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Flux coordinates: Difference between revisions
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→Covariant Form
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If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as | If we consider an equilibrium magnetic field such that <math> \mathbf{j}\times\mathbf{B} \propto \nabla\psi</math>, where <math> \mathbf{j}</math> is the current density , then both <math> \mathbf{B}\cdot\nabla\psi = 0</math> and <math> \nabla\times\mathbf{B}\cdot\nabla\psi = 0</math> and the magnetic field can be written as | ||
:<math> | :<math> | ||
\mathbf{B} = -\eta\nabla\psi | \mathbf{B} = \nabla\chi -\eta\nabla\psi | ||
</math> | </math> | ||
where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is | where <math>\chi</math> is identified as the magnetic ''scalar'' potential. Its general form is | ||
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\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | \int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | ||
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l} | = \int_{\partial S}\mathbf{B}\cdot d\mathbf{l} | ||
= \oint(-\eta\nabla\psi | = \oint(\nabla\chi-\eta\nabla\psi)\cdot d\mathbf{l} | ||
= \oint(-\eta d\psi | = \oint(d\chi-\eta d\psi ) | ||
</math> | </math> | ||
and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get | and choosing an integration circuit contained within a flux surface <math>(d\psi = 0)</math>. Then we get | ||