Light of wavelength 500nm traveling with a speed of `2.0xx10^(8)ms^(-1)` in a certain medium enters another medium of refractive index `5//4` times that of the first medium. What are the wavelength and speed in the second medium?

Options

A. Wavelength `(nm)` speed`(ms^(-1))` 400 `1.6xx10^(8)`
B. Wavelength `(nm)` speed`(ms^(-1))` 400 `2.5xx10^(8)`
C. Wavelength `(nm)` speed`(ms^(-1))` 500 `2.5xx10^(8)`
D. Wavelength `(nm)` speed`(ms^(-1))` 625 `1.6xx10^(8)`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the refractive index of the first medium (n1)
The speed of light in the first medium (v1) is given as \(2.0 \times 10^8 \, \text{m/s}\). The speed of light in vacuum (c) is approximately \(3.0 \times 10^8 \, \text{m/s}\).

The refractive index (n1) is given by the formula:
\[ n_1 = \frac{c}{v_1} \]
Substituting the values:
\[ n_1 = \frac{3.0 \times 10^8 \, \text{m/s}}{2.0 \times 10^8 \, \text{m/s}} = 1.5 \]

### Step 2: Calculate the actual wavelength in the first medium (λ1)
The wavelength in the first medium is given as \(500 \, \text{nm}\). The actual wavelength (λ) in the medium can be calculated using:
\[ \lambda = \lambda_0 \cdot n_1 \]
where \( \lambda_0 \) is the wavelength in vacuum. Thus,
\[ \lambda = 500 \, \text{nm} \cdot 1.5 = 750 \, \text{nm} \]

### Step 3: Determine the refractive index of the second medium (n2)
The refractive index of the second medium (n2) is given as \( \frac{5}{4} \) times that of the first medium:
\[ n_2 = \frac{5}{4} \cdot n_1 = \frac{5}{4} \cdot 1.5 = \frac{15}{8} = 1.875 \]

### Step 4: Calculate the speed of light in the second medium (v2)
The speed of light in the second medium can be calculated using the formula:
\[ v_2 = \frac{c}{n_2} \]
Substituting the values:
\[ v_2 = \frac{3.0 \times 10^8 \, \text{m/s}}{1.875} \approx 1.6 \times 10^8 \, \text{m/s} \]

### Step 5: Calculate the wavelength in the second medium (λ2)
The wavelength in the second medium can be calculated using:
\[ \lambda_2 = \frac{\lambda}{n_2} \]
Substituting the values:
\[ \lambda_2 = \frac{750 \, \text{nm}}{1.875} \approx 400 \, \text{nm} \]

### Final Results
- Speed of light in the second medium (v2): \(1.6 \times 10^8 \, \text{m/s}\)
- Wavelength in the second medium (λ2): \(400 \, \text{nm}\)