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Frozen-in Aproximation





The equation which governs the behavior of magnetic fields in a plasma is the induction equation:
 
 

\begin{displaymath}{\partial {\bf B}\over\partial t}=\nabla \times ({\bf v\times B })+ \lambda \nabla^2 {\bf B}\end{displaymath}






In order to know which of the right side terms dominates the ratio between the two can be found which is aproximately:
 
 

\begin{displaymath}Rm= {{{VB}\over L}\over{{\lambda B}\over L^2}} = {VL\over \lambda}\end{displaymath}







which  if greater than one will mean that the first term rules the behavior of the magnetic field so the induction equation can be written as:
 
 

\begin{displaymath}{\partial {\bf B}\over\partial t}=\nabla \times ({\bf v\times B })\end{displaymath}







Which  has the same form of the vorticity equation, so in analogy with the Kelvin's vorticity theorem we can show that the magnetic field satisfies:
 
 

\begin{displaymath}{d\over dt}\int_{S}{\bf B}\cdot d{\bf S}=0\end{displaymath}






Which means that the magnetic field remains "frozen" and moves with the plasma material.
 
 
 
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