The focal length of a convex lens of glass `(mu = 1.5)` in air is 30 cm. Find its focal length when the lens is placed in water `(mu = 4//3)`.

Answer

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

### Step 1: Understand the Lensmaker's Formula
The lensmaker'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 material,
- \( r_1 \) and \( r_2 \) are the radii of curvature of the lens surfaces.

### Step 2: Focal Length in Air
Given that the focal length of the lens in air is \( f_A = 30 \, \text{cm} \) and the refractive index of the glass lens is \( \mu_g = 1.5 \), we can express the lensmaker's formula for air:
\[ \frac{1}{f_A} = (\mu_g - 1) \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]
Substituting the known values:
\[ \frac{1}{30} = (1.5 - 1) \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]
\[ \frac{1}{30} = 0.5 \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]
From this, we can find:
\[ \frac{1}{r_1} + \frac{1}{r_2} = \frac{1}{30 \times 0.5} = \frac{1}{15} \]

### Step 3: Focal Length in Water
Now, we need to find the focal length of the lens when it is placed in water. The refractive index of water is \( \mu_w = \frac{4}{3} \). We can use the lensmaker's formula again:
\[ \frac{1}{f_W} = \left( \frac{\mu_g}{\mu_w} - 1 \right) \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]
Substituting the values:
\[ \frac{1}{f_W} = \left( \frac{1.5}{\frac{4}{3}} - 1 \right) \left( \frac{1}{15} \right) \]
Calculating \( \frac{1.5}{\frac{4}{3}} \):
\[ \frac{1.5 \times 3}{4} = \frac{4.5}{4} = 1.125 \]
Thus:
\[ \frac{1}{f_W} = (1.125 - 1) \left( \frac{1}{15} \right) = 0.125 \left( \frac{1}{15} \right) \]
Calculating this gives:
\[ \frac{1}{f_W} = \frac{0.125}{15} = \frac{0.125}{15} = \frac{1}{120} \]

### Step 4: Calculate the Focal Length in Water
Now, we find \( f_W \):
\[ f_W = 120 \, \text{cm} \]

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

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