A convex lens focal lenghts 30 cm made of glass of refractive index `1.5` is immersed in water having refrctive `1.33`. The change in the focal length of lens is

Options

A. `62.3` cm
B. `87.4 cm`
C. `58.2 cm`
D. `73.6 cm`

Answer

To solve the problem of finding the change in focal length of a convex lens when it is immersed in water, we can follow these steps:

### Step 1: Understand the given data
- Focal length of the lens in air, \( F_{air} = 30 \, \text{cm} \)
- Refractive index of glass, \( \mu_{glass} = 1.5 \)
- Refractive index of water, \( \mu_{water} = 1.33 \)

### Step 2: Use the lens maker's formula
The lens maker's formula is given by:
\[ \frac{1}{F} = (\mu_{relative} - 1) \cdot k \]
where \( \mu_{relative} = \frac{\mu_{lens}}{\mu_{medium}} \).

### Step 3: Calculate the constant \( k \)
In air, the effective refractive index is 1 (for air):
\[ \mu_{relative, air} = \frac{\mu_{glass}}{\mu_{air}} = \frac{1.5}{1} = 1.5 \]
Using the lens maker's formula:
\[ \frac{1}{F_{air}} = (1.5 - 1) \cdot k \]
Substituting \( F_{air} = 30 \, \text{cm} \):
\[ \frac{1}{30} = 0.5 \cdot k \]
Thus,
\[ k = \frac{1}{15} \]

### Step 4: Calculate the focal length in water
Now, we need to find the focal length of the lens when it is in water:
\[ \mu_{relative, water} = \frac{\mu_{glass}}{\mu_{water}} = \frac{1.5}{1.33} \]
Calculating \( \mu_{relative, water} \):
\[ \mu_{relative, water} \approx 1.127 \]
Now, substituting into the lens maker's formula:
\[ \frac{1}{F_{water}} = (1.127 - 1) \cdot k \]
Substituting \( k = \frac{1}{15} \):
\[ \frac{1}{F_{water}} = 0.127 \cdot \frac{1}{15} \]
Calculating:
\[ \frac{1}{F_{water}} \approx \frac{0.127}{15} \approx 0.00847 \]
Thus,
\[ F_{water} \approx \frac{1}{0.00847} \approx 118.7 \, \text{cm} \]

### Step 5: Calculate the change in focal length
Now, we find the change in focal length:
\[ \Delta F = F_{water} - F_{air} = 118.7 \, \text{cm} - 30 \, \text{cm} \approx 88.7 \, \text{cm} \]

### Final Answer
The change in the focal length of the lens when immersed in water is approximately \( 88.7 \, \text{cm} \).

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