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Toroidal coordinates: Difference between revisions
→Covariant Form
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= \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~. | = \Delta \chi = \frac{I_{tor}}{2\pi}\Delta\theta + \frac{I_{pol}^d}{2\pi}\Delta\phi~. | ||
</math> | </math> | ||
[[Image:CurrentIntegrationCirtuits.png|thumb|right|alt=Sample integration circuits for the definitions of currents.|Sample integration circuits for the current definitions.]] | |||
If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get | If we now chose a ''toroidal'' circuit <math>(\Delta\theta = 0, \Delta\phi = 2\pi)</math> we get | ||
:<math> | :<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)~. | 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> | ||
Similarly | |||
:<math> | |||
I_{tor} = \int_S \mu_0\mathbf{j}\cdot d\mathbf{S}\; ; ~\mathrm{with}~ \partial S ~\mathrm{such~that}~ (\Delta\theta = 2\pi, \Delta\phi = 0)~. | |||
</math> | |||
== Magnetic coordinates == | == Magnetic coordinates == | ||