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

[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).

 

[Dale]

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.

 

[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.

 

[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.

 

[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.

 

[Meir Achuz]

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.

 

[Dale]

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?

 

[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).

 

[Meir Achuz]

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?

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

 

[Dale]

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.

 

[atyy]

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.

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.

 

[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.

 

[weejee]

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.

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.

 

[atyy]

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.

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.

 

[weejee]

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.

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.

 

[Andy Resnick]

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.

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?

 

[Meir Achuz]

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.

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.

 

[Hurkyl]

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

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.

 

[weejee]

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.

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.

 

[Andy Resnick]

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.

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.

 

[cesiumfrog]

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.

If we have complete knowledge of the global dynamics of the fields, Maxwell's equations uniquely specify the global dynamics of charge ("to produce those fields"), correct?

So if we consider the time reflection of a radiating charge (caused by a brief non-EM acceleration), we must know exactly the trajectory along which the charge will accelerate (such as to exactly annihilate the fields). You seem to be arguing that what we don't know is how much of this acceleration will actually be caused by EM forces (as opposed to being done by external work)? Or are you just pointing out that there will exist a trivial possibility (zero Coulomb's constant)?

 

[Hurkyl]

If we have complete knowledge of the global dynamics of the fields, Maxwell's equations uniquely specify the global dynamics of charge ("to produce those fields"), correct?

Yes, if you know the values of a field that was produced by matter evolving according to the Lorentz force law and Maxwell's equations, then you should be able to recover the Lorentz force law from that data.

But that's not the question asked in the thread: the question was whether you can derive the Lorentz force law "strictly from maxwell's equations". And since the global dynamics of the fields cannot be uniquely determined from Maxwell's laws...

You seem to be arguing that what we don't know is how much of this acceleration will actually be caused by EM forces (as opposed to being done by external work)? Or are you just pointing out that there will exist a trivial possibility (zero Coulomb's constant)?

My point is that Maxwell's equations are consistent with absolutely any force law you can imagine, as long as that force law ensures matter obeys the charge continuity equation.

 

[samalkhaiat]

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).

You cann't, only the continuity equation is contained within Maxwell's equations. Indeed,
Electrodynamics = Maxwell's equations + Newton's 2nd law.
However, I would like to invite attention to the fact that Maxwell's equations, in a certain sense, contain "the equations of motion" for mass(energy) of purely electromagnetic origin, if the inertial effects of that mass are negligible; consider an electric circuit consisting of an inductance and a capacitance, but with an open switch. Suppose the capacitor is initially charged, then the switch is closed. Now Maxwell's equations alone give a description of the future motion of the electrical energy. For all practical purposes, this description is complete because the inertial effects of the electron mass are small.

When Einstein first formulated his General Relativity, he postulated the geodesic as the equations of motion. In 1927, he realized that the equations of motion are in fact contained within his field equations and therefore do not need to be postulated separately. Unlike Maxwell's, the field equations of GR contain the equations of motion of mass elements made up of ALL kinds of energy. Indeed, we can derive the equations of motion for both gravitational and non-gravitational forces from Einstein's field equations alone. This fact has always been considered as the most attractive feature of Einstein's equations.

regards

sam

 

[Count Iblis]

In the textbook by Ohanian, everything is derived from Coulomb's Law only and by demanding covariance under Lorentz transformation.
So, you don't even need the Maxwell equations, except div E = rho. You also don't need to postulate the existence of a magnetic field. It all follows from Lorentz invariance.

 

[Peeter]

In the textbook by Ohanian, everything is derived from Coulomb's Law only and by demanding covariance under Lorentz transformation.
So, you don't even need the Maxwell equations, except div E = rho. You also don't need to postulate the existence of a magnetic field. It all follows from Lorentz invariance.

I have a text (electrodynamics, Dover), that has the similar approach, but there's still the $\dot{\mathbf{p}} = q\mathbf{E}$ to relate that $div \mathbf{E} = \rho$ back to forces and masses.

My original question, which I believe has been answered (with a lot of additional interesting debate), was whether an additional law is required to relate the fields to the dynamics is required. ie: there is an additional postulate required on top of maxwell's equations (or something like what you describe that can be used to generate Maxwells). That additional something could be the $\dot{\mathbf{p}} = qE$ term plus relativistic arguments, or a Lagrangian $A \cdot v$ term.

Now, I've seen hints that one can express both the field equations and the Lorentz equation in a single Lagrangian, and that's probably the best way to look at the big picture view. However, I haven't yet seen a treatment of this that I understood or tried yet to figure it out for myself. It is definitely on my TODO list for study;)

Thanks to everybody who chimed in, providing info and discussion on this thread!

 

[fizzle]

Here's an interesting bit of trivial on the Lorentz force:

Assume a charge is moving along the positive x-axis and is overtaken by a constant electromagnetic wave also moving along the positive x-axis (think of a battery connected to an infinitely long parallel plate transmission line). In this case, E + v X B becomes E (1 - v/c). That's equivalent to a "Doppler Shift" of the electric field's strength. Does this have any physical significance or is it just coincidence? How would our understanding of electromagnetic theory change if we thought of the electric field strength as a measure of the frequency of something else?

 

[PhilDSP]

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.

Mass doesn't enter explicitly, no. However I believe Maxwell defined the dimensions of the electric field intensity to be the same dimensions as force which is of course mass×acceleration (ML/T^2)

If that seems strange, it probably should. A photon, for instance, has no mass and yet it has momentum and can exert a force. I think we're looking at the de Broglie hypothesis here which implies a type of electromagnetic mass that is not currently codified.

Otherwise it seems the Lorentz force equation arises trivially as the vector addition of the components of E and B fields (forces) times the electric charge. I assume the electric charge is a gauge condition that specifies the relative strength of the force as measured per electron (but could well be wrong about that)

 

[PhilDSP]

Here's an interesting bit of trivial on the Lorentz force:

Assume a charge is moving along the positive x-axis and is overtaken by a constant electromagnetic wave also moving along the positive x-axis (think of a battery connected to an infinitely long parallel plate transmission line). In this case, E + v X B becomes E (1 - v/c). That's equivalent to a "Doppler Shift" of the electric field's strength.

That's pretty much what Count Iblis was alluding to. (1 - v^2/c^2) is the pattern of Lorentz Invariance.

 

[PhilDSP]

It's interesting too that Maxwell, in the same treatise, published an equation which he called The Mechanical Force Equation. It is simply F = current x B.

His idea was that the stress of the electromagnetic medium resulted in a restoring force which can be interpreted in mechanical terms.