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 solve the problem, we need to extract the frequency (f) and speed (v) of the electromagnetic wave from the given electric field equation:

### Given:
The electric field of the electromagnetic wave is given by:
\[ E = 10^{-4} \sin(6 \times 10^5 t - 0.01 x) \]

### Step 1: Identify the Angular Frequency (ω) and Propagation Constant (k)
The general form of the electric field for an electromagnetic wave can be expressed as:
\[ E = E_0 \sin(\omega t - kx) \]
where:
- \( \omega \) is the angular frequency,
- \( k \) is the wave number (or propagation constant).

From the given equation, we can identify:
- \( \omega = 6 \times 10^5 \) (from the coefficient of \( t \))
- \( k = 0.01 \) (from the coefficient of \( x \))

### Step 2: Calculate the Frequency (f)
The relationship between angular frequency (ω) and frequency (f) is given by:
\[ \omega = 2\pi f \]
To find the frequency, we rearrange this equation:
\[ f = \frac{\omega}{2\pi} \]

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

Calculating this:
\[ f \approx \frac{6 \times 10^5}{6.2832} \approx 95500 \, \text{Hz} \]
or
\[ f \approx 95.5 \, \text{kHz} \]

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

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

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

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