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I know that the Lorentz force formula is $$F=\iiint \rho(E+J\times B)dV$$ for continuous charge distribution. But is E the electric field that creates the current (density) J, or this is another electric field that is pushing the charge distribution in a direction? If I have a rectangular pipe with a pair of electrodes on opposite walls submerged in salt water, and an electromagnet on the wall of the pipe creating a magnetic field perpendicular to the current direction between the electrodes, how to use this formula to calculate the force exerted on the salt water inside the pipe between the electrodes?

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The Lorentz force on a charge is caused by the E and B fields at that point from all other charges/currents. You integrate to calculate the total force on a collection of charges.

So yes, the current creates a B field, and that B field can affect charges comprising the current. That is you can think of the current in the tank as being distributed among a lot of parallel wires. Each wire generates a magnetic field that exerts forces on all the other wires.

Currents in parallel wires attract each other. You might expect the currents in the tank to bunch toward the center.

But it isn't so simple. Design of electroplating tanks for uniform distribution is something of an art. Electrode placement, resistivities of solution and electrodes, and shape are important. There are patents and commercial software packages for this.

For example, See Electrolytic Tanks, this 1954 article Current Density Distribution in Electroplating by Use of Models

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