Determining the direction and origin of the Hall current

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TheCanadian
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A Hall current arises when electric currents transverse to a magnetic field exist. In this figure (found online), the Faraday current appears to be going upwards due to a magnetic field pointing into the page, but is this correct?

I'm largely just trying to understand the origin of the Hall current here: The Faraday current density, ##{\bf{J_F}} = \sigma \bf{v} \times \bf{B}##, is upwards since ##\bf{v}## is to the right and ##\bf{B}## into the page. (I realize the conductivity tensor is anisotropic due to the Hall effect, but I'm trying to understand how the Hall effect itself arises here.) Thus the force density from the Hall effect is ##\bf{F_H} = \bf{J_F} \times \bf{B}##, but this force associated to the Hall effect is pointing to the left and independent of charge, correct? It thus slows down the overall quasi neutral initial fluid flow. Thus why is a Hall effect current in the rightward direction created?

Also, perhaps this is a separate question but possibly related: why are the external current links (orange) in this configuration necessary? Why are they even connected in the first place for the generator? This appears to be a consequence of the Hall effect, but I'm unsure. Shouldn't only the electrodes connected by the light blue arrows be externally connected in a circuit since that's the direction of net current flow?

mhd_current.gif
 

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