Find the speed of light of wavelength `lambda =780nm` (in air) in a medium of refractive index `mu=1.55.`
(b) What is the wavelength of this light in the given medium ?

Explanation

To solve the problem step by step, we will break it down into two parts as requested.

### Part (a): Finding the speed of light in the medium

1. **Identify the known values**:
- Wavelength of light in air, \( \lambda = 780 \, \text{nm} = 780 \times 10^{-9} \, \text{m} \)
- Refractive index of the medium, \( \mu = 1.55 \)
- Speed of light in vacuum (or air), \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Use the formula for the speed of light in a medium**:
\[ v = \frac{c}{\mu} \]
where \( v \) is the speed of light in the medium.

3. **Substitute the values into the formula**:
\[ v = \frac{3 \times 10^8 \, \text{m/s}}{1.55} \]

4. **Calculate the speed**:
\[ v \approx 1.935 \times 10^8 \, \text{m/s} \approx 1.94 \times 10^8 \, \text{m/s} \]

### Part (b): Finding the wavelength of light in the medium

1. **Use the relationship between the refractive indices and wavelengths**:
\[ \frac{\mu_1}{\mu_2} = \frac{\lambda_2}{\lambda_1} \]
where:
- \( \mu_1 = 1 \) (refractive index of air)
- \( \mu_2 = 1.55 \) (refractive index of the medium)
- \( \lambda_1 = 780 \, \text{nm} \) (wavelength in air)
- \( \lambda_2 \) is the wavelength in the medium (unknown)

2. **Rearranging the formula to find \( \lambda_2 \)**:
\[ \lambda_2 = \lambda_1 \cdot \frac{\mu_1}{\mu_2} \]

3. **Substitute the known values**:
\[ \lambda_2 = 780 \, \text{nm} \cdot \frac{1}{1.55} \]

4. **Calculate the wavelength in the medium**:
\[ \lambda_2 \approx 503.23 \, \text{nm} \approx 503 \, \text{nm} \]

### Final Answers:
- (a) Speed of light in the medium: \( v \approx 1.94 \times 10^8 \, \text{m/s} \)
- (b) Wavelength of light in the medium: \( \lambda_2 \approx 503 \, \text{nm} \)

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