The electric field portion of an electromagnetic wave is given by ( all variables in SI units) `E=10^(-4) sin (6xx10^(5)t -0.01x)` The. Frequency (f) and the speed (v) of electromagnetic wave are

Options

A. `f=30/pikHz and v=1.5xx10^(7)ms^(-1)`
B. `f=90/pikHz and v=6.0xx10^(7)ms^(-1)`
C. `f=300/pikHz and v=6.0xx10^(7)ms^(-1)`
D. `f=600/pikHz and v=7.5xx10^(7)ms^(-1)`

Answer

To find the frequency (f) and the speed (v) of the electromagnetic wave given the electric field equation \( E = 10^{-4} \sin(6 \times 10^{5} t - 0.01 x) \), we will follow these steps:

### Step 1: Identify the angular frequency (ω) and wave number (k)
The general form of the electric field of an electromagnetic wave is given by:
\[ E = E_0 \sin(\omega t - kx) \]

From the given equation:
\[ E = 10^{-4} \sin(6 \times 10^{5} t - 0.01 x) \]

We can identify:
- \( \omega = 6 \times 10^{5} \) radians/second (angular frequency)
- \( k = 0.01 \) radians/meter (wave number)

### Step 2: Calculate the frequency (f)
The frequency (f) is related to the angular frequency (ω) by the formula:
\[ f = \frac{\omega}{2\pi} \]

Substituting the value of ω:
\[ f = \frac{6 \times 10^{5}}{2\pi} \]

Calculating this gives:
\[ f \approx \frac{6 \times 10^{5}}{6.2832} \approx 95500.5 \text{ Hz} \]

To convert this to kilohertz (kHz):
\[ f \approx 95.5 \text{ kHz} \]

### Step 3: Calculate the speed (v)
The speed (v) of the wave can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]

Substituting the values of ω and k:
\[ v = \frac{6 \times 10^{5}}{0.01} \]

Calculating this gives:
\[ v = 6 \times 10^{7} \text{ m/s} \]

### Summary of Results
- Frequency \( f \approx 95.5 \text{ kHz} \)
- Speed \( v = 6 \times 10^{7} \text{ m/s} \)