For the plane electromagnetic wave given by `E =E_0 sin (omegat -kx) and B=B_0sin (omegat -kx)` , the ratio of average electric energy density to average magnetic energy density is

Options

A. `1/2`
B. 2
C. 4
D. 1

Answer

To find the ratio of average electric energy density to average magnetic energy density for the given plane electromagnetic wave, we can follow these steps:

### Step 1: Understand the given equations
We are given the electric field \( E \) and magnetic field \( B \) of an electromagnetic wave as:
\[ E = E_0 \sin(\omega t - kx) \]
\[ B = B_0 \sin(\omega t - kx) \]

### Step 2: Write the formulas for energy densities
The average electric energy density \( u_E \) and the average magnetic energy density \( u_B \) are given by the formulas:
\[ u_E = \frac{1}{2} \epsilon_0 E^2 \]
\[ u_B = \frac{1}{2} \frac{B^2}{\mu_0} \]

### Step 3: Calculate average electric energy density
Substituting \( E = E_0 \):
\[ u_E = \frac{1}{2} \epsilon_0 (E_0)^2 \]

### Step 4: Calculate average magnetic energy density
Substituting \( B = B_0 \):
\[ u_B = \frac{1}{2} \frac{(B_0)^2}{\mu_0} \]

### Step 5: Find the ratio of average electric energy density to average magnetic energy density
We need to find the ratio \( \frac{u_E}{u_B} \):
\[ \frac{u_E}{u_B} = \frac{\frac{1}{2} \epsilon_0 (E_0)^2}{\frac{1}{2} \frac{(B_0)^2}{\mu_0}} = \frac{\epsilon_0 (E_0)^2}{\frac{(B_0)^2}{\mu_0}} \]

### Step 6: Simplify the ratio
This simplifies to:
\[ \frac{u_E}{u_B} = \frac{\epsilon_0 (E_0)^2 \mu_0}{(B_0)^2} \]

### Step 7: Use the relation between \( E_0 \) and \( B_0 \)
For electromagnetic waves in free space, the relationship between the electric field and magnetic field is given by:
\[ E_0 = c B_0 \]
where \( c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \).

Substituting \( B_0 = \frac{E_0}{c} \) into the ratio:
\[ \frac{u_E}{u_B} = \frac{\epsilon_0 (E_0)^2 \mu_0}{\left(\frac{E_0}{c}\right)^2} = \frac{\epsilon_0 (E_0)^2 \mu_0 c^2}{(E_0)^2} \]

### Step 8: Cancel \( (E_0)^2 \) and simplify
This simplifies to:
\[ \frac{u_E}{u_B} = \epsilon_0 \mu_0 c^2 \]

### Step 9: Substitute \( c^2 \)
Since \( c^2 = \frac{1}{\epsilon_0 \mu_0} \):
\[ \frac{u_E}{u_B} = \epsilon_0 \mu_0 \cdot \frac{1}{\epsilon_0 \mu_0} = 1 \]

### Final Answer
Thus, the ratio of average electric energy density to average magnetic energy density is:
\[ \frac{u_E}{u_B} = 1 \]