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My electromagnetics textbook has a picture of an RC circuit with a current source.

screenshot of textbook

As you can see, there are three current density vectors, J_i, J_c, and J_d. The book labels them as follows:

J_i = impressed (source) electric current density

J_c = conduction electric current density

J_d = displacement electric current density = partial D / partial t, where D is electric flux density

It then uses this figure to explain the following Maxwell equation:

curl of H = J_i + J_c + J_d, where H is magnetic field intensity

What I don't understand is... Isn't J_i the same as J_c? I'm getting this from the idea that if you do a Kirchhoff Current Law evaluation at the top left node (between the source and resistor), then J_i is going in and J_c is going out, thereby making them equal? ... So wouldn't this mean that J_i and J_c are the same current? And if they're the same current, then doesn't that mean that the Maxwell equation is counting it twice?

I'm looking for a solid explanation of why J_i is different than J_c, despite Kirchhoff's Current Law. Thanks!

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  • $\begingroup$ $J_i$ might have a different value then $J_c$ if the source and the conductor(resistor) have different cross sectional areas. I suspect the point is to try to make displacement current less mysterious by pointing out that there are other different kinds of current that you might not normally think of. You're not likely to ever hear about "impressed current" again. $\endgroup$ – The Photon Feb 8 '17 at 5:22
  • $\begingroup$ Even if the cross sectional areas are different, KCL should still make J_i = J_c, right? $\endgroup$ – Mark Betters Feb 8 '17 at 19:39
  • $\begingroup$ No, KCL makes $I_i=I_c$. Think about a thick pipe running into a skinny pipe. The total amount of water flowing out of the thick pipe has to equal the total amount of water flowing in to the skinny pipe. For that to happen, the current per mm^2 of cross section has to be more in the skinny pipe. $\endgroup$ – The Photon Feb 8 '17 at 19:59
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It seems that this picture is from Balanis Advance Engr E&M. Here is a somewhat intuitive explanation taken from Harrington's book:

"In source-free media, We identify $J = J_c$ as the conduction current 'caused by' the field"

An antenna would be a good example for conduction current density, which has no source current density.

In your picture, the three currents are all equal. They have different names because they are different types of current density, but they would have been the same thing if this picture appeared in a circuits textbook.

As for $J_{ic}$. Physicists defined too many names and they got sick of tracking them. So they add the two so that they have one convenient quantity to work with.

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