Magnetohydrodynamic Generator (POPTOR 2005)

[EFuzzy]

There was an interesting problem that appeared in POPTOR 2005:

A generator consists of a parallel-plate capacitor immersed in a stream of conductive liquid with conductivity $$\sigma$$. The surface area of a capacitor's plate is S, the distance between the plates is d. A liquid flows with constant velocity v parallel to the plates. The capacitor is in a uniform magnetic field B, which is perpendicular to the velocity and parallel to the plates. If the plates are connected to a resistor R, what is the current that flows through the resistor?

I understand that the free charges inside the conductive liquid will feel a magnetic force, and thus begin "build up" on the plates. "Build up" is in quotation marks because I think that the charges are still moving with the liquid flow, but are right up against the plates. This will continue until an electrostatic field is established that perfectly cancels out the magnetic forces acting on the free charges in the moving fluids. Thus the voltage of the capacitor by itself would be vBd. I would then divide that voltage by R to get the current through the resistor.
However, the solution that POPTOR gives is that the capacitor is in series with both R and the "internal resistance" of the fluid. Why does the resistance of the fluid matter? The voltage between the two plates is vBd regardless of the fluid inside.

 

[Troels]

However, the solution that POPTOR gives is that the capacitor is in series with both R and the "internal resistance" of the fluid. Why does the resistance of the fluid matter? The voltage between the two plates is vBd regardless of the fluid inside.

It would be because the fluid is the final resistor that completes the circuit. Every voltage source has an "internal resistance" in series with it's electromotive force and it should be included for Ohm's law to accurately determine the resistance of the circuit.

 

[EFuzzy]

I guess the problem is that I don't fully understand how the internal resistance works. The voltage difference from one plate to the other is going to be vBd, so why does it matter what's going on between the plates? All the resistor will notice is that the capacitor provides some constant voltage.

 

[Troels]

I guess the problem is that I don't fully understand how the internal resistance works. The voltage difference from one plate to the other is going to be vBd, so why does it matter what's going on between the plates? All the resistor will notice is that the capacitor provides some constant voltage.

I should perhaps say, that I'm just having a shrewd guess at thinks here, so I welcome other opinions as well.

It is correct that the capacitor supplies a constant voltage, but the emf still has to overcome the resistance in the fluid, just like a conventional generator has to overcome the resistance in the copper wires in the coils. It matters not very much where in the circut the emf is generated, the total resistance is still the sum of all serial resistances in the circiut.