A hollow double concave lens is made of very thin transparent material. It can be filled with air or either of two liquids `L_(1)` or `L_(2)` having refractive indices `n_(1)` and `n_(2)`, respectively `(n_(2) gt n_(1) gt 1)`. The lens will diverge parallel beam of light if it is filles with

Options

A. air and placed in air
B. air and immersed in `L_(1)`
C. `L_(1)` and immersed in `L_(2)`
D. `L_(2)` and immersed in `L_(1)`

Explanation

To determine which liquid will allow a hollow double concave lens to diverge a parallel beam of light, we need to analyze the refractive indices of the liquids and their relationship with the lens.

### Step-by-Step Solution:

1. **Understanding the Lens**: A hollow double concave lens is designed to diverge light rays. This lens has two inward-curving surfaces.

2. **Refractive Indices**: We have two liquids, L1 and L2, with refractive indices \( n_1 \) and \( n_2 \) respectively, where \( n_2 > n_1 > 1 \). This means that liquid L2 is denser than liquid L1.

3. **Filling the Lens**: The lens can be filled with either air, liquid L1, or liquid L2. The behavior of light as it passes through the lens will depend on the refractive index of the medium inside the lens compared to the medium outside.

4. **Condition for Divergence**: For a double concave lens to diverge a parallel beam of light, the refractive index of the medium inside the lens must be less than that of the medium outside. This is because light will bend away from the normal when it moves from a denser medium to a less dense medium.

5. **Case Analysis**:
- **If filled with air** (refractive index ≈ 1): The lens will diverge light since air (1) is less dense than the lens material (which is typically greater than 1).
- **If filled with L1** (where \( n_1 < n_2 \)): If L1 is inside and the outside medium is L2 (which is denser), the light will diverge as it goes from the denser medium (L2) to the less dense medium (L1).
- **If filled with L2**: If L2 is inside and the outside medium is L1, the light will converge as it goes from the less dense medium (L1) to the denser medium (L2).

6. **Conclusion**: The lens will diverge a parallel beam of light if it is filled with liquid L1 (less dense) and surrounded by liquid L2 (denser). Therefore, the answer is that the lens will diverge light if it is filled with liquid L1 and immersed in liquid L2.

### Final Answer:
The lens will diverge a parallel beam of light if it is filled with liquid L1 and immersed in liquid L2.