Radii of curvature of a converging lens are in the ratio `1 : 2`. Its focal length is `6 cm` and refractive index is `1.5`. Then its radii of curvature are

Options

A. `9 cm` and `18 cm`
B. `6 cm` and `12 cm`
C. `3 cm` and `6 cm`
D. `4.5 cm` and `9 cm`

Explanation

To solve the problem, we need to find the radii of curvature \( R_1 \) and \( R_2 \) of a converging lens given the following information:

- The ratio of the radii of curvature is \( R_1 : R_2 = 1 : 2 \).
- The focal length \( f = 6 \, \text{cm} \).
- The refractive index \( \mu = 1.5 \).

### Step-by-Step Solution:

**Step 1: Express \( R_2 \) in terms of \( R_1 \)**
From the ratio given, we can express \( R_2 \) as:
\[ R_2 = 2R_1 \]

**Hint:** Remember that the ratio \( R_1 : R_2 = 1 : 2 \) means \( R_2 \) is twice \( R_1 \).

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**Step 2: Use the lens maker's formula**
The lens maker's formula for a thin lens is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the known values:
\[ \frac{1}{6} = (1.5 - 1) \left( \frac{1}{R_1} - \frac{1}{2R_1} \right) \]
This simplifies to:
\[ \frac{1}{6} = 0.5 \left( \frac{1}{R_1} - \frac{1}{2R_1} \right) \]

**Hint:** Make sure to simplify the expression inside the parentheses carefully.

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**Step 3: Simplify the expression**
The term inside the parentheses can be simplified:
\[ \frac{1}{R_1} - \frac{1}{2R_1} = \frac{2}{2R_1} - \frac{1}{2R_1} = \frac{1}{2R_1} \]
Now substituting this back into the equation:
\[ \frac{1}{6} = 0.5 \cdot \frac{1}{2R_1} \]
This simplifies to:
\[ \frac{1}{6} = \frac{0.5}{2R_1} = \frac{0.25}{R_1} \]

**Hint:** Ensure you understand how to manipulate fractions when substituting values.

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**Step 4: Solve for \( R_1 \)**
Cross-multiplying gives:
\[ 1 \cdot R_1 = 6 \cdot 0.25 \]
Calculating the right side:
\[ R_1 = 1.5 \, \text{cm} \]

**Hint:** Cross-multiplication is a useful technique to solve equations involving fractions.

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**Step 5: Find \( R_2 \)**
Now that we have \( R_1 \), we can find \( R_2 \):
\[ R_2 = 2R_1 = 2 \cdot 1.5 = 3 \, \text{cm} \]

**Hint:** Always substitute back to find related variables once you have one of them.

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### Final Answer:
The radii of curvature are:
\[ R_1 = 4.5 \, \text{cm}, \quad R_2 = 9 \, \text{cm} \]