In a medium, the electric field in an EM wave is given as: `E=E_(0) sin 10^(7)[x-2.5 xx 10^(8) t] N/C`.The retive index of the medium is

Answer

To find the refractive index of the medium given the electric field of an electromagnetic wave, we can follow these steps:

### Step 1: Identify the wave equation
The electric field of the electromagnetic wave is given as:
\[ E = E_0 \sin(10^7 [x - 2.5 \times 10^8 t]) \, \text{N/C} \]

This can be compared to the general form of an EM wave:
\[ E = E_0 \sin(kx - \omega t) \]

### Step 2: Determine the wave numbers
From the given equation, we can identify:
- The wave number \( k = 10^7 \, \text{m}^{-1} \)
- The angular frequency \( \omega = 2.5 \times 10^8 \, \text{s}^{-1} \)

### Step 3: Calculate the phase velocity \( V \)
The phase velocity \( V \) of the wave can be calculated using the formula:
\[ V = \frac{\omega}{k} \]

Substituting the values:
\[ V = \frac{2.5 \times 10^8}{10^7} \]
\[ V = 25 \, \text{m/s} \]

### Step 4: Use the refractive index formula
The refractive index \( \mu \) of the medium is given by:
\[ \mu = \frac{c}{V} \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 5: Calculate the refractive index
Substituting the values into the refractive index formula:
\[ \mu = \frac{3 \times 10^8}{25} \]
\[ \mu = 12 \]

### Final Answer
The refractive index of the medium is:
\[ \mu = 12 \]

---