A convex lens has the same radius of curvature (R ) for both the surfaces. For what value of the refractive index of the material of the convex lens, the numerical values of f and R are equal ?

Options

A. 1.4
B. 1.5
C. 1.55
D. 1.6

Explanation

To solve the problem, we need to find the value of the refractive index \( n \) of a convex lens, given that the numerical values of the focal length \( f \) and the radius of curvature \( R \) are equal.

### Step-by-Step Solution:

1. **Understand the Lens Maker's Formula**:
The lens maker's formula for a lens is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( f \) is the focal length, \( n \) is the refractive index, and \( R_1 \) and \( R_2 \) are the radii of curvature of the two surfaces of the lens.

2. **Identify the Radii of Curvature**:
For a convex lens with the same radius of curvature \( R \) for both surfaces, we have:
\[ R_1 = R \quad \text{and} \quad R_2 = -R \]

3. **Substitute the Values into the Formula**:
Plugging in the values of \( R_1 \) and \( R_2 \) into the lens maker's formula, we get:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) \]
Simplifying the right-hand side:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R} + \frac{1}{R} \right) = (n - 1) \left( \frac{2}{R} \right) \]
Therefore, we can rewrite the equation as:
\[ \frac{1}{f} = \frac{2(n - 1)}{R} \]

4. **Set the Condition \( f = R \)**:
We are given that the numerical values of \( f \) and \( R \) are equal, so we set \( f = R \):
\[ \frac{1}{R} = \frac{2(n - 1)}{R} \]

5. **Simplify the Equation**:
Cancel \( R \) from both sides (assuming \( R \neq 0 \)):
\[ 1 = 2(n - 1) \]

6. **Solve for \( n \)**:
Rearranging the equation gives:
\[ 2(n - 1) = 1 \implies n - 1 = \frac{1}{2} \implies n = \frac{3}{2} \]

7. **Final Answer**:
The value of the refractive index \( n \) for which the numerical values of \( f \) and \( R \) are equal is:
\[ n = 1.5 \]