Toroidal coordinates: Difference between revisions
→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>, 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>, 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} = \ | \mathbf{B} = \beta\nabla\psi + \nabla\chi | ||
</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 | ||
:<math> | :<math> | ||
\chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) | \chi(\psi, \theta, \phi) = \frac{I_{tor}}{2\pi}\theta + \frac{I_{pol}^d}{2\pi}\phi + \tilde\chi(\psi, \theta, \phi) | ||
</math> | |||
In fact, noting that | |||
:<math> | |||
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | |||
= \int_{\partial S}\mathbf{B}\cdot d\mathbf{l} | |||
= \oint(\beta\nabla\psi + \nabla\chi)\cdot d\mathbf{l} | |||
= \oint(\beta d\psi + d\chi) | |||
</math> | |||
and choosing an integration circuit contained within a flux surface we get | |||
:<math> | |||
\int_S \mu_0\mathbf{j}\cdot d\mathbf{S} | |||
= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~. | |||
</math> | |||
If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get | |||
:<math> | |||
I_{pol}^d = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 0, \Delta\phi = 2\pi)~. | |||
</math> | </math> | ||