The radii of curvature of a lens are + 20 cm and + 30 cm. The material of the lens has a refracting index 1-6. Find the focal length of the lens (a) if it is placed in air, and (b) if it is placed in water (mu = 1.33).

Explanation

To find the focal length of the lens using the lens maker's formula, we will follow these steps:

### Given Data:
- Radius of curvature \( R_1 = +20 \, \text{cm} \)
- Radius of curvature \( R_2 = +30 \, \text{cm} \)
- Refractive index of the lens \( \mu_l = 1.6 \)
- Refractive index of air \( \mu_m = 1 \)
- Refractive index of water \( \mu_m = 1.33 \)

### Step 1: Use the Lens Maker's Formula
The lens maker's formula is given by:

\[ \frac{1}{f} = \left( \frac{\mu_l}{\mu_m} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

### Step 2: Calculate the Focal Length in Air
#### (a) When the lens is placed in air:
1. Substitute the values into the formula:
\[ \frac{1}{f} = \left( \frac{1.6}{1} - 1 \right) \left( \frac{1}{20} - \frac{1}{30} \right) \]

2. Simplify the expression:
\[ \frac{1}{f} = (1.6 - 1) \left( \frac{1}{20} - \frac{1}{30} \right) \]
\[ = 0.6 \left( \frac{3 - 2}{60} \right) = 0.6 \left( \frac{1}{60} \right) \]
\[ = \frac{0.6}{60} = \frac{1}{100} \]

3. Therefore, the focal length \( f \) is:
\[ f = 100 \, \text{cm} \]

### Step 3: Calculate the Focal Length in Water
#### (b) When the lens is placed in water:
1. Substitute the values into the formula:
\[ \frac{1}{f} = \left( \frac{1.6}{1.33} - 1 \right) \left( \frac{1}{20} - \frac{1}{30} \right) \]

2. Simplify the expression:
\[ \frac{1}{f} = \left( \frac{1.6}{1.33} - 1 \right) \left( \frac{1}{20} - \frac{1}{30} \right) \]
\[ = \left( 1.202 - 1 \right) \left( \frac{1}{20} - \frac{1}{30} \right) \]
\[ = 0.202 \left( \frac{1}{20} - \frac{1}{30} \right) = 0.202 \left( \frac{3 - 2}{60} \right) = 0.202 \left( \frac{1}{60} \right) \]
\[ = \frac{0.202}{60} = \frac{1}{296} \]

3. Therefore, the focal length \( f \) is:
\[ f = 296 \, \text{cm} \]

### Final Answers:
- (a) Focal length in air: \( f = 100 \, \text{cm} \)
- (b) Focal length in water: \( f = 296 \, \text{cm} \)