A double convex lens of refractive index `m_(1)` is immersed in a liquid of refractive index `m_(2)`. This lens will act as:

Options

A. diverging lens If `m_(1) gt m_(2)`
B. diverging lens if `m_(1) lt m_(2)`
C. converging lens if `m_(1) = m_(2)`
D. converging lens if `m_(1) lt m_(2)`

Answer

To determine how a double convex lens behaves when immersed in a liquid of a different refractive index, we can use the lens maker's formula. Here’s the step-by-step solution:

### Step 1: Understand the Lens Maker's Formula
The lens maker's formula is given by:
\[ \frac{1}{F} = \left(\frac{m_1}{m_2} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]
where:
- \( F \) is the focal length of the lens,
- \( m_1 \) is the refractive index of the lens,
- \( m_2 \) is the refractive index of the surrounding medium (liquid),
- \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

### Step 2: Analyze Different Cases
We will consider three cases based on the relationship between \( m_1 \) and \( m_2 \):

1. **Case 1: \( m_1 < m_2 \)**
- If the refractive index of the lens is less than that of the liquid, then:
\[ \frac{m_1}{m_2} < 1 \implies \frac{m_1}{m_2} - 1 < 0 \]
- This results in a negative focal length, indicating that the lens behaves as a diverging lens.

2. **Case 2: \( m_1 = m_2 \)**
- If the refractive indices are equal, then:
\[ \frac{m_1}{m_2} = 1 \implies \frac{m_1}{m_2} - 1 = 0 \]
- This means the lens will not converge or diverge light, effectively acting as a transparent plane sheet.

3. **Case 3: \( m_1 > m_2 \)**
- If the refractive index of the lens is greater than that of the liquid, then:
\[ \frac{m_1}{m_2} > 1 \implies \frac{m_1}{m_2} - 1 > 0 \]
- This results in a positive focal length, indicating that the lens behaves as a converging lens.

### Step 3: Conclusion
Based on the analysis:
- The lens acts as a **diverging lens** if \( m_1 < m_2 \).
- It acts as a **transparent plane sheet** if \( m_1 = m_2 \).
- It acts as a **converging lens** if \( m_1 > m_2 \).

Thus, the answer to the question is:
- The lens will act as a diverging lens if \( m_1 < m_2 \).