A double convex lens of focal length 6 cm is made of glass of refractive index 1.5. The radius of curvature of one surface is double that of the other surface. The value of smaller radius of curvature is

Options

A. 6 cm
B. 4.5 cm
C. 9 cm
D. 4 cm

Explanation

To find the smaller radius of curvature of a double convex lens given its focal length and refractive index, we can follow these steps:

### Step 1: Understand the problem
We have a double convex lens with:
- Focal length (F) = 6 cm
- Refractive index (n) = 1.5
- The radius of curvature of one surface (R1) is smaller than the other surface (R2), and R2 = 2R1.

### Step 2: Use the lens maker's formula
The lens maker's formula for a thin lens is given by:
\[ \frac{1}{F} = (n - 1) \left( \frac{1}{R1} - \frac{1}{R2} \right) \]

### Step 3: Substitute the values
Let R1 = R (the smaller radius of curvature). Then R2 = 2R. Substitute these into the lens maker's formula:
\[ \frac{1}{6} = (1.5 - 1) \left( \frac{1}{R} - \frac{1}{2R} \right) \]
This simplifies to:
\[ \frac{1}{6} = 0.5 \left( \frac{1}{R} - \frac{1}{2R} \right) \]

### Step 4: Simplify the equation
Now simplify the expression inside the parentheses:
\[ \frac{1}{R} - \frac{1}{2R} = \frac{2}{2R} - \frac{1}{2R} = \frac{1}{2R} \]
Substituting this back into the equation gives:
\[ \frac{1}{6} = 0.5 \cdot \frac{1}{2R} \]
\[ \frac{1}{6} = \frac{0.5}{2R} = \frac{1}{4R} \]

### Step 5: Cross multiply to solve for R
Cross-multiplying gives:
\[ 1 \cdot 4R = 6 \cdot 1 \]
\[ 4R = 6 \]
\[ R = \frac{6}{4} = 1.5 \text{ cm} \]

### Step 6: Find the smaller radius of curvature
Since R1 = R, the smaller radius of curvature is:
\[ R1 = 1.5 \text{ cm} \]

### Conclusion
The value of the smaller radius of curvature is **1.5 cm**.

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