lengths of the lenses.

1. **Focal length of the convex lens \( f_c \)**:
\[ f_c = \frac{1}{P_c} = \frac{1}{3p} = \frac{1}{3 \times \frac{10}{3}} = \frac{1}{10} \text{ m} = 10 \text{ cm} \]

2. **Focal length of the concave lens \( f_n \)**:
\[ f_n = \frac{1}{P_n} = \frac{1}{-2p} = \frac{1}{-2 \times \frac{10}{3}} = \frac{1}{-\frac{20}{3}} = -\frac{3}{20} \text{ m} = -15 \text{ cm} \]

### Final Result
Thus, the focal lengths of the lenses are:
- Focal length of the convex lens \( f_c = 10 \text{ cm} \)
- Focal length of the concave lens \( f_n = -15 \text{ cm} \)

### Summary
The focal lengths of the thin convex and thin concave lenses are \( 10 \text{ cm} \) and \( -15 \text{ cm} \), respectively.

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