Radius of curvature of a equi convex lens is R. Find focal length (`n = 3/2`)

Options

A. `R/2`
B. `-R/2`
C. R
D. -R

Answer

To find the focal length of an equi-convex lens with a radius of curvature \( R \) and a refractive index \( n = \frac{3}{2} \), we will use the lensmaker's formula:

### Step-by-Step Solution:

1. **Understanding the Lensmaker's Formula**:
The lensmaker's formula is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
where:
- \( f \) is the focal length,
- \( n \) is the refractive index of the lens material,
- \( r_1 \) is the radius of curvature of the first surface,
- \( r_2 \) is the radius of curvature of the second surface.

2. **Identifying the Radii of Curvature**:
For an equi-convex lens:
- The radius of curvature for the first surface \( r_1 = R \) (positive, as it is convex).
- The radius of curvature for the second surface \( r_2 = -R \) (negative, as it is also convex but the surface is oriented in the opposite direction).

3. **Substituting Values into the Formula**:
Substitute \( n = \frac{3}{2} \), \( r_1 = R \), and \( r_2 = -R \) into the lensmaker's formula:
\[ \frac{1}{f} = \left( \frac{3}{2} - 1 \right) \left( \frac{1}{R} - \left(-\frac{1}{R}\right) \right) \]

4. **Simplifying the Equation**:
- Calculate \( n - 1 \):
\[ \frac{3}{2} - 1 = \frac{1}{2} \]
- Calculate \( \frac{1}{R} - \left(-\frac{1}{R}\right) \):
\[ \frac{1}{R} + \frac{1}{R} = \frac{2}{R} \]
- Substitute these results back into the equation:
\[ \frac{1}{f} = \frac{1}{2} \cdot \frac{2}{R} \]

5. **Final Calculation**:
\[ \frac{1}{f} = \frac{1}{R} \]
Therefore, taking the reciprocal gives:
\[ f = R \]

### Conclusion:
The focal length \( f \) of the equi-convex lens is:
\[ f = R \]