harrylin said:
That may be so, but I found that such textbooks simply cite Poynting on this:
https://en.wikisource.org/wiki/On_the_Transfer_of_Energy_in_the_Electromagnetic_Field
He did however add the precision that it "expresses the energy transformed by the conductor into heat, chemical energy, and so on".
I think the writer is technically correct the way he said it. However, the thought expressed is a incomplete.
When the electric current satisfies the original Ohm's Law, where conductivity is always a real quantity, then in some sense E.j is the rate at which internal energy is changing. "Internal energy" is often referred to as "heat", although this is inconsistent with the way the word "heat" is used in the laws of thermodynamics.
A reason that this thought is incomplete is that not all electric currents satisfy Ohms Law. Here is an example. Suppose one had an object which was an insulator. and the center of the insulator was electrically charged. The object is immersed in salt water. When I say insulator, I mean that it is an insulator both electrically an thermally. Therefore, electrical current can't pass through the object. The object can't contain its own internal energy.
A potential difference is applied across the tank which contains the charged object. A constant electric field is applied both to the salt water and the object. The electric field is applied to both the ions in the water and the insulating object.
The electric current passing through the salt water may satisfy Ohm's Law. I don't think it does precisely due to electrolytic chemistry. However, I hypothesize the the current passing through the salt water satisfies Ohm's Law where the salt water has a constant conductivity.
The electric field applies a force to the insulated object. Therefore, the insulated object moves through the water. I hypothesize that the object was initially stationary. The velocity of the object is small enough that viscosity doesn't play a role. The object accelerates in response to the electric field.
There are two different types of electric current here. The current that passes through the salt water and the current caused by the moving charge density in the center of the object.
E.j in the salt water probably does turn into internal energy of the water. However, the insulated object is acting like any electrically charge body. The movement of the object doesn't immediately turn into internal energy. Some of E.j goes into the kinetic energy of the insulating object.
So the part of the current that satisfies the original Ohm's law really does heat the water, in the sense of internal energy. Part of the current that doesn't satisfy Ohm's Law turns into kinetic energy. So what happened?
Poynting's theorem includes kinetic energy. However, it does not discriminate between the part of the kinetic energy that is in the internal energy, and the part of the kinetic energy that is macroscopic. So the decision on how to partition the kinetic energy has to be made by the constitutive equations and the force laws.
Conductivity includes information on the microscopic states of the salt water. Conductivity is a macroscopic property that is merely an ensemble average of microscopic properties. So in a sense, conductivity is defined in terms of the internal energy. So any time you use a conductivity as a parameter, you are deciding what part of the kinetic energy is internal energy. The assignment of conductivity is part of the definition of internal energy.
The insulated object has no internal energy. The most important parameter with regards to the insulating object is the center of mass. So the kinetic energy of the insulating object is primarily a macroscopic quantity. So the kinetic energy of the insulating object can not be part of the internal energy.
The rate at which the insulated object is gaining kinetic energy is determined by the Lorentz force law. Applying the Lorentz force law to the insulating object implies a length scale. Large objects are not part of the conductivity. Large objects by definition can be characterized by the Lorentz force law. So Ohm's Law and the Lorentz force law are basically constitutive equations that imply a length scale.
There has to be a length scale that determines how the kinetic energy is partitioned. The constitutive parameters implicitly contain the length scale.
Maybe in our discussion we should discriminate between macroscopic kinetic energy and microscopic kinetic energy. Conductivity tells us how fast the microscopic kinetic energy is changing. However, the Lorentz force law tells us how the macroscopic kinetic energy is changing.
Poynting's theorem has no length scale. It has no thermodynamics. In a sense, it has no "heat". Extra hypotheses have to be thrown in if you want a meaningful analysis of "heat".
Anyway, that's not my problem and neither does it matter for this thread - as long as everyone agrees that currents can be induced! 
