The electric field of a plane electromagnetic wave varies with time of amplitude `2Vm^-1` propagating along z-axis. The average energy density of the magentic field is (in `Jm^-3`)

Options

A. `13.29 xx 10^(-12)`
B. `8.86 xx 10^(-12)`
C. `17.72 xx 10^(-12)`
D. `4.43 xx 10^(-12)`

Explanation

To find the average energy density of the magnetic field in a plane electromagnetic wave with a given electric field amplitude, we can follow these steps:

### Step 1: Identify the given values
The amplitude of the electric field \( E_0 \) is given as \( 2 \, \text{V/m} \).

### Step 2: Use the relationship between electric and magnetic fields
In an electromagnetic wave, the relationship between the electric field \( E_0 \) and the magnetic field \( B_0 \) is given by:
\[ B_0 = \frac{E_0}{c} \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 3: Calculate the amplitude of the magnetic field
Substituting the values:
\[ B_0 = \frac{2 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} = \frac{2}{3 \times 10^8} \approx 6.67 \times 10^{-9} \, \text{T} \]

### Step 4: Calculate the average energy density of the magnetic field
The average energy density \( u_B \) of the magnetic field is given by:
\[ u_B = \frac{1}{2} \frac{B_0^2}{\mu_0} \]
where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 5: Substitute \( B_0 \) into the energy density formula
First, calculate \( B_0^2 \):
\[ B_0^2 = (6.67 \times 10^{-9})^2 = 4.4489 \times 10^{-17} \, \text{T}^2 \]

Now, substitute \( B_0^2 \) into the energy density formula:
\[ u_B = \frac{1}{2} \frac{4.4489 \times 10^{-17}}{4\pi \times 10^{-7}} \]

### Step 6: Calculate \( u_B \)
Calculating the denominator:
\[ 4\pi \times 10^{-7} \approx 1.2566 \times 10^{-6} \]
Now substituting back:
\[ u_B = \frac{1}{2} \frac{4.4489 \times 10^{-17}}{1.2566 \times 10^{-6}} \approx \frac{1}{2} \times 3.55 \times 10^{-11} \approx 1.775 \times 10^{-11} \, \text{J/m}^3 \]

### Final Answer
The average energy density of the magnetic field is approximately:
\[ u_B \approx 1.775 \times 10^{-11} \, \text{J/m}^3 \]

---