An electromagnetic wave travelling through a transparent medium is given by `E_(x)(y,t)=E_(ax)sin2pi[y/(5xx10^(-7))-3xx10^(14)t]` in SI unit then the refractive index of medium is,

Options

A. `1.2`
B. `1.50`
C. `1.33`
D. `2`

Explanation

To find the refractive index of the medium from the given electromagnetic wave equation, we can follow these steps:

### Step 1: Identify the wave equation
The given electromagnetic wave is:
\[ E_x(y,t) = E_a \sin\left(2\pi\left(\frac{y}{5 \times 10^{-7}} - 3 \times 10^{14} t\right)\right) \]

### Step 2: Compare with standard wave equation
The standard form of a wave equation is:
\[ y = A \sin(kx - \omega t) \]
From the given equation, we can identify:
- \( k = \frac{2\pi}{5 \times 10^{-7}} \)
- \( \omega = 2\pi \times 3 \times 10^{14} \)

### Step 3: Calculate wave number \( k \) and angular frequency \( \omega \)
1. Calculate \( k \):
\[ k = \frac{2\pi}{5 \times 10^{-7}} \approx 1.25664 \times 10^{7} \, \text{m}^{-1} \]

2. Calculate \( \omega \):
\[ \omega = 2\pi \times 3 \times 10^{14} \approx 1.88496 \times 10^{15} \, \text{s}^{-1} \]

### Step 4: Calculate the wave velocity \( V \)
The wave velocity \( V \) can be calculated using the formula:
\[ V = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ V = \frac{1.88496 \times 10^{15}}{1.25664 \times 10^{7}} \approx 1.5 \times 10^{8} \, \text{m/s} \]

### Step 5: Calculate the refractive index \( \mu \)
The refractive index \( \mu \) is given by:
\[ \mu = \frac{c}{V} \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^{8} \, \text{m/s} \). Substituting the values:
\[ \mu = \frac{3 \times 10^{8}}{1.5 \times 10^{8}} = 2 \]

### Conclusion
The refractive index of the medium is:
\[ \mu = 2 \]