I'm a physicist, when I'm working on the quantum spin hall effect, I recollected the high-school knowledge on Lorentz force and try to explain the origin of it, but didn't get it in the first glance. Can anyone explain how the magnetic field interact with the moving charge more fundamentally, or, let's say, derive the equation: $\mathbf F=q\mathbf v\times \mathbf B$ in a more fundamental way?

# [Physics] Origin of the Lorentz force from the point view of relativity

coulombs-lawelectromagnetismspecial-relativity

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Trying to create an analogy with common experiences seems useless; if I were running north through a west-flowing "field" of some sort, I wouldn't expect to suddenly go flying into the sky.

This is a reasonable expectation, since the electric and gravitational fields do make forces that are in the direction of the field. So let's try to see what goes wrong if we write down a force law for magnetism that behaves in the same way. The first thing we could try would be

$$ \textbf{F}=q\textbf{B} \qquad (1) $$

Well, this doesn't work, because such a force would behave in exactly the same way as the electric force, and it would therefore *be* the electric force, not a separate phenomenon. Magnetic forces are supposed to be interactions of moving charges with moving charges, so clearly we need to include $\textbf{v}$ on the right-hand-side. One way to do this would be the standard Lorentz force law, but we're looking for some alternative that is in the direction of the field. So we could write down this:

$$ \textbf{F}=q\textbf{B}|\textbf{v}| \qquad (2) $$

As an example of what's wrong with this one, suppose we have identical charges $q$ bound together with a spring. If they're sitting at rest in equilibrium, equation (2) says there's no magnetic force on them. But suppose we start them vibrating just a little bit. Now they're going to start shooting off in the direction of the magnetic field. This violates conservation of energy and momentum.

Fundamentally, this comes down to an algebraic issue. The vector cross product has the distributive property $(\textbf{v}_1+\textbf{v}_2)\times\textbf{B}=\textbf{v}_1\times\textbf{B}+\textbf{v}_2\times\textbf{B}$, and in the example of the charges on a spring, with the actual Lorentz force, this guarantees that the magnetic forces on the two charges cancel out. We really need this distributive property, and in fact it can be proved that the vector cross product is the only possible form of vector multiplication (up to a multiplicative constant) that produces a vector result, is rotationally invariant, is distributive, and commutes with scalar multiplication. (See my book http://www.lightandmatter.com/area1sn.html , appendix 2.)

It is very difficult task to actually explain the Lorentz force to 12 year old. In fact, the concept of force alone is not that simple, let alone Lorentz guy.

I've worked in Science Museum for a while, and I came to conclusion that there is no need to explain everything. You must address feelings and imagination of that aged audience, not their brain. But you already know it.

Now, I don't know how to explain Lorentz force in a cosmological framework. However, there are few simple demonstrations of Lorentz force you can do. The best one is demonstrated in this video by (the greatest) Prof. Walter Lewin of MIT (around 12:00). Maybe you can use this demonstration and somehow draw a parallel between the current in the wire and solar wind.

You may also want to check out this site. Maybe you could apply some of the concepts you find there. There even an instruction for cosmological demonstration at the bottom, though the equipment is not that simple.

Hope this helps.

## Best Answer

The definition of the Lorentz force is fundamental already, it can't really be derived.

However, a useful treatment/thought experiment is to consider the electric/magnetic fields due to a current carrying wire in the stationary frame and a frame moving parallel to the wire.

In the stationary frame there is only a circulating magnetic field. In the moving frame there is a transformed magnetic field and an electric field radial to the wire, caused by length contraction of the positive and negative charges (or you can just think of the field transforms).

This electric field would exert a radial force on a test charge originally at rest with respect to the wire in the stationary frame. But given that there is no radial force in the stationary frame, there cannot be a net radial force on the charge when it is in the moving frame either. The force that counteracts the radial electric field in the moving frame is the Lorentz force. Its direction is perpendicular to the velocity and the magnetic field and it's strength must be equal to $qvB$, where $B$ is the magnitude of the magnetic field in the frame moving with velocity $v$ parallel to the wire.

Likely not general enough for you.