Why is there a cross product in Biot-Savart law?

Question

In middle school, we are told about the right hand thumb rule which helps us determine the direction of the magnetic field around a current carrying wire.

In high school, we are taught the Biot-Savart law, which pretty much explains why the right hand thumb rule works (right hand thumb rule is the result of the cross product in the expression of the Biot-Savart law).

But why is there a cross product in Biot-Savart law in the first place? Can we explain the existence of the cross product? And is there a way to sidestep Biot-Savart law and still be able to explain (maybe by using symmetry arguments or fundamental intuitive arguments) the right hand thumb rule?

Answer 1

It's possible to prove the Biot-Savart law from Maxwell's equations. That doesn't mean that the Biot-Savart law is unique, either in the sense of the notation used or in the sense of what the integrand is. Similarly, the Pythagorean theorem can be expressed as $A^2+B^2=C^2$, and it can be proved from Euclid's five postulates, but that doesn't mean that there is no other way of expressing it than by using modern algebra notation with exponents. Euclid expressed it using the areas of squares.

It does make sense that the Biot-Savart law involves a cross product, because the integrand involves taking the product of two vectors to find a vector, and there is only one rotationally invariant way to do that, which is the cross product.

Answer 2

The physics behind the cross product is the empirical observation, that the magnetic field "curls around" the wire. Nobody is able to tell us why this is the case -- it's the way nature does it. Thus, our job is to describe it.

Once we accept that we have to find a mathematical expression to describe the curling, it's rather natural to try a cross product: The cross product of two vectors $\vec a$ and $\vec b$ is perpendicular to both. Thus, by using the direction of the current flow $I d\vec \ell$ and the radius vector $\vec r$ their cross product yields the correct direction of the magnetic field $\vec B$. An other way to describe this direction is to define the vector potential $\vec A$ and to define $\vec B = \vec \nabla \times \vec A$ (maybe with an additional sign). However, this is mathematically more complex.

Answer 3

The experimental fact behind is the Oersted observation that a compass needle is deflected by a conducting wire. Before the invention of the battery by Volta, it was not possible to have a continuous current in a wire.

If the wire is aligned north-south, (that is aligned with the needle), after closing the circuit the needle deflects from the north, and the angle of deflection depends on the intensity and polarization of the current.

Because compass alignment is a magnetic phenomena, the dominant theory was to postulate a magnetic field created by the current. And the magnetic field should be transverse to the wire to explain what happens. The cross product fits well, because the deflection angle changes direction (east or west) if the wire is above or behind the compass, for the same polarization.

It is worth to remark that the French mathematician Ampère notices, very soon afterwards, that two conduction wires were attracted or repelled depending on the direction, intensity and polarization of the current. He developed a quantitative theory to calculate that forces. And postulates that magnets were only the side effect of microscopic currents. So, the compass deflection effect was for him basically a force between currents. There was no such a thing as a magnetic field.

But the existence of that invisible current was probably considered too speculative. And Faraday idea of using small iron pieces to show the magnetic field lines was a good argument for the existence of a magnetic field.