Electric field strength from AC conducting wire in form of a loop along z-axis?

without a core you won't have a magnetic force to define a charge constants. the magnetization will propagate at the speed of light thus produce the desired effects of the loop which could be a near zero drop of E through the field. thus you can't possibly measure E within a free space loop without several orders of magnitude of currents. lets say dozens to hundred of amps. in that case the conductor itself produces such a resistance that you can measure E from the magnetic field it creates in the conductor itself.

I view a loop of wire, at low frequencies, as an excellent provider of displacement currents. I don't view the shape of the wire as a major influence (at low frequencies) on the interference generated.

Two parallel wires, radius 1mm and 1meter long and 1meter apart, have capacitance (ignoring the finite length) of

C = Eo * Er * 2 * PI * Length / natural_log[X * sqrt(X^2 -1)]

where

X = Num/Denom, Num is (Center-to-Center-separation)^2 for large separation, Denom is 2 * Radius2 * Radius2 [notice this is dimensionally neutral, thus is a fine argument to feed into a log-function]

Poking in the numbers, you'll see about 4pF between the 2 wires.

You start out with question about general shapes, then move to a specific (loop) shape.

Does this answer give you some insight?

I use this method to provide WORST CASE displacement currents from AC_power_cords that couple into sensitive nodes of signal chains. Or from switching power supply nodes (the drain of the MOSFET, switching 200 volts in 200 nanoSeconds).

If same amount of Q is flowing in the loop wire as out of it (I_in = I_out), then net charge of the loop is Q=0 and there is no E-field (But there is a H/B-field).

There can only be Q in the wire if I_in != I_out for some time. Then the loop wire is behaving like a capacitor and there is some Q != 0 left in the loop of wire.

The site you linked shows a setup where there is "a ring of charge". So there is no current I involved. (There maybe was a current I_in to bring that charge Q onto the ring).