A convex glass lens `(mu_(g) = 1.5)` has a focal length of 8 cm when placed in air. What is the focal length of the lens when it is immersed in water ?
`(mu_(omega) = (4)/(3))`

Options

A. 32 cm
B. 6 cm
C. 16 cm
D. 30 cm

Answer

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

### Step 1: Understand the given data
- The refractive index of the glass lens, \( \mu_g = 1.5 \).
- The focal length of the lens in air, \( f = 8 \, \text{cm} \).
- The refractive index of water, \( \mu_w = \frac{4}{3} \).

### Step 2: Calculate the relative refractive index when the lens is in air
When the lens is in air, the relative refractive index \( \mu_{relative} \) is given by:
\[ \mu_{relative} = \frac{\mu_g}{\mu_{air}} = \frac{1.5}{1} = 1.5 \]

### Step 3: Use the lens maker's formula in air
The lens maker's formula is given by:
\[ \frac{1}{f} = \mu_{relative} - 1 \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting the values we have:
\[ \frac{1}{8} = (1.5 - 1) \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
\[ \frac{1}{8} = 0.5 \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
This simplifies to:
\[ \frac{1}{R_1} - \frac{1}{R_2} = \frac{1}{4} \quad \text{(Equation 1)} \]

### Step 4: Calculate the relative refractive index when the lens is in water
Now, when the lens is immersed in water, the relative refractive index becomes:
\[ \mu_{relative} = \frac{\mu_g}{\mu_w} = \frac{1.5}{\frac{4}{3}} = \frac{1.5 \cdot 3}{4} = \frac{4.5}{4} = \frac{9}{8} \]

### Step 5: Use the lens maker's formula in water
Now, we can use the lens maker's formula again for the lens in water:
\[ \frac{1}{f'} = \left( \frac{9}{8} - 1 \right) \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
This simplifies to:
\[ \frac{1}{f'} = \left( \frac{1}{8} \right) \cdot \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Substituting from Equation 1:
\[ \frac{1}{f'} = \frac{1}{8} \cdot \frac{1}{4} = \frac{1}{32} \]

### Step 6: Calculate the focal length in water
Taking the reciprocal gives us the focal length in water:
\[ f' = 32 \, \text{cm} \]

### Conclusion
The focal length of the lens when it is immersed in water is \( 32 \, \text{cm} \).

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