Most fundamental quantity in magnetostatics and magnetodynamics [closed]

Question

In both elctrostatics and electrodynamics, the electric charge is defined as the most fundamental quantity.

What is the most fundamental quantity in case of magnetostatics and magnetodynamics ?

Answer 1

Short vision: The most fundamental is the fact that we get same physics in all inertial frames. In this light, the most fundamental quantity in classical electrodynamics is Charge,$^1$ measured and defined as: $Q=\epsilon_0\oint_S\vec {E}\cdot d\vec {a}$.

Magnetic force is the consequence of charge invariance under Lorentz transformation and principles of special relativity. In other words, force between currents $^2$ is a result of the facts of charge invariance and special relativity.$^3$

Long vision: I am trying that explain it in a way high school students will understand, so please notice that the language I use is not precise, don't memories any of it! Just get a feeling of what is going on.

We build this understanding on two facts:

a.) charge is invariant: it doesn't matter how you move, you get the same charge measured and defined as$Q=\epsilon_0\oint_S\vec {E}\cdot d\vec {a}$; and $\vec{E}$ is directly measured quantity.

b.) Any moving object's length will get shorter in your frame. The faster it moves, the shorter it becomes.

Consider a charge $q$ moving parallel to a current in a wire with net charge density zero. Now if you move with the same speed of charge $q$. You will observe something funny: the net charge density of the wire is no longer zero.$^4$ And hence the charge $q$ now experiences a electric field $E'$; therefore the charge $q$ gets a force acting on itself : $F'=qE'$.

Now sure the charge $q$ will accelerate and move if you move with the same speed and direction of $q$. What will happen if we switch the channel to the old frame (that you don't move, the charge $q$ move)? Will charge $q$ behave as if nothing happened?

No.

In fact it will have funny motion as a result of it experiencing a force $F$ transformed from $F'$. $^5$

We give that force $F$ some fancy name: magnetic force. $^6$

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$1:$ Assuming that A is more fundamental than B means B can be deduced from A, but not vice versa.

$2:$ In other words: moving charges.

$3:$ See: Electricity and Magnetism by Purcell.

$4:$ In the old frame (that you don't move, charge $q$ moves), there are electrons and positive ions (nuclei) in the wire. Given there is a current, which means the electrons are moving, the ions are at rest in the old frame. Now if you move right next to the charge $q$, you will observe that ions and electrons get new but very different speeds. By fact b.) their distance between electrons A and electrons B will be shorter, and by fact a.) their charges are the same, hence imply a different density. Similarly for ions, but as it turns out, ion's new speed give it a new density that is no longer equal to the new density of electrons; hence these two new densities no longer cancel each other and thus have a net density not equal to zero in the new frame (that you move right next to the charge $q$ such that you observe $q$ at rest.)

$5:$ That's a Lorentz transformation for force, or more precisely, Lorentz transformation of the rate of change of momentum.

$6:$ but historically, we discovery magnetic force before relativity.

$P.S.$ Feel free to ask more details, I don't mind adding some math at all.

Answer 2

In both electrostatics and electrodynamics, the electric charge is defined as the most fundamental quantity.

That's what people tend to say, but don't forget that electromagnetic waves propagate regardless of charge. Displacement current is not the motion of charged particles. Check out Taming Light at the Nanoscale:

"Look around, and you will probably see numerous electronic and optical gadgets, such as mobile phones, personal digital assistants, laptops, TVs and digital cameras. These may all do different things but they have one thing in common: in the electronic circuits that drive these devices, charged particles flow through components and impart power via what is known as the conduction current. But is the motion of charged particles the only current we have available?"

Light is in essence alternating displacement current. And don't forget pair production. We can quite literally make charged particles out of light. Then note that in atomic orbitals electrons "exist as standing waves". Standing wave, standing field. Look at the Poynting vector and check out the Einstein-de Haas effect. All in all I think it's better say electromagnetic waves are more fundamental than electromagnetic charge. And then after that, take a look at this:

"In fact Richard Feynman complained^{[citation needed]} that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of the electromagnetic potential instead, as this would be more fundamental".

What is the most fundamental quantity in case of magnetostatics and magnetodynamics ?

Electricity and magnetism are merely two "aspects" of electromagnetism, so there aren't two different fundamental things here. If pushed, one is tempted to say moving charge but I think's better to again say electromagnetic waves or the underlying four-potential. See the wiki electromagnetic radiation article where you can read that "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time". One's the spatial derivative, the other's the time derivative. If it was a water wave and you were in a canoe, the tilt of your canoe is E and the rate of change of tilt is B. There's only one wave present. The sinusoidal "electric" wave is the derivative of it.