I try to solve this problem symbolically. Here is a diagram to visualize the geometry:
($\vec{a}$, $\vec{R}$ and $\vec{B}$ are in the same (yellow) plane.)
This is the magnetic field $B$ of a cable with two conductors along the vector $\vec n$. Vector $\vec a$ is the Dipol moment resulting from the current in the conductors. The equation describes the magnetic field at $\vec R$ relative to the cable.
$$ B= -\frac{k}{R^4}\left( a -2\left( a \cdot R \right) R\right) \times \hat n $$
This equation is only valid if the origin of $R$ is the center of the $B$ field. If that is not the case, let $R_0$ be the vector from the origin of the coordinate system to the center of the $B$-field. Then the field equation is:
$$ B(R)= -\frac{k}{(R-R_0)^4}\left( a -2\left( a \cdot (R-R_0) \right) (R-R_0)\right) \times \hat n $$
Here R is from the origin of the $B$ measurement coordinate system to the measurement point. Not including a superimposed background field (earth's magnetic field), the number of free parameters are:
- $\vec{R_0}$ has three parameters.
- $\vec{a}$ has 3 parameters.
- $\hat n$ has 2 parameters. If we let $R_0 \cdot \vec {\hat{n}} = 0\,$ all the equations for the system are: $$ R_0 \cdot {\hat{n}} = 0 $$ $$ \hat a \cdot \hat n = 0 $$ $$ B(R)= -\frac{k}{(R-R_0)^4}\left( a -2\left( a \cdot (R-R_0) \right) (R-R_0)\right) \times \hat n $$
Since the last equation is a 3D vector equation it counts for three scalar equations, so we can solve for all our parameters with $B$ measurement in two different locations. If we include the constant superimposed background field, there would be three more parameters and we would have to measure $B$ in three locations.
Is that correct, even though the field is invariant in z-direction? Or would the measurements not give independent z-values, and I could only use one measurement in z-direction?
I want to find primarily $R_0$, and secondarily $n$ and $a$, contingent on the measured $B_x$, $B_y$, $B_z$, and the distances between the points of measurement.