An electromagneti wave propagationg through vacuum is given `B=B_(0)sin(kx-omega)` . Which of the following is/are independent of frequency?

Options

A. k
B. `(omega)/(k)`
C. `k omega`
D. `omega`

Answer

To solve the problem, we need to analyze the given electromagnetic wave equation and determine which quantities are independent of frequency.

### Given:
The electromagnetic wave is represented as:
\[ B = B_0 \sin(kx - \omega) \]

Where:
- \( B_0 \) is the amplitude of the magnetic field.
- \( k \) is the wave number.
- \( \omega \) is the angular frequency.

### Step-by-Step Solution:

1. **Identify the Wave Number \( k \)**:
The wave number \( k \) is defined as:
\[ k = \frac{2\pi}{\lambda} \]
where \( \lambda \) is the wavelength. The wavelength is related to the frequency \( f \) by:
\[ \lambda = \frac{c}{f} \]
where \( c \) is the speed of light. Substituting for \( \lambda \):
\[ k = \frac{2\pi f}{c} \]
This shows that \( k \) is directly proportional to the frequency \( f \). Thus, \( k \) is **dependent** on frequency.

2. **Identify the Ratio \( \frac{\omega}{k} \)**:
The angular frequency \( \omega \) is given by:
\[ \omega = 2\pi f \]
Therefore, the ratio \( \frac{\omega}{k} \) can be calculated as follows:
\[ \frac{\omega}{k} = \frac{2\pi f}{\frac{2\pi f}{c}} = c \]
The speed of light \( c \) is a constant and does not depend on frequency. Thus, \( \frac{\omega}{k} \) is **independent** of frequency.

3. **Identify the Product \( k \cdot \omega \)**:
The product \( k \cdot \omega \) can be expressed as:
\[ k \cdot \omega = \left(\frac{2\pi f}{c}\right) \cdot (2\pi f) = \frac{(2\pi)^2 f^2}{c} \]
This expression clearly shows that \( k \cdot \omega \) is dependent on \( f^2 \) and thus is **dependent** on frequency.

4. **Identify \( \omega \)**:
Since \( \omega = 2\pi f \), it is directly proportional to frequency \( f \). Therefore, \( \omega \) is **dependent** on frequency.

### Conclusion:
From the analysis:
- Option 1: \( k \) - **Dependent on frequency**
- Option 2: \( \frac{\omega}{k} \) - **Independent of frequency** (Correct option)
- Option 3: \( k \cdot \omega \) - **Dependent on frequency**
- Option 4: \( \omega \) - **Dependent on frequency**

Thus, the only quantity that is independent of frequency is:
**Option 2: \( \frac{\omega}{k} \)**.