A concave lens of glass, refractive index 1.5, has both surfaces of same radius of curvature R. On immersion in a medium of refractive index 1.75, it will behave as a

Options

A. convergent lens of focal length 3.5R
B. convergent lens of focal length 3.0R
C. divergent lens of focal length 3.5R
D. divergent lens of focal length 3.0R

Explanation

To determine how a concave lens behaves when immersed in a medium with a higher refractive index, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Refractive index of the glass lens, \( n_g = 1.5 \)
- Refractive index of the medium, \( n_m = 1.75 \)
- Both surfaces of the lens have the same radius of curvature, \( R \).

2. **Understand the Behavior of Lenses in Different Media**:
- A concave lens (divergent lens) has a negative focal length when in air (or a medium with a lower refractive index).
- When a lens is immersed in a medium with a higher refractive index than its own, its behavior reverses. Thus, a concave lens will behave like a convex lens (convergent).

3. **Use the Lens Maker's Formula**:
The lens maker's formula in the context of a lens in a medium is given by:
\[ \frac{1}{F} = \left( \frac{n_g}{n_m} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a concave lens:
- \( R_1 = -R \) (the first surface)
- \( R_2 = +R \) (the second surface)

4. **Substituting Values into the Formula**:
\[ \frac{1}{F} = \left( \frac{1.5}{1.75} - 1 \right) \left( \frac{1}{-R} - \frac{1}{R} \right) \]
Simplifying the right side:
- The term \( \frac{1}{-R} - \frac{1}{R} = -\frac{2}{R} \)
- Thus, we have:
\[ \frac{1}{F} = \left( \frac{1.5 - 1.75}{1.75} \right) \left( -\frac{2}{R} \right) \]
\[ \frac{1}{F} = \left( -\frac{0.25}{1.75} \right) \left( -\frac{2}{R} \right) \]

5. **Calculating the Focal Length**:
\[ \frac{1}{F} = \frac{0.25 \times 2}{1.75 R} = \frac{0.5}{1.75 R} \]
Therefore, the focal length \( F \) is:
\[ F = \frac{1.75 R}{0.5} = 3.5 R \]

6. **Conclusion**:
Since the focal length \( F \) is positive, this indicates that the lens behaves like a convex lens, which is convergent in nature.

### Final Answer:
The concave lens will behave as a convergent lens when immersed in a medium of refractive index 1.75.