Nuclear Fusion Physics and Technology/Example page

Note: Electromagnetic field

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

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

and may be represented by field lines defined as

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

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 ${\vec {E}}({\vec {r}}),{\vec {B}}({\vec {r}})$ with field lines. Then

${\frac {dl_{x}}{B_{x}}}={\frac {dl_{y}}{B_{y}}}={\frac {dl_{z}}{B_{z}}}$

Proof:
The theorem results from field line definition directly

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

which is a vector equation of three scalar equations

${\frac {dl_{x}}{B_{x}}}=\alpha .ds\qquad {\frac {dl_{y}}{B_{y}}}=\alpha .ds\qquad {\frac {dl_{z}}{B_{z}}}=\alpha .ds$

and thus may be written

${\frac {dl_{x}}{B_{x}}}={\frac {dl_{y}}{B_{y}}}={\frac {dl_{z}}{B_{z}}}\qquad Q.E.D.$