The electric field part of an electromagnetic wave in vacuum is E=3.1cos[(1.8 rad/m ​ )y+(5.4×10^8 rad/s ​ )t] i^ . The frequency corresponding to the given part of electromagnetic wave is

Options

A. `5.4xx10^(8)Hz`
B. `8.6xx10^(7)Hz`
C. `3.2xx10^(8)Hz`
D. `4.8xx10^(7)Hz`.

Explanation

To find the frequency corresponding to the given electric field part of an electromagnetic wave, we can follow these steps:

### Step 1: Identify the given electric field equation
The electric field is given as:
\[ E = 3.1 \cos(1.8 \, \text{rad/m} \cdot y + 5.4 \times 10^8 \, \text{rad/s} \cdot t) \hat{i} \]

### Step 2: Compare with the standard form of the wave equation
The standard form of the electric field of an electromagnetic wave is:
\[ E = E_0 \cos(k y + \omega t) \]
where:
- \( E_0 \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

From the given equation, we can identify:
- \( k = 1.8 \, \text{rad/m} \)
- \( \omega = 5.4 \times 10^8 \, \text{rad/s} \)

### Step 3: Use the relationship between angular frequency and frequency
The relationship between angular frequency \( \omega \) and frequency \( f \) is given by:
\[ f = \frac{\omega}{2\pi} \]

### Step 4: Substitute the value of \( \omega \)
Now, substituting the value of \( \omega \):
\[ f = \frac{5.4 \times 10^8 \, \text{rad/s}}{2\pi} \]

### Step 5: Calculate the frequency
Using the value of \( \pi \approx 3.14 \):
\[ f = \frac{5.4 \times 10^8}{2 \times 3.14} \]
\[ f = \frac{5.4 \times 10^8}{6.28} \]
\[ f \approx 8.6 \times 10^7 \, \text{Hz} \]

### Final Answer
Thus, the frequency corresponding to the given part of the electromagnetic wave is:
\[ f \approx 8.6 \times 10^7 \, \text{Hz} \]

---