Imagine 2 parallel antennas (wires) of equal length (a) with a distance r between them.

Both have AC currents with identical sine wave forms (equal frequencies and amplitudes) . They are also in-phase with each other.

Edit: The distance between them (r) is equal to half the wavelength due to the frequency of AC, (r=λ/2) so that there's no phase delay between them.

Editting the edit above: It turned out that (thanks to Void for making me aware of this), another restriction concerning the "near field equation" for the antennas is necessary here. The distance r should be far enough to let the EM fields to "settle down" and become more like plane waves before they hit the other antenna/wire. The minimum distance for this is calculated as,

enter image description here unless I misunderstood anything.

For the distance between the wires therefore, I assume the two conditions below are satisfied:

enter image description here

I’m trying to calculate the average force between these wires (which must be an attractive force) by using very simple math: I know the magnetic field at point p, caused by a current of finite length wire is given by,

enter image description here

By integrating the magntetic field of I1 over a length of an opposite wire and multiplying with the current I2 at each point of that opposite finite wire, I think I found the magnetic force acting on each wire:

enter image description here

Now I have to use this equation in an AC situation. Currents are in phase, so I thought their average multiplication (averaged by time) could be calculated by dividing the area under the curve of current squared, by the time of half a period (which is simply pi).

enter image description here

So the average force between wires should be:

enter image description here

No need to complicate this equation any further by defining the angle alpha in terms of lengths a and r.

I brought different pieces of knowledge together to form a new one and when I do this, it usually turns out to be wrong so I can’t be confident about this equation.

I have two concerns (edit: now a third one):

1- I'm not sure if I calculated the area under curve correctly.

2- If two such antennas applied a force to each other, this force should somehow be related to the antenna power or antenna gain. But when calculating the antenna power, I see that not only the magnetic field but also the electric field is taken into account. It becomes more than a simple DC "Amperian situation”. It’s electromagnetic. But where is the electric field in this method? Am I missing something?

3- About the "near field" approach; as far as I understand, because the instantaneous current and voltage is not the same at various segments of the antenna during the oscillations, the EM field is also complicated at close range relative to the antenna length and also relative to the gradients of instantaneous voltage and current values over the antenna (which is realted to the frequency and wavelength). But I can't be certain about the equation I used. According to the equation, the antennas could be as close as about 1/2 or 1 wavelength. However in practice, the minimum distance is often told to be greater than 2 to 10 wavelengths to stay away from the near-field (aka. the reactive field). So how would be the correct equation for the "near field" boundary condition?

  • $\begingroup$ This might be more complicated than you think. When you have AC current flowing, the force is not necessarily attractive. If they are odd integer wavelength apart, their phases might be inverted. I believe that it would be possible to find an analytical result for the force per unit length of two infinitely long wires. $\endgroup$ Nov 22 '16 at 1:20
  • $\begingroup$ I meant half integer wavelength apart. $\endgroup$ Nov 22 '16 at 1:26
  • $\begingroup$ Right. I forgot about the influnce of time and phase delay. Thanks for pointing that. I will edit now. $\endgroup$
    – Xynon
    Nov 22 '16 at 9:16
  • 1
    $\begingroup$ Interesting question. 1) I would suggest evaluating the integral of the electrical waveform by expressing the signal as function of sine with the limits being a function of it's argument. 2) It is indeed electromagnetic. Although the magnetic field around the wire wont just be defined by the gradient of the electric field as it extends from the wire, but also by the oscillation from the AC current. So we apply boundary conditions for the electric field to Maxwell's equations to express the amplitude of the AC current. The E and B fields would then be a function of the AC phase. $\endgroup$
    – user97261
    Nov 24 '16 at 17:44
  • $\begingroup$ Your computation is correct if the quasistationary approximation holds. However, this approximation holds only as far as the typical wavelength $\lambda=c/f$ is much larger than the dimension of the system, i.e. when the wave nature of the electromagnetic field is negligible. But the exact point of antennae is that they are devices trapping and producing waves. What you must then do is to include the full set of Maxwell equations in computing both the E and B fields. $\endgroup$
    – Void
    Nov 25 '16 at 10:20

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