Nuclear Fusion Physics and Technology/Example page

Note: Electromagnetic field

As summarized in previous chapters, electromagnetic field is mathematical abstraction of two projections $\overrightarrow{E}(\overrightarrow{r}):{ℝ}^3\to {ℝ}^3$ and $\overrightarrow{B}(\overrightarrow{r}):{ℝ}^3\to {ℝ}^3$, which meets Maxwell equations

$rot\overrightarrow{H}=\overrightarrow{j}+\frac{\partial \overrightarrow{D}}{\partial t}rot\overrightarrow{E}+\frac{\partial \overrightarrow{B}}{\partial t}=0$
$div\overrightarrow{B}=0\overrightarrow{D}=\rho$

and may be represented by field lines defined as

$\frac{d\overrightarrow{x}}{ds}=\alpha \overrightarrow{B}(\overrightarrow{r})$

Definition: Open field line

Field line is open, when it is not closed in plasma.

Definition: Closed field line

Field line is closed, when it is closed in plasma.

Theorem: Magnetic field line equation

Lets assume electromagnetic field $\overrightarrow{E}(\overrightarrow{r}),\overrightarrow{B}(\overrightarrow{r})$ with field lines. Then

$\frac{d{l}_x}{B_x}=\frac{d{l}_y}{B_y}=\frac{d{l}_z}{B_z}$

Proof:
The theorem results from field line definition directly

$\frac{d\overrightarrow{x}}{ds}=\alpha \overrightarrow{B}(\overrightarrow{r})/.\frac{ds}{\overrightarrow{B}}$ $\frac{d\overrightarrow{x}}{\overrightarrow{B}}=\alpha .ds$

which is a vector equation of three scalar equations

$\frac{d{l}_x}{B_x}=\alpha .ds\frac{d{l}_y}{B_y}=\alpha .ds\frac{d{l}_z}{B_z}=\alpha .ds$

and thus may be written

$\frac{d{l}_x}{B_x}=\frac{d{l}_y}{B_y}=\frac{d{l}_z}{B_z}Q.E.D.$