A biconvex lens has focal length `(2)/(3)` times the radius of curvature of either surface. Calculate refraction index f material of the lens.

Options

A. `1.75`
B. `1.33`
C. `1.5`
D. `1.0`

Explanation

To solve the problem of finding the refractive index of a biconvex lens with a focal length that is \( \frac{2}{3} \) times the radius of curvature, we will use the lensmaker's formula. Here’s a step-by-step solution:

### Step 1: Understand the parameters
Given:
- Focal length \( f = \frac{2}{3} r \)
- For a biconvex lens, the radius of curvature for the first surface \( r_1 = +r \) and for the second surface \( r_2 = -r \).

### Step 2: Write down the lensmaker's formula
The lensmaker's formula is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

### Step 3: Substitute the values into the formula
Substituting the values of \( f \), \( r_1 \), and \( r_2 \):
\[ \frac{1}{f} = \frac{1}{\frac{2}{3} r} = \frac{3}{2r} \]
And substituting \( r_1 \) and \( r_2 \):
\[ \frac{1}{r_1} - \frac{1}{r_2} = \frac{1}{r} - \left(-\frac{1}{r}\right) = \frac{1}{r} + \frac{1}{r} = \frac{2}{r} \]

### Step 4: Set up the equation
Now, substituting these into the lensmaker's formula:
\[ \frac{3}{2r} = (n - 1) \left( \frac{2}{r} \right) \]

### Step 5: Simplify the equation
Cancelling \( r \) from both sides:
\[ \frac{3}{2} = 2(n - 1) \]

### Step 6: Solve for \( n \)
Now, divide both sides by 2:
\[ \frac{3}{4} = n - 1 \]
Adding 1 to both sides gives:
\[ n = 1 + \frac{3}{4} = \frac{7}{4} = 1.75 \]

### Conclusion
The refractive index \( n \) of the material of the lens is \( 1.75 \).