Relation between energy density of electric field and magnetic field in em wave.

Options

A. `U_E = U_B`
B. `U_E gt U_B`
C. `U_E lt U_B`
D. `U_E != U_B`

Answer

To derive the relation between the energy density of the electric field and the magnetic field in an electromagnetic wave, we can follow these steps:

### Step 1: Define Energy Density
The energy density (u) of an electric field (E) and a magnetic field (B) is given by the following formulas:

- For the electric field:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]

- For the magnetic field:
\[ u_B = \frac{1}{2} \frac{B^2}{\mu_0} \]

Where:
- \( \epsilon_0 \) is the permittivity of free space.
- \( \mu_0 \) is the permeability of free space.

### Step 2: Relate Electric Field and Magnetic Field
In an electromagnetic wave, the electric field (E) and magnetic field (B) are related by the equation:
\[ B = \frac{E}{c} \]
Where \( c \) is the speed of light in vacuum, given by:
\[ c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \]

### Step 3: Substitute B in the Energy Density Equation
Substituting \( B = \frac{E}{c} \) into the equation for magnetic energy density:
\[ u_B = \frac{1}{2} \frac{B^2}{\mu_0} = \frac{1}{2} \frac{\left(\frac{E}{c}\right)^2}{\mu_0} \]

### Step 4: Simplify the Magnetic Energy Density
Now substituting \( c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \):
\[ u_B = \frac{1}{2} \frac{E^2}{\mu_0 c^2} = \frac{1}{2} \frac{E^2}{\mu_0 \left(\frac{1}{\epsilon_0 \mu_0}\right)} = \frac{1}{2} \epsilon_0 E^2 \]

### Step 5: Compare Energy Densities
Now we can compare the energy densities:
- Electric field energy density:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]
- Magnetic field energy density:
\[ u_B = \frac{1}{2} \epsilon_0 E^2 \]

### Conclusion
From the above calculations, we can conclude that:
\[ u_E = u_B \]
Thus, the average energy density of the electric field is equal to the average energy density of the magnetic field in an electromagnetic wave.

### Final Answer
The relation between the energy density of the electric field and the magnetic field in an electromagnetic wave is:
\[ \text{Energy density of electric field} = \text{Energy density of magnetic field} \]

---