A plane `E.M.` wave of frequency `40 MHz` travels along `X`-axis. At same point at same instant, the electric field `E` has maximum value of `750 N//C` in `Y`- direction. The magnitude and direction of magnetic field is

Options

A. `2.5 muT` in x - direction
B. `2.5 muT ` in y - direction
C. `2.5 muT ` in z - direction
D. None

Answer

To find the magnitude and direction of the magnetic field associated with a plane electromagnetic wave, we can follow these steps:

### Step 1: Identify the given parameters
- Frequency of the electromagnetic wave, \( f = 40 \, \text{MHz} = 40 \times 10^6 \, \text{Hz} \)
- Maximum value of the electric field, \( E_0 = 750 \, \text{N/C} \)
- Direction of the electric field, \( \hat{j} \) (Y-direction)

### Step 2: Calculate the speed of light
The speed of light \( c \) is approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 3: Calculate the amplitude of the magnetic field
The relationship between the electric field \( E \) and the magnetic field \( B \) in an electromagnetic wave is given by the equation:
\[ B_0 = \frac{E_0}{c} \]
Substituting the values:
\[ B_0 = \frac{750 \, \text{N/C}}{3 \times 10^8 \, \text{m/s}} = \frac{750}{3 \times 10^8} = 2.5 \times 10^{-6} \, \text{T} \]

### Step 4: Determine the direction of the magnetic field
In an electromagnetic wave, the electric field \( \mathbf{E} \), magnetic field \( \mathbf{B} \), and the direction of wave propagation \( \mathbf{k} \) are mutually perpendicular. Given:
- The wave travels along the X-axis (direction of propagation).
- The electric field is in the Y-direction.

Using the right-hand rule:
1. Point your thumb in the direction of the wave propagation (X-axis).
2. Your index finger points in the direction of the electric field (Y-axis).
3. Your middle finger will then point in the direction of the magnetic field (Z-axis).

Thus, the magnetic field will be in the Z-direction.

### Final Answer
- Magnitude of the magnetic field: \( B_0 = 2.5 \times 10^{-6} \, \text{T} \)
- Direction of the magnetic field: Z-direction

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