A plane electromagenic wave having a frequency `v= 23.9 GHz` propagates along the positive z-direction in free space. The peak value of the Electric field is `60 V//m` Which among the following is the acceptable magnetic field component in the electromagnetic wave ?

Options

A. `vec(B)=2 xx 10^(7)sin(0.5 xx10^(3) z+1.5 xx 10^(11) t)hat(i)`
B. `vec(B)=2xx 10^(-7)sin(0.5 xx10^(3)z-1.5 xx 10^(11)t)hat(i)`
C. `vec(B)=60 sin(0.5 xx 10^(3) x+1.5 xx10^(11)t)hat(k)`
D. `vec(B)=2 xx 10^(-7)sin(1.5 xx 10^(2) x+0.5 xx10^(11) t)hat(j)`

Answer

To find the acceptable magnetic field component of the electromagnetic wave given the frequency and peak electric field, we can follow these steps:

### Step 1: Identify the known values
- Frequency \( v = 23.9 \, \text{GHz} = 23.9 \times 10^9 \, \text{Hz} \)
- Peak electric field \( E_0 = 60 \, \text{V/m} \)

### Step 2: Calculate the speed of light \( c \)
The speed of light in free space is given by:
\[ c = 3 \times 10^8 \, \text{m/s} \]

### Step 3: Use the relationship between electric field and magnetic field
The relationship between the peak electric field \( E_0 \) and the peak magnetic field \( B_0 \) in an electromagnetic wave is given by:
\[ \frac{E_0}{B_0} = c \]
From this, we can express \( B_0 \) as:
\[ B_0 = \frac{E_0}{c} \]

### Step 4: Substitute the known values into the equation
Substituting the values of \( E_0 \) and \( c \):
\[ B_0 = \frac{60 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} \]

### Step 5: Perform the calculation
Calculating \( B_0 \):
\[ B_0 = \frac{60}{3 \times 10^8} = 2 \times 10^{-7} \, \text{T} \]

### Step 6: Conclusion
The acceptable magnetic field component \( B_0 \) in the electromagnetic wave is:
\[ B_0 = 2 \times 10^{-7} \, \text{T} \]