The focal length of lens of refractive index `1.5`in air is`30cm`When it is immersed in water of refractive index`(4)/(3)`,then its focal length will be

Options

A. 120cm
B. `60cm`
C. `30`cm
D. `240cm`

Explanation

To find the new focal length of a lens when it is immersed in water, we can use the lens maker's formula. Let's go through the solution step by step.

### Step-by-Step Solution:

**Step 1: Understand the lens maker's formula.**
The lens maker's formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
where:
- \( f \) is the focal length of the lens,
- \( \mu \) is the refractive index of the lens with respect to the medium,
- \( r_1 \) and \( r_2 \) are the radii of curvature of the lens surfaces.

**Step 2: Calculate the focal length in air.**
Given:
- Focal length in air, \( f_1 = 30 \, \text{cm} \)
- Refractive index of the lens in air, \( \mu = 1.5 \)

Using the lens maker's formula for air:
\[ \frac{1}{f_1} = (1.5 - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
This simplifies to:
\[ \frac{1}{f_1} = 0.5 \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
Let this be Equation (1).

**Step 3: Calculate the refractive index of the lens in water.**
When the lens is immersed in water (refractive index \( \mu_w = \frac{4}{3} \)):
\[ \mu_{\text{lens, water}} = \frac{\mu_{\text{lens, air}}}{\mu_w} = \frac{1.5}{\frac{4}{3}} = \frac{1.5 \times 3}{4} = \frac{4.5}{4} = 1.125 \]

**Step 4: Calculate the focal length in water.**
Using the lens maker's formula for water:
\[ \frac{1}{f_2} = (1.125 - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
This simplifies to:
\[ \frac{1}{f_2} = 0.125 \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
Let this be Equation (2).

**Step 5: Relate the two equations.**
From Equation (1) and Equation (2):
\[ \frac{f_2}{f_1} = \frac{0.5}{0.125} \]
Calculating the ratio:
\[ \frac{f_2}{f_1} = 4 \implies f_2 = 4 f_1 \]

**Step 6: Substitute the value of \( f_1 \).**
Now substituting \( f_1 = 30 \, \text{cm} \):
\[ f_2 = 4 \times 30 \, \text{cm} = 120 \, \text{cm} \]

### Final Answer:
The focal length of the lens when immersed in water is \( 120 \, \text{cm} \).

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