1

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?

Qmechanic
  • 201,751
doublecl
  • 11
  • 1
    There's a textbook I own somewhere that has a decent explanation of it. The short answer is that the magnetic force is a consequence of relativity. The speed limit of the universe and things like length contraction mean that where one observer might see an electrostatic force between two charged objects, another observer might see one of the objects has no charge at all. For things to be consistent, there has to be some other field present. If I can find my book I'll try giving a more complete answer. – MonkeysUncle Feb 20 '15 at 05:16
  • 1
    You could start from the wiki article http://en.wikipedia.org/wiki/Relativistic_electromagnetism . – anna v Feb 20 '15 at 05:38
  • Related: http://physics.stackexchange.com/q/3618/2451 – Qmechanic Feb 20 '15 at 08:10
  • The wiki article matches exactly what I remember. As an interesting side note, a gravitational field looks a lot like an electric field, and you can make an analogy between the forces and predict there's a similar effect, called gravitomagnetism. There are groups looking at matter around black holes to see if they can confirm gravitomagnetic effects. – MonkeysUncle Feb 20 '15 at 22:49
  • Like this one. I'm actually looking for the origin of magnetic field, which perform phenomenologicaly follow the Lorentz equation. I thought it should has something to do with relativity, unfortunately, I'm totally blind on this subject. It may worth a deep thinking when I got time. Thanks guys. – doublecl Feb 22 '15 at 09:01

3 Answers3

2

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.

ProfRob
  • 130,455
1

derive the equation … in a more fundamental way?

There is no more fundamental way. That equation defines the magnetic field: it is the field that produces forces in moving charges such that the force is proportional to the charge and to velocity. Also, the force must be perpendicular to a given direction which can vary from point to point. More succinctly, $\mathbb{F}=q\,\mathbb{v}\times\mathbb{B}$.

Relativity tells you how to compare the measurement from different observers. Special relativity does not derive the magnetic field. Special relativity does make the point that while one observer might see only electric field, another observer might observe also magnetic field. Something similar to: while an observer might see the blue face of a Rubik's cube, another observer might see parts of the red and blue faces. However, the unity of the electric field and the magnetic field in the electromagnetic field was established before (think Maxwell).

Quantum mechanics tells you how the energy is absorbed in quanta. Something similar to: $\mathbb{F}\cdot \mathbb{x}$ can only be 1, 2, 3 but no 1.112343. You know that.

Before you can talk about the properties (say energy carried by, momentum of, etc.) you need a definition. Don't fight the definitions!

I am aware that you could also define the $\mathbb{B}$ field as the field produced by a current so and so. Then, $\mathbb{F}=q\,\mathbb{v}\times\mathbb{B}$ would be a theorem rather than an axiom. We would then be chasing our tails looking for fundamental. In any case, why do you mean as more fundamental?

Probably you are asking for a better mental picture. That might not be the same as a framework to make calculations more efficiently. For example, Faraday (I think) had this mental image of the electromagnetic field being small rubber bands. While that mental picture helped him to set up certain calculations, modern calculations in electromagnetism use gauge theory.

Hector
  • 439
-1

I'm sorry guys, but Lorentz force could be derived from Maxwell equations, written in the integral form.

Nicc
  • 7
  • 1