While attempting to express the magnetic field induced by a single coil of current (at any point in space, not just on the coil's axis), I tried visualising the set of the infinitesimal contributions $\text{d}\mathbf{B}$ given by the Biot-Savart law at specific points. The goal is to study the shapes and curves produced, and potentially help me come up with a geometrical solution to the integral, as the integral would be the sum of all those vectors.

I was kind of expecting to get oblique or elliptic cones (which would've been encouraging to express the resulting vector), since on the axis the sum of the $\text{d}\mathbf{B}$s form a perfect cone, as shown partially here, to help comprehension. One has to imagine the element of current doing one revolution, and adding all the magnetic fields.

Magnetic field vectors for a point on the coil's axis

Instead, depending on the point we're evaluating the field at, we obtain a variety of shapes (or curves, if we only look at the trace of the tip of the vector's arrow), four of which are shown below. Those curves are not planar (i.e. not a subset of a plane), which was a bit surprising to me.

Since a coil of current is one of the most basic systems we can design, I expect those curves to have been studied, hence my question:

Have those curves been studied at all and, if so, what are their properties and what are they called?

Field at random point 1 Field at random point 2 Field at random point 3



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