Why does generalized Ohm's law give non-zero divergence of current density?

[mertcan]

initially my attachment/picture has been cut off that link http://sun.stanford.edu/~sasha/PHYS780/PLASMA_PHYSICS/phys780_2014_l13.pdf page 6

Also I would like to put into words that divergence of current density is accepted as 0 in continuous loop( no capacitors exist...). But if you look at the generalized ohm law divergence of current density seems to be not zero. How is that possible?

 

[NFuller]

Also I would like to put into words that divergence of current density is accepted as 0 in continuous loop( no capacitors exist...). But if you look at the generalized ohm law divergence of current density seems to be not zero.

Wouldn't the generalized law need to work for capacitors (or other current sources and sinks) in order to be general? In those cases, you would need a term with non-vanishing divergence.