How do we Know Something is a Cross Product?

Question

I have always had this doubt in my mind that when things like force due to moving electric and magnetic field were being discovered, how did physicists know that the Force acting on a charged particle moving in a Magnetic Field (or a moving magnetic field in an electric field) will be a Cross Product?

Since, from what I know these terms had to be previously defined.

To summarize, I can understand intuitively what is a dot product and why it is needed. Sometimes we need a component of a vector acting along the direction of another vector (that is being dotted with) to get the value of a resultant vector.

For eg, while considering Work, we know that it is always supposed to be in the Direction of Displacement, and so the Force needs to be in the direction of displacement, so when we have a force acting in a direction not equal to the direction of the displacement, we just have to take the component of force acting in the direction of displacement. and that's why we have the formula

W=FScos(theta) force and displacement always have to be in the same direction

In order to project one vector on to another and then just multiply with that component, which is very similar to one projecting a vector onto x y or z-axis in a right-handed cartesian coordinate system.

But there is no intuitive way I can imagine why two coplanar vectors can ever result in a third dimension. And how do we know whether 3 more such quantities in the physical world are actually related in such a manner?

like I said I didn't understand why force on a moving charged particle in a magnetic field would be in some other dimension.

I hope I am clear with my explanation. I appreciate every answer.

Answer 1

Almost exactly 200 years ago, Ørsted discovered that a compass needle pointed along circular paths around a current-carrying wire. Shortly afterwards, Ampère found that two wires carrying currents in the same direction attract each other. Whereas Ampère saw the phenomenon as action-at-a-distance between the wires, the notion of field-as-agent came to supersede action-at-a-distance. Ørsted's compass was now seen as giving the direction of the field due to a wire, and the force on each of Ampère's current-carrying wires was seen as being at right angles to the wire and to the (causative) field due to the other wire. That the force is at right angles to both the magnetic field and current density vectors was confirmed and generalised by further experiments.

The names 'cross product' or 'vector product', and the vector algebra notation, to describe such a relationship came later. The experimental findings helped to motivate the mathematical concept, not the other way round. And, of course, it wasn't just electromagnetism that gave rise to the cross product idea. Torque and angular momentum cry out for the treatment.

Answer 2

It was realized long ago that the idea of a "cross product" $(A_x,A_y,A_z)\times(B_x,B_y,B_z)=(A_yB_z-A_zB_y,A_zB_x-A_xB_z,A_xB_y-A_yB_x)$ had surprising relevance in the real world (e.g. with torque and angular momentum). So when J.J. Thomson was trying to mathematically describe the force imparted on charged corpuscles by a constant magnetic field, the idea of a cross-product was already at his disposal (but he probably would have arrived at the correct result in any case).

From a modern perspective though (I don't expect you to understand the following paragraph), the cross product actually emerges quite naturally from the covariant formalism of classical electrodynamics. The electromagnetic field is the field-strength 2-form $F_{\mu\nu}$ (omitting basis 2-forms in standard physics fashion). The equation of motion for a charged point-particle in an electromagnetic field is:

$$f_{\mu}=ma_{\mu}=F_{\mu\nu}p^{\nu}$$

Now if you just look at the spatial component of this equation $\mu=1,2,3$, you will get the standard Lorentz force formula with the magnetic field and a cross-product.

Answer 3

It seems that Thomson was one of the first that wrote the Lorentz force formula as we know it today. He was interested in determining the motion of charged particles (in particular electrons) in cathode rays. Around 1881 he gave the force on the particles due to an external magnetic field as $f = (q/2) B \times v$: the form is correct but he made a little mistake (that factor of $2$).

Why this "cross product" form?

With the experiments of Thompson it was easy to realize that the force must have this form. In fact, with the Thompson's apparatus you can literally see the trajectory of electrons in a uniform magnetic field by naked eye (see this nice picture Which cyan colored line is produced in the Thomson e/m apparatus? ): electrons move in a circular path, and so the force must act as a centripetal force (namely, a force transverse to the direction of motion). Form this basic fact Thompson correctly found the form of the Lorentz force: the analogy with the circular motion is important to understand the "geometry" of the Lorentz force: $$ \text{Lorentz force} \propto B \times \text{velocity of the particle} $$ $$ \text{centripetal force} \propto \text{angular velocity} \times \text{velocity of the particle} $$ So, loosely speaking, you can interpret the action of the magnetic field as that of ``imparting an angular velocity to the particle'', in the plane orthogonal to the direction of the field (in fact there is a notion called "radius of gyration", see https://en.wikipedia.org/wiki/Gyroradius).

At that time (1880 circa) the cross product was already there, and was already part of the mathematical tools used in physics and engineering. It seems it was Lagrange, in his studies of three-dimensional geometry ("Solutions analytiques de quelques problèmes sur les pyramides triangulaires", 1773), that introduced the modern notions of "dot" and "cross" products. So, when Thompson saw the electrons moving in a circular path, the mathematical notion was already there, ready to be used.