## 3 Transfer and Conversion of Field Power

### 3.1 Poynting’s Theorem: Poynting Vector^{*}

The basis for investigating energy motion in electromagnetic fields is Poynting’s theorem [3.3] and the Poynting vector (Turowski [1.15], [3.4]). Poynting’s theorem says that:

The electromagnetic power *P*_{s} flowing into (or flowing out of) a closed volume, across an enclosing surface *A*, equals to the surface integral of the normal component of the Poynting vector *S*_{n} over the entire enclosed surface *A*:

$${\mathit{P}}_{\text{s}}=\underset{A}{\u222f}\mathit{S}\cdot \text{d}A=\underset{A}{\u222f}{\mathit{S}}_{\text{n}}\text{d}A\left(3.1\right)$$

where

$$\mathit{S}=\mathit{E}\times \mathit{H}\left(3.2\right)$$

is the Poynting vector, which determines the power and direction of the electromagnetic power flux passing through a surface unit perpendicular to the energy flow direction. Of course,

$${\mathit{S}}_{\text{n}}={\mathit{E}}_{\text{t}}\times {\mathit{H}}_{\text{t}}\left(\mathrm{3.2a}\right)$$

Poynting’s theorem can ...

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