During the propagation of e.m. wave in a medium

Options

A. Electric energy density is double of the magnetic energy density
B. Electric energy density is half of the magnetic energy density.
C. Electric energy density is equal to the magnetic energy density.
D. Both electric and magnetic energy densities are zero

Explanation

To solve the problem regarding the relationship between electric energy density and magnetic energy density during the propagation of an electromagnetic (EM) wave in a medium, we can follow these steps:

### Step 1: Understand the Definitions
- **Electric Energy Density (U_E)**: Given by the formula \( U_E = \frac{1}{2} \epsilon E^2 \), where \( \epsilon \) is the permittivity of the medium and \( E \) is the electric field intensity.
- **Magnetic Energy Density (U_B)**: Given by the formula \( U_B = \frac{1}{2} \mu B^2 \), where \( \mu \) is the permeability of the medium and \( B \) is the magnetic field intensity.

### Step 2: Write the Ratio of Energy Densities
We need to find the ratio \( \frac{U_E}{U_B} \):
\[ \frac{U_E}{U_B} = \frac{\frac{1}{2} \epsilon E^2}{\frac{1}{2} \mu B^2} = \frac{\epsilon E^2}{\mu B^2} \]

### Step 3: Relate Electric and Magnetic Fields
From the properties of electromagnetic waves, we know that the speed of light \( c \) is related to the electric and magnetic fields:
\[ c = \frac{E}{B} \]
Also, we can express \( c \) in terms of permittivity and permeability:
\[ c = \frac{1}{\sqrt{\mu \epsilon}} \]

### Step 4: Substitute for \( E \) and \( B \)
From the equation \( c = \frac{E}{B} \), we can express \( E \) in terms of \( B \):
\[ E = cB \]
Now, substitute \( E \) in the ratio:
\[ \frac{U_E}{U_B} = \frac{\epsilon (cB)^2}{\mu B^2} = \frac{\epsilon c^2 B^2}{\mu B^2} \]
The \( B^2 \) terms cancel out:
\[ \frac{U_E}{U_B} = \frac{\epsilon c^2}{\mu} \]

### Step 5: Substitute the Value of \( c^2 \)
Using the relationship \( c^2 = \frac{1}{\mu \epsilon} \):
\[ \frac{U_E}{U_B} = \frac{\epsilon \cdot \frac{1}{\mu \epsilon}}{\mu} = \frac{1}{\mu} \cdot \frac{1}{\mu} = 1 \]

### Step 6: Conclusion
Thus, we find that:
\[ U_E = U_B \]
This means the electric energy density is equal to the magnetic energy density during the propagation of an electromagnetic wave in a medium.

### Final Answer
The correct option is that the electric energy density is equal to the magnetic energy density.

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