When a capacitor of capacitance C after charging with a charge Q is connected to inductor of self inductance L, the oscillation of charge takes place with time between the two plates of capacitor. If one plate of capacitor is connected to antenna and other plate is earthed, then em wave are produced, which are sinusoidal variation of electric and magnetic field vectors, perpendicular to each other as well as perpendicular to the direction of propagation of wave. The velocity of these waves depends upon the electric and magnetic properties of the medium. the em wave were produced experimentally by Hertz in 1888 using Hertz Oscillator, which were of wavelength 6m. Jagdish chander bose in 1895 produced these waves which were of wave length 5mm to 25mm and in 1896, G. Marconi established a wireless communication between two stations 50km apart using em waves. In an em wave, the amplitude of electric field is `10Vm^-1`. The frequency of wave is `5xx10^(14)Hz .` the wave is propagating along z-axis.
In em wave, the average energy density due to magnetic field is

Options

A. `8.85xx10^(-10)Jm^-3`
B. `4.42xx10^(-10)Jm^-3`
C. `2.21xx10^(-10)Jm^-3`
D. `6.63xx10^(-10)Jm^-3`

Explanation

To find the average energy density due to the magnetic field in an electromagnetic wave, we can follow these steps:

### Step 1: Understand the relationship between electric field (E) and magnetic field (B)
In an electromagnetic wave, the relationship between the amplitude of the electric field (E₀) and the amplitude of the magnetic field (B₀) is given by:
\[ E₀ = c \cdot B₀ \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 2: Calculate the value of B₀
Given that the amplitude of the electric field \( E₀ = 10 \, \text{V/m} \), we can rearrange the equation to find \( B₀ \):
\[ B₀ = \frac{E₀}{c} = \frac{10 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} \]
Calculating this gives:
\[ B₀ \approx 3.33 \times 10^{-8} \, \text{T} \]

### Step 3: Find the average energy density due to the magnetic field
The average energy density \( U_B \) due to the magnetic field in an electromagnetic wave is given by the formula:
\[ U_B = \frac{1}{2} \cdot \frac{B^2}{\mu_0} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 4: Substitute B₀ into the energy density formula
Substituting \( B₀ \) into the formula:
\[ U_B = \frac{1}{2} \cdot \frac{(3.33 \times 10^{-8})^2}{4\pi \times 10^{-7}} \]
Calculating \( (3.33 \times 10^{-8})^2 \):
\[ (3.33 \times 10^{-8})^2 \approx 1.11 \times 10^{-15} \, \text{T}^2 \]
Now substituting this value into the energy density formula:
\[ U_B = \frac{1.11 \times 10^{-15}}{2 \cdot 4\pi \times 10^{-7}} \]

### Step 5: Calculate the average energy density
Calculating the denominator:
\[ 2 \cdot 4\pi \times 10^{-7} \approx 2.51 \times 10^{-6} \, \text{T m/A} \]
Now, substituting this back into the equation for \( U_B \):
\[ U_B \approx \frac{1.11 \times 10^{-15}}{2.51 \times 10^{-6}} \approx 4.42 \times 10^{-10} \, \text{J/m}^3 \]

### Step 6: Final result
Thus, the average energy density due to the magnetic field is approximately:
\[ U_B \approx 4.42 \times 10^{-10} \, \text{J/m}^3 \]