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Two part question:

  1. If I have two coils of wire facing each other with all parameters for both being equal: Current, radius, # of turns, current traveling in same direction, etc. and there is a space between the two of them (fig 1), is the magnetic field geometry always going to just be a superposition of the dipole fields generated by each one? Or do the field lines merge into a single dipole field and thus make them equivalent to fig 2. If they do approximate to fig 2, is this determined by some limit for each parameter mentioned earlier? If so, how would that be determined mathematically? I know from electrodynamics that we can solve for electric fields of discrete charge density distributions using Greene's function. but I never applied it for B-fields and am pretty sure that doesn't work. If fig 1 does approximate to fig 2, what mathematical tools can we use to try to solve for that geometry similarly to the Greene's function for charges?

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  1. If I have an even more complicated system now (fig 3), where I have two pairs of coils facing each other and each pair is perpendicular wrt each other, will those field lines merge and combine into some uniform, single continuous geometry, or will it still be a superposition of now, 4 dipoles?
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I think there is some confusion in this question. There is a distinction being made between a "superposition" of fields and a "single continuous geometry." The word superposition has a different meaning depending on whether one is talking about classical fields (like the electromagnetic fields of Maxwell's equations) and the quantum states of quantum mechanics. This question belongs in the first category. There is no uncertainty in the magnetic field as in a quantum superposition.

A superposition of classical fields results in a single continuous geometry. There is no difference between these two concepts. In Figure 1 of the question, the fields created by the two dipoles will sum together into a continuous field. At every point in space, there will be a single magnetic field vector with a single direction and a single magnitude. That vector can be found by adding the magnetic field vectors that would be created by each coil of wire alone.

Given the configuration of the coils in Figure 1, it will create a different magnetic field than Figure 2 because of there being two coils with a gap instead of a single coil. But, if you look at points far away from the coils at distances much larger than the gap between the coils, the field will be well approximated by a single-coil dipole field. This is just like the fact that the electric field of any charge distribution that takes up finite space looks like a point charge from far enough away. The same goes for the gravitational field of a finite arbitrary mass distribution.

For the second question, the fields from the four individual coils will form a single continuous geometry. That single continuous geometry will be the result of adding the fields of the individual coils in superposition. This will form a field geometry called a quadrupole field. This arrangement is often used in particle accelerators to focus the beam like a lens.

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  • $\begingroup$ Thank you for the explanation! After looking at the quadrapole link, I delved further and found that the name for figure 1's config is Helmholtz coils. I was able to understand this a little more with the math involved for that too what's going on $\endgroup$
    – Sophia
    Mar 24, 2023 at 2:26
  • $\begingroup$ Also, I'm pretty sure I can use the term superposition outside of just the qm context and it still being valid. I double checked the definition just to make sure I was using the correct term and it seems that way: "the action of placing one thing on or above another, especially so that they coincide" $\endgroup$
    – Sophia
    Mar 24, 2023 at 2:26
  • $\begingroup$ @Sophia Glad I could help. I didn't mean to say that you were wrong about using the term "superposition." I just didn't understand the distinction you were making between superposition and a continuous geometry. $\endgroup$
    – Mark H
    Mar 24, 2023 at 7:33

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