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I guess this is maybe more of a vector calculus question, but here it goes. Say I have an arbitrary current distribution [itex]\vec{J}[/itex] with the corresponding magnetic field given by

[tex]\vec{\nabla}\times\vec{H} = \vec{J}[/tex]

[tex]\vec{\nabla}\cdot\vec{H} = 0[/tex]

What is the best way to calculate the field/flow lines of [itex]\vec{H}[/itex] (contours to which [itex]\vec{H}[/itex] is always tangent)? Do I need to calculate the magnetic field and then find the field lines from that, or is there an easier way to directly extract information about the geometry of the field lines from the equations above if I don't care about the magnitude of the magnetic field?

Edit: If it makes things easier, I'd still be interested to see the answer to this question with the added assumption that everything is uniform in the z-direction and [itex]\vec{J}[/itex] only has a z component (i.e. the magnetic field is 2D: it has no z-dependence and no z-component)

[tex]\vec{\nabla}\times\vec{H} = \vec{J}[/tex]

[tex]\vec{\nabla}\cdot\vec{H} = 0[/tex]

What is the best way to calculate the field/flow lines of [itex]\vec{H}[/itex] (contours to which [itex]\vec{H}[/itex] is always tangent)? Do I need to calculate the magnetic field and then find the field lines from that, or is there an easier way to directly extract information about the geometry of the field lines from the equations above if I don't care about the magnitude of the magnetic field?

Edit: If it makes things easier, I'd still be interested to see the answer to this question with the added assumption that everything is uniform in the z-direction and [itex]\vec{J}[/itex] only has a z component (i.e. the magnetic field is 2D: it has no z-dependence and no z-component)

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