A plane electromagnetic wave of frequency 35 MHz travels in free space along the X-direction. At a particular point (in space and time) `vec E = 9.6 hat jV//m`. The value of magnetic field at this point is :

Options

A. `3.2 times 10^(-8) hat k T`
B. `3.2 times 10^(-8) hat i T`
C. `9.6 hat j T`
D. `9.6 times 10^(-8) hat k T`

Answer

To solve the problem of finding the magnetic field at a point where the electric field of an electromagnetic wave is given, we can follow these steps:

### Step 1: Identify the given values
- Frequency (f) = 35 MHz = \(35 \times 10^6\) Hz
- Electric field (E) = \(9.6 \hat{j}\) V/m

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 35 \times 10^6 = 70\pi \times 10^6 \text{ rad/s} \]

### Step 3: Calculate the wave number (k)
The wave number (k) is related to the wavelength (λ) and is given by:
\[ k = \frac{2\pi}{\lambda} \]
To find λ, we can use the relationship between frequency, wavelength, and the speed of light (c):
\[ c = f \lambda \implies \lambda = \frac{c}{f} \]
Where \(c = 3 \times 10^8\) m/s. Thus:
\[ \lambda = \frac{3 \times 10^8}{35 \times 10^6} = \frac{3 \times 10^8}{35 \times 10^6} \approx 8.57 \text{ m} \]
Now substituting λ into the equation for k:
\[ k = \frac{2\pi}{8.57} \approx \frac{2\pi}{8.57} \approx 0.733 \text{ rad/m} \]

### Step 4: Calculate the maximum magnetic field (B₀)
The relationship between the maximum electric field (E₀) and the maximum magnetic field (B₀) in an electromagnetic wave is given by:
\[ B_0 = \frac{E_0}{c} \]
Substituting the values:
\[ B_0 = \frac{9.6}{3 \times 10^8} \approx 3.2 \times 10^{-8} \text{ T} \]

### Conclusion
The value of the magnetic field at the given point is:
\[ B_0 \approx 3.2 \times 10^{-8} \text{ T} \]