# Derivation of Lorentz force Law strictly from maxwell's equations?

1. Aug 21, 2008

### Peeter

Basic electrostatics (as I've seen it presented) usually starts off with an implicit $\mathbf{F} = q q_2 \hat{\mathbf{r}}/r^2 = q \mathbf{E}$ definition of the electric field. Then with a limiting volume argument, you can then show that this can be expressed $div \mathbf{E} = \rho$. Eventually one builds up to a complete picture where you have all the maxwell's equations together describing the true picture.

Now, you take the $\mathbf{F} = q \mathbf{E}$ that's correct for electrostatic, and lorentz transform appropriately sure enough you get the Lorentz force law.

From Maxwell's equations, assuming one is sufficiently talented mathematically, once given any particular charge and current distribution, you get these six position and time dependent numbers that are associated with that distribution. Now, is that really enough to describe the dynamics? Do you need this something extra like that statics $\mathbf{F} = q \mathbf{E}$ condition, or the $-q \phi + \mathbf{v} \cdot \mathbf{B}$ Lagrangian term to connect this to the dynamics?

I'm just trying to identify for myself the root laws that I'm working from while studying E&M. Would I be able to start with the 4 vector equations of maxwell and deduce everything else (if so that isn't obvious to me how to do so).

2. Aug 21, 2008

### Staff: Mentor

I don't think you need anything to connnect it to the dynamics other than the definitions of E and B (and of course Newton's 2nd law). The forces are in the definition of the field.

3. Aug 21, 2008

### Andy Resnick

IIRC, the Lorentz force law is postulated in addition to Maxwell'e equations.

However, Post's "Formal Structure of Electromagnetics" presents a derivation of the Loretz force law, beginning with Minkowski's formulation for the stress tensor constructed from the fields E and B. Even so, the Maxwell equations are then derived from the Minkowski equation and are separate from the (derived) Lorentz force equation.

4. Aug 21, 2008

### weejee

I think Maxwell's equations are insufficient. The Lorentz force law, or equivalently, the term $-q \phi + \mathbf{v} \cdot \mathbf{B}$, amounts to the 'interaction of the EM field with a matter', where as the Maxwell's equations describe the dynamics of the EM field itself.

5. Aug 21, 2008

### weejee

Andy Resnick, that's awesome!

How does the mass arise in the Lorentz force equation according to the Minkowski's formulation? Is it related to the general relativity? Otherwise I don't see how 'the equation of motion of a particle in EM fields', which inevitably involves its mass, is derived from the stress tensor constructed from just E and B.

6. Aug 21, 2008

### clem

The Maxwell stress tensor is derived starting with the Lorentz force law, so using it to derive the Lorentz force law is circular.
The Lorentz force law is an additional law needed to give the effect of the EM fields on matter.

7. Aug 21, 2008

### Staff: Mentor

I don't agree with all of that. How do you define the E and B fields if not in terms of their force on a charge?

8. Aug 21, 2008

### cesiumfrog

Hmm.. Maxwell equations define a relationship between the dynamics of charge and the dynamics of the field. And those equations have certain symmetries, which dictate the symmetries of the solutions. So the fact that an accelerating charge radiates an EM wave must imply that an equivalently focussed EM wave provides the same accelerating force on a charge (which seems only a frame-change away from deriving the Lorentz force law and all of electrodynamics, without even resorting to guessing a field-action to minimise over).

Last edited: Aug 21, 2008
9. Aug 21, 2008

### clem

You have just stated the substance of the Lorentz force law.

10. Aug 21, 2008

### Staff: Mentor

That is exactly my point. The "substance of the Lorentz force law" is implied in the definition of E and B, without which Maxwell's equations have no physical meaning.

Although I think cesiumfrog makes a much better point.

11. Aug 22, 2008

### atyy

As cesiumfrog said, Maxwell's equations relate E,B,q,j.

As you said, you also need Newton's 2nd law F=ma.

Without the Lorentz force law F=q(E + v X B), there is no way to relate the quantities in Maxwell's equations to the quantities in Newton's 2nd law.

12. Aug 22, 2008

### atyy

There's an interesting point from the Feynman Lectures.

In Maxwell's equations, you make an arbitrary choice of the 'right hand rule'. For example, in Ampere's law, the direction of the circular B field lines about an infinitely long straight wire is conventionally given by the direction in which your fingers point, if you grasp the wire with your right hand such that your thumb points in the direction of the current. Using your left hand will give you an answer that would be inappropriate for the AP tests, A-levels or IB. Very strange, a law of nature that distinguishes your right and left hands wasn't discovered until Wu and colleagues did their experiment after WWII. How can we be using a right hand rule here?

Feynman notes that two applications of the right hand rule give the same direction as two applications of the left hand rule. Where is the second application coming from? From the cross product v X B in the Lorentz force law.

13. Aug 22, 2008

### weejee

The 'handedness in the physical law'(such as that in the weak interaction) is a different one from the 'handedness in the vector product rule'.

What Lee and Yang first suggested and Wu subsequently confirmed is that the physical law governing the weak interaction changes when you invert the space(x->-x, y->-y, z->-z). This is not the case for most physcial laws we experience normally. If that kind of thing happens in the classical mechanics or in the classical electromagnetic theory, F=ma or Maxwell's equations doesn't hold anymore once you invert the space. Yet, this is not the case.

14. Aug 22, 2008

### atyy

Seems the same to me. Ampere's law in local form (del X B = j) changes sign under (x->-x, y->-y, z->-z). The reason why classical electromagnetism doesn't give predictions that change sign under (x->-x, y->-y, z->-z) is that there's another cross product when you relate B to the motion of a charge via the magnetic part of the Lorentz force law (F = v X B).

Try i X j (unit cartesian vectors), and you'll find that the outcome depends on your choice of right or left hand rule. But i X j X k doesn't. Similarly, classical electromagnetism needs 2 cross products in order for its predictions not to depend on the choice of right or left hand rule.

15. Aug 22, 2008

### weejee

You are absolutely right in this point. What makes the Maxwell's equation invariant under spatial inversion is the fact that B is a 'pseudovector'(=a quantity which exactly behaves like a vector under rotation but oppositely under spatial inversion) by its definition from the Biot-Savart law. Or we can say that Maxwell chose B to be a pseudovector quantity to make his equations invariant under spatial inversion.

For an equation to have 'handedness', it should equate two quantities which differ from each other in their transformation properties under spatial inversion. A simple example is k=ixj. This clearly doesn't hold once you invert the space.

16. Aug 22, 2008

### Andy Resnick

I wanted to look through Jackson before answering this, because the OP raised a very interesting question. Mass only gets introduced into electromagnetics in the context of special relativity- and specifically in the context of 4-vectors. That's Chapter 11 in Jackson, and prior to that (10 previous chapters), the mass of a charged particle is not mentioned, AFAIK.

But again, the Loretz force equation is introduced as a postulate, then generalized to 4-momentum and a covariant formulation. From there, we can write down a Lagrangian and Hamiltonian to talk about moving (massive) charges and the resultant radiation field.

So the Lorentz force law appears to be much more fundamental than I initially thought- I always considered it some sort of "add-on". Has anyone seen a derivation or development of this? For example, can anyone cite Lorentz's original presentation?

17. Aug 23, 2008

### clem

1. Mass doesn't enter the Lorentz force equation which relates dp/dt to q, v, E, and B.
Relating dp/dt to mass is part of mechanics, not EM.
2. The stress tensor is derived from the force equation and Maxwell's equations.

18. Aug 23, 2008

### Hurkyl

Staff Emeritus
No, that doesn't follow. In fact, the only constraint that Maxwell's equations place on the matter distribution is the charge continuity equation -- and you even get that 'for free' if charges are assigned to particles! This is clear mathematically simply by writing down down the time evolution of the E and B fields and plugging into the equations. IMHO it's clear physically because Maxwell's equations have no idea what sort of other dynamics can be pushing charges around! You really do need something extra for your conclusion to follow.

Last edited: Aug 23, 2008
19. Aug 23, 2008

### weejee

You're right. I was mistaken.

Then it seems to me that the Lorentz force equation can be derived from Maxwell's equations.

1. First construct the Lagrangian which gives Maxwell's equations as the equation of motion.
2. Using Noether's theorem, get the energy-stress tensor as the conserved current for spacetime translation.
3. derive the Lorentz force law from the energy-stress tensor.

20. Aug 23, 2008

### Andy Resnick

That's not exactly true- the fact that electrodynamics must be covariant means that the momentum p (or dp/dt) transforms as a 4-vector, three components of which have to do with mass. That's where the mass comes in- via Lorentz transformations of the Lorentz force equation.

As for part 2, one can go either way- either construct a stress tensor from the field components, or begin with the tensor which has certain transformation properties.