There is one bi-concave lens of material having refractive index of 1.3. This lens is now dipped in a transparent liquid of refractive index 1.4. Lens will now behave as :

Options

A. convergent lens
B. divergent lens
C. prism
D. any of the above depending upon the radius of curvature of both the surfaces of lens

Explanation

To determine how the bi-concave lens behaves when dipped in a transparent liquid, we can follow these steps:

### Step 1: Understand the Lens and Medium
We have a bi-concave lens with a refractive index (n_lens) of 1.3, and it is placed in a liquid with a refractive index (n_liquid) of 1.4.

### Step 2: Identify the Relative Refractive Index
The relative refractive index (n_relative) can be calculated using the formula:
\[ n_{\text{relative}} = \frac{n_{\text{lens}}}{n_{\text{liquid}}} \]

### Step 3: Calculate the Relative Refractive Index
Substituting the values:
\[ n_{\text{relative}} = \frac{1.3}{1.4} \]

### Step 4: Determine the Sign of the Relative Refractive Index
Since \( n_{\text{relative}} < 1 \), this indicates that the lens will behave differently when placed in the liquid.

### Step 5: Use the Lens Maker's Formula
The lens maker's formula is given by:
\[ \frac{1}{f} = n_{\text{relative}} - 1 \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a bi-concave lens, both \( R_1 \) and \( R_2 \) are negative, leading to:
\[ \frac{1}{f} = (n_{\text{relative}} - 1) \cdot \left( -\frac{2}{R} \right) \]

### Step 6: Substitute the Values
Substituting the relative refractive index:
\[ \frac{1}{f} = \left( \frac{1.3}{1.4} - 1 \right) \cdot \left( -\frac{2}{R} \right) \]
Calculating \( n_{\text{relative}} - 1 \):
\[ n_{\text{relative}} - 1 = \frac{1.3}{1.4} - 1 = \frac{1.3 - 1.4}{1.4} = \frac{-0.1}{1.4} \]

### Step 7: Analyze the Result
Since \( n_{\text{relative}} - 1 \) is negative, the overall expression for \( \frac{1}{f} \) becomes positive when multiplied by \( -\frac{2}{R} \).

### Step 8: Conclusion
A positive focal length indicates that the lens will behave as a convergent lens when submerged in the liquid.

### Final Answer
The bi-concave lens will behave as a convergent lens when dipped in the transparent liquid.

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