How do we know the direction of magnetic field that we assign to it?

Question

In my class, I was taught about Biot Savart Law and how to calculate direction of magnetic field as a cross product of radial vector from current element and the length vector of current element.

But I am not able to understand why magnetic field have been assigned such a direction, how did Biot and Savart know that it is perpendicular to both the current element vector and the position vector of the point where field is to be determined?

We define the direction of electric field as the force acting on a positive test charge. Is there some definition like that with magnetic field too?

Do vector fields have an 'inbuilt' direction of their own or its just by convention?

If vector fields really have some inbuilt direction, how did Biot and Savart determined that of magnetic field?

Answer 1

I’m not sure if this helps you with the history with Biot and Savart...

One fact that might help you is the magnetic field isn’t an ordinary vector field... it’s an axial vector field or a psuedovector field. You can’t take the sum of the electric field and a magnetic field.... note that the velocity is an ordinary vector but the cross product of a vector with a psuedovector gives a vector.

Rather than axial vectors or psuedovector, it might be helpful to think of bivectors (for now, oriented parallelograms.... which underlie the cross product). Focus on the parallelogram formed by $d\vec l$ and $\hat r$.

Possibly useful reading:

“Magnetic field surfaces” Bernard Jancewicz and Piotr Brzeski Published 6 May 2005 European Journal of Physics, Volume 26, Number 4

https://www.researchgate.net/publication/231130688_Magnetic_field_surfaces

“ A new way of visualizing magnetostatic fields is presented with the aid of field surfaces. They are counterparts of the electric field lines. Their characteristic property is that the Lorentz forces are tangent to them. Moreover, they have their boundaries on electric currents.”

Answer 2

The original notion of the "direction" of the magentic field came from suspending a compass needle near the wire. The direction that the compass pointed was, by definition, the direction of the magnetic field. Later it was discovered that the force on a moving charge of magnitude $q$ was ${\bf F}= q({\bf E}+{\bf v}\times {\bf B})$, and this is today's definition of the direction of ${\bf B}$ --- but this Lorentz force was discovered much later (1895) than Biot and Savart, who wrote in 1820.