Calculating Spatial Current in Resonant Inductors

[decaf14]

Hello,

I would like to replicate the results from the paper "wireless power transfer via strongly coupled magnetic resonances", but I'm having trouble interpreting their equations. I'm creating a MATLAB script to do so. In the paper, they get to a point where L and C are to be calculated in a self-resonating inductor. The equation is of the followng form:

Where J(r) is the spatial current density and p(r) is the charge current density. I'm confused on how to get spatial current density and charge current density. Previously, they mention that "the time dependent current profile has the form I0 * cos(pi * s / L) exp(i * w * t) where I0 is (I'm assuming) current input, s is a "parameterization coefficient" whatever that means (all i know is it varies in value from -l/2 to l/2 so it gives you the spatial location on the coil). "i" is imaginary number 1, L is total length, and t is time.

They jump right from this time dependent current profile into equations 3 and 4 I've listed above. Because they did not discuss methods for calculating J(r), I'm assuming the solution is trivial. However, I do not know where to even start for calculating J(r). Theoretical electromagnetics is not my strong suit. Any help would be greatly appreciated.

EDIT: to clarify a bit more, I understand that current density is simply current / area. However, the current varies at every point in space along the coil. The authors indicate that current is a function of radius, but previously they denote it as being a function of length or time, not radius. If it were to be a function of space, then time would need to be fixed. Even then, it should be a function of length, not radius, according to the equation they have given.

The paper can be found here: https://science.sciencemag.org/content/317/5834/83

 

[Delta2]

There is something wrong looking at those equations. I expect them to be volume integrals and not line integrals. The units of the LHS and RHS of the equations don't match . So are the differentials $dr,dr'$ that appear in those equations actually $d^3r,d^3r'$?

Unfortunately I don't have account access to check the paper myself.

 

[decaf14]

Nope. There is no mention of what kind of integral. They don't even name what " r' "actually means, but I'm assuming it's a derivative.

They sort of just introduce the equation out of nowhere. It's a very highly cited (2.6k citations) paper in the field, so it's a reputable paper. I just can't seem to figure what they're doing.

 

[Delta2]

No r' is not the derivative, its just the name of the second integration variable.

What does the paper say about L, is it the self inductance of the inductor and measured in units of Henry, or is it something else (self inductance per unit of length or per unit of volume perhaps?)

 

[decaf14]

No r' is not the derivative, its just the name of the second integration variable.

What does the paper say about L, is it the self inductance of the inductor and measured in units of Henry, or is it something else (self inductance per unit of length or per unit of volume perhaps?)

L is referred to simply as the "effective inductance ... for each coil" . No units are listed, but based on the context, I think it's simply units of henries. Especially since they list this equation for resonant frequency that we know to be true (taken directly as a .jpeg from the text):