High School General solution for the magnetic field of a current of arbitrary shap

[Ievgen2000]

TL;DR

Despite the fact that the Biot–Savart law, which makes it possible to calculate the magnetic field generated by a conductor of arbitrary shape, has been known for a long time, only a limited number of particular solutions exist. Why is there no general solution for arbitrary current curves described by mathematical functions? The result of solving this problem should be integral expressions. If a definite integral does not have a solution in terms of elementary functions, it can always be evalua

Despite the fact that the Biot–Savart law, which makes it possible to calculate the magnetic field generated by a conductor of arbitrary shape, has been known for a long time, only a limited number of particular solutions exist. Why is there no general solution for arbitrary current curves described by mathematical functions? The result of solving this problem should be integral expressions. If a definite integral does not have a solution in terms of elementary functions, it can always be evalua

 

[Baluncore]

Welcome to PF.

There is software that solves the problem for AC currents using moment methods. It is called NEC, and is often used for antenna modelling.
https://en.wikipedia.org/wiki/Numerical_Electromagnetics_Code

 

[Baluncore]

Why is there no general solution for arbitrary current curves described by mathematical functions?

Each current in a conductor, will generate a magnetic field, that will, in turn, induce another current in all other nearby conductors, changing the hypothetical initial current.

Each of the many different, function defined conductors, in 3D space, will interact with every other function defined conductor, with many different functions, and in an infinity of possible orientations.

In order to compute the resultant current and field, each mathematical function defined conductor, needs to be factored into a minimum subset of geometries, say short straight segments, and sheets of conductor that will act like mirrors. The problem then becomes a simple, but massive, matrix inversion. That is NEC.

 

[Ievgen2000]

We're talking about a constant magnetic field. And the guys seem to have solved it.
https://bfieldapp.com/

 

[Ievgen2000]

A constant magnetic field does not induce current in adjacent conductors.

 

[Baluncore]

We're talking about a constant magnetic field.

You did not specify that you assumed the arbitrary current was constant.

A constant magnetic field does not induce current in adjacent conductors.

In reality, there is no such thing, as a current generated constant magnetic field. There may be only very low frequency components, with the higher frequency components excited by the turn on and turn off events.

The link you posted, only shows field computation methods for the geometry of simple loops, closed current sheets, or coils such as solenoids. It assumes that all current flows continuously in closed loops, without external connections. It ignores self inductance. Those circular, static, applied mathematical solutions appeared between about 1860 and 1930.

If a catenary wire, hangs between two arbitrary points, or a sinusoidal track runs across a circuit board, then the vector fields that result from the mathematically defined conductors, and all the connections beyond, that make up the closed circuit, must be summed. Those solutions are more difficult.

 

[phyzguy]

The solution for the magnetic field of an arbitrary distribution of currents is well known. The attached summary is from the Feynman Lectures on Physics, Chapter II-21. So if you know an arbitrary distribution of currents J(x,y,z,t), you calculate the integral for A, than take curl(A) to get B.