A concave lens with unequal radii of curvature made of glass `(mu_(g)=1.5)` has a focal length of 40cm. If it is immersed in a liquid of refractive index `mu_(1)=2`, then

Options

A. it behaves like a convex lens of 80cm focal length
B. it behaves like a convex lens of 20cm focal length
C. its focal length becomes 60cm
D. nothing can be said

Explanation

To solve the problem, we need to determine the new focal length of a concave lens when it is immersed in a liquid with a different refractive index. Here are the steps to arrive at the solution:

### Step 1: Understand the Given Information
- The refractive index of glass, \( \mu_g = 1.5 \)
- The focal length of the concave lens in air, \( f = -40 \, \text{cm} \) (negative because it is a concave lens)
- The refractive index of the liquid, \( \mu_1 = 2 \)

### Step 2: Use the Lensmaker's Formula
The lensmaker's formula for a lens in air is given by:
\[ \frac{1}{f} = \left( \mu_g - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
For a concave lens, \( f \) is negative. Thus, we can write:
\[ \frac{1}{-40} = (1.5 - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
This simplifies to:
\[ \frac{1}{-40} = 0.5 \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
From this, we can find:
\[ \frac{1}{R_1} - \frac{1}{R_2} = -\frac{1}{20} \]

### Step 3: Calculate the New Focal Length in the Liquid
When the lens is immersed in a liquid, the lensmaker's formula changes to:
\[ \frac{1}{f} = \left( \frac{\mu_g}{\mu_1} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the values:
\[ \frac{1}{f} = \left( \frac{1.5}{2} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Calculating \( \frac{1.5}{2} - 1 \):
\[ \frac{1.5}{2} = 0.75 \quad \Rightarrow \quad 0.75 - 1 = -0.25 \]
Thus, we have:
\[ \frac{1}{f} = -0.25 \left( -\frac{1}{20} \right) \]
This simplifies to:
\[ \frac{1}{f} = \frac{0.25}{20} = \frac{1}{80} \]
Therefore, the focal length \( f \) is:
\[ f = 80 \, \text{cm} \]

### Step 4: Determine the Nature of the Lens
Since the focal length is positive, the lens behaves like a convex lens when immersed in the liquid.

### Final Answer
The new focal length of the concave lens when immersed in the liquid is \( 80 \, \text{cm} \) and it behaves like a convex lens.

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