Poynting vectors `vec(S)` is defined as `vec(S)=(1)/(mu_(0))vec(E)xx vec(B)`. The average value of `'vec(S)'` over a single period 'T' is given by

Options

A. `E_(0)^(2)//2cmu_(0)`
B. `E_(0)^(2)//cmu_(0)`
C. `2E_(0)2^(2)//c mu_(0)`
D. `E_(0)^(2)//cmu_(0)^(2)`

Explanation

To find the average value of the Poynting vector \(\vec{S}\) over a single period \(T\), we start with the definition of the Poynting vector:

\[ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \]

### Step 1: Express the Electric and Magnetic Fields
Assuming the electric field \(\vec{E}\) and magnetic field \(\vec{B}\) can be expressed as:

\[ \vec{E} = E_0 \cos(kx - \omega t) \hat{i} \]
\[ \vec{B} = B_0 \cos(kx - \omega t) \hat{j} \]

where \(E_0\) and \(B_0\) are the maximum values of the electric and magnetic fields, respectively.

### Step 2: Calculate the Cross Product
Now, we calculate the cross product \(\vec{E} \times \vec{B}\):

\[ \vec{E} \times \vec{B} = (E_0 \cos(kx - \omega t) \hat{i}) \times (B_0 \cos(kx - \omega t) \hat{j}) \]

Using the right-hand rule, we find:

\[ \vec{E} \times \vec{B} = E_0 B_0 \cos^2(kx - \omega t) \hat{k} \]

### Step 3: Substitute into the Poynting Vector
Substituting this result into the expression for \(\vec{S}\):

\[ \vec{S} = \frac{1}{\mu_0} (E_0 B_0 \cos^2(kx - \omega t)) \hat{k} \]

### Step 4: Average Value Over One Period
To find the average value of \(\vec{S}\) over one period \(T\), we need to calculate:

\[ \langle \vec{S} \rangle = \frac{1}{T} \int_0^T \vec{S} \, dt \]

Substituting for \(\vec{S}\):

\[ \langle \vec{S} \rangle = \frac{1}{\mu_0} E_0 B_0 \left( \frac{1}{T} \int_0^T \cos^2(kx - \omega t) \, dt \right) \hat{k} \]

### Step 5: Evaluate the Integral
The average value of \(\cos^2\) over one period is:

\[ \int_0^T \cos^2(kx - \omega t) \, dt = \frac{T}{2} \]

Thus, we have:

\[ \langle \vec{S} \rangle = \frac{1}{\mu_0} E_0 B_0 \left( \frac{1}{2} \right) \hat{k} \]

### Step 6: Relate \(B_0\) to \(E_0\)
Using the relationship between the electric and magnetic fields in electromagnetic waves:

\[ B_0 = \frac{E_0}{c} \]

where \(c\) is the speed of light. Substituting this into the equation gives:

\[ \langle \vec{S} \rangle = \frac{1}{\mu_0} E_0 \left(\frac{E_0}{c}\right) \left( \frac{1}{2} \right) \hat{k} \]

### Step 7: Final Expression
This simplifies to:

\[ \langle \vec{S} \rangle = \frac{E_0^2}{2 \mu_0 c} \hat{k} \]

### Conclusion
Thus, the average value of the Poynting vector over a single period \(T\) is:

\[ \langle \vec{S} \rangle = \frac{E_0^2}{2 \mu_0 c} \hat{k} \]