A double convex lens (`R_(1)=R_(2)=10 cm`) having focal length equal to the focal length of a concave mirror. The radius of curvature of the concave mirror

Options

A. 10 cm
B. 20 cm
C. 40 cm
D. 15 cm

Explanation

To find the radius of curvature of the concave mirror given that the focal length of a double convex lens is equal to that of the concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- The radius of curvature of the double convex lens:
- \( R_1 = 10 \, \text{cm} \) (for the first surface)
- \( R_2 = -10 \, \text{cm} \) (for the second surface, negative because it is a convex surface)
- The refractive index of the lens material (glass):
- \( \mu = 1.5 \)

2. **Use the Lens Maker's Formula:**
The Lens Maker's Formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the values:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \frac{1}{-10} \right) \]
Simplifying this:
\[ \frac{1}{f} = 0.5 \left( \frac{1}{10} + \frac{1}{10} \right) = 0.5 \left( \frac{2}{10} \right) = 0.5 \times 0.2 = 0.1 \]

3. **Calculate the Focal Length of the Lens:**
Now, taking the reciprocal to find \( f \):
\[ f = \frac{1}{0.1} = 10 \, \text{cm} \]

4. **Relate the Focal Length of the Lens to the Mirror:**
Since the focal length of the double convex lens is equal to the focal length of the concave mirror, we have:
\[ f_{\text{mirror}} = 10 \, \text{cm} \]

5. **Use the Mirror Formula to Find Radius of Curvature:**
The focal length \( f \) of a concave mirror is related to its radius of curvature \( R \) by the formula:
\[ f = -\frac{R}{2} \]
Rearranging gives:
\[ R = -2f \]
Substituting the value of \( f \):
\[ R = -2 \times 10 \, \text{cm} = -20 \, \text{cm} \]

6. **Conclusion:**
The radius of curvature of the concave mirror is:
\[ R = 20 \, \text{cm} \quad (\text{taking magnitude, as radius is positive}) \]

### Final Answer:
The radius of curvature of the concave mirror is \( 20 \, \text{cm} \).