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  1. In a microscopic classic picture, ohmic heating occurs due to collisions. Let's start with the simplest case: a cooper wire and a battery that induces a current. Electrons start flowing and the energy gained by the field that is accelerating them is lost through collisions with the cooper atoms. I think we all agree that the energy transfer to the Cu atoms through these collisions is what makes the wire heats up.

  2. Now suppose we have a plasma, namely a fusion plasma in a magnetic confinement device. It is well known that because the plasma resitivity goes with $T_e^{-3/2}$, then ohmic heating is not really efficient to heat the plasma up to the desired temperature. I have two questions regarding this:

    2a) I suppose plasma current is mainly carried by electrons which are much more mobile. Then, it is the electrons that are being accelerated by the field. They make (coulomb) collisions with the ions, so ions will gain energy and increase their temperature. Because these electrons are, in the overall process, not gaining any energy, shouldn't their temperature stay the same? Why is it that $T_e$ also increases?

    2b) Why is it said that ohmic heating is a loss process when one makes the plasma energy balance? Because all the energy that is being given to the elctrons in the current by the field stays in the plasma because electrons transfer their energy to the ions and ultimately also to other electrons with whom they can also collide.

EDIT: I realized that my second question (2b) didn't make sense: I was making confusion with something else (see comments below). The first question is perfectly answered by @honeste_vivere

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2a) I suppose plasma current is mainly carried by electrons which are much more mobile. Then, it is the electrons that are being accelerated by the field. They make (coulomb) collisions with the ions, so ions will gain energy and increase their temperature.

First some background information is necessary based upon some of your comments. A resistivity is a type of drag force that acts to inhibit or limit the relative drift between oppositely charged species, i.e., currents. The drift that leads to currents is a bulk flow, i.e., the first velocity moment of the constituent species (e.g., see discussion at https://physics.stackexchange.com/a/341352/59023 or https://physics.stackexchange.com/a/235549/59023).

The current density in a kinetic gas like a plasma is defined as: $$ \sum_{s} \ q_{s} \ n_{s} \mathbf{v}_{s} = \mathbf{j} \tag{1} $$ where $q_{s}$ is the charge of species $s$, $n_{s}$ is the number density of species $s$, and $\mathbf{v}_{s}$ is the bulk flow velocity of species $s$ in the center of momentum frame for the entire plasma (well, technically one can define the current in any reference frame they wish but this would be the physically meaningful one).

Because these electrons are, in the overall process, not gaining any energy, shouldn't their temperature stay the same?

Not necessarily, as I illustrated at https://physics.stackexchange.com/a/268594/59023 it is not required for a plasma to have $T_{e} = T_{i}$ and in many plasmas, there are multiple components per species (e.g., counter-streaming proton beams).

Why is it that $T_{e}$ also increases?

If we assume that the only inhibiting force on the particles is due to Coulomb collisions, then the bulk flow velocities responsible for the currents will be converted to increased thermal speeds to conserve energy. That is, the electrons scatter during the Coulomb collisions as well as the ions, thus it is not just that their bulk flow velocity reduces.

2b) Why is it said that ohmic heating is a loss process when one makes the plasma energy balance?

Because the current shows up on the right-hand side of the electromagnetic energy continuity equation, called Poynting's theorem. The sign of the inner product between the current density and electric field determines whether it is a loss or source mechanism, i.e., if negative(positive) one can say that the particles lose(gain) energy/momentum to(from) the fields.

Because all the energy that is being given to the elctrons in the current by the field stays in the plasma because electrons transfer their energy to the ions and ultimately also to other electrons with whom they can also collide.

Currents act as the source for magnetic fields much like electric charge is a source for electric fields. If the plamsa in which a current exists has a high resistivity, then it will be difficult to maintain currents without significant external work. If the currents are maintained in such a system, then the plasma will heat up (e.g., see https://physics.stackexchange.com/a/348211/59023).

I go into a little more detail about Joule/Ohmic heating at https://physics.stackexchange.com/a/180680/59023.

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    $\begingroup$ 2a) So, the bulk velocity is being converted into thermal velocity due to collisions. And because Te is a measure of the "thermal energy", then Te will increase. Is that it? 2b) Just like you refer: "the bulk flow velocities will be converted to increased thermal speeds to conserve energy". So if energy is conserved, I still don't get why ohmic heating is a loss process when making PLASMA power balance. I understand that the field is loosing energy to the plasma by accelerating particles and then the enerrgy stays in the system after the collisions. So the energy doesn't leave the plasma... $\endgroup$ – AJHC Feb 28 '18 at 15:26
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    $\begingroup$ @AJHC - (2a) Yes, that is it. (2b) It is called a loss process because it is irreversible and generates entropy. You are taking EM energy and converting it to particle heat. Does that make more sense? $\endgroup$ – honeste_vivere Feb 28 '18 at 16:31
  • $\begingroup$ But from the point of view of the PLASMA (not the all system plasma+transformer, whatever that drives the current), isn't that a gain? $\endgroup$ – AJHC Feb 28 '18 at 18:27
  • $\begingroup$ @AJHC - Well, I am not sure what you are asking but I think the answer is "not really." The use of the word "loss" vs. "gain" or "source" derives from the general form and purpose of the continuity equation. One has the time variation of a density plus the divergence of a flux through a surface equal to the net total of sources and losses on the right-hand side. $\endgroup$ – honeste_vivere Feb 28 '18 at 19:16
  • $\begingroup$ I was actually referring to the energy equation: the 2nd moment of Boltzman equation. In the rhs of this equation you have the sources $S$. For a plasma in a fusion reactor you usually will wave: $S=S_h+S_f-S_b-S_k-S_{\Omega}$ where $S_h$ is the heating power, $S_f$ is the fusion power (the energy that comes from the fusion reactions), $S_b$ are the radition losses, $S_k$ the conduction losses and $S_{\Omega}$ the losses due to ohmic heating. And it's precisely $S_{\Omega}$ that I don't understand why is it a loss. $\endgroup$ – AJHC Feb 28 '18 at 19:31

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