In a plane e.m. wave, the electric field oscillates withe amplitude `20Vm^-1`. Find (a) energy density of electric field (b) energy density of magnetic field.

Explanation

To solve the problem step by step, we will calculate the energy density of the electric field and the magnetic field in a plane electromagnetic wave where the electric field oscillates with an amplitude of \(20 \, \text{V/m}\).

### Step 1: Understanding Energy Density of Electric Field
The energy density \(U_e\) of the electric field in an electromagnetic wave is given by the formula:
\[ U_e = \frac{1}{2} \epsilon_0 E_{\text{rms}}^2 \]
where \(\epsilon_0\) is the permittivity of free space and \(E_{\text{rms}}\) is the root mean square (RMS) value of the electric field.

### Step 2: Calculate RMS Value of Electric Field
The RMS value \(E_{\text{rms}}\) is related to the amplitude \(E_0\) of the electric field by the formula:
\[ E_{\text{rms}} = \frac{E_0}{\sqrt{2}} \]
Given that the amplitude \(E_0 = 20 \, \text{V/m}\), we can calculate \(E_{\text{rms}}\):
\[ E_{\text{rms}} = \frac{20}{\sqrt{2}} = 20 \times \frac{1}{\sqrt{2}} \approx 14.14 \, \text{V/m} \]

### Step 3: Substitute into Energy Density Formula
Now we substitute \(E_{\text{rms}}\) into the energy density formula:
\[ U_e = \frac{1}{2} \epsilon_0 E_{\text{rms}}^2 = \frac{1}{2} \epsilon_0 \left(14.14\right)^2 \]

### Step 4: Calculate \(E_{\text{rms}}^2\)
Calculating \(E_{\text{rms}}^2\):
\[ E_{\text{rms}}^2 = (14.14)^2 \approx 200 \, \text{(V/m)}^2 \]

### Step 5: Substitute \(\epsilon_0\) Value
The permittivity of free space \(\epsilon_0\) is approximately:
\[ \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{F/m} \]
Now substituting this value into the energy density formula:
\[ U_e = \frac{1}{2} \times 8.85 \times 10^{-12} \times 200 \]

### Step 6: Calculate Energy Density of Electric Field
Calculating \(U_e\):
\[ U_e = \frac{1}{2} \times 8.85 \times 10^{-12} \times 200 = 8.85 \times 10^{-10} \, \text{J/m}^3 \]

### Step 7: Energy Density of Magnetic Field
In a plane electromagnetic wave, the energy density of the magnetic field \(U_b\) is equal to the energy density of the electric field:
\[ U_b = U_e = 8.85 \times 10^{-10} \, \text{J/m}^3 \]

### Final Answer
(a) Energy density of electric field \(U_e = 8.85 \times 10^{-10} \, \text{J/m}^3\)

(b) Energy density of magnetic field \(U_b = 8.85 \times 10^{-10} \, \text{J/m}^3\)