# Lorentz Force replaces 2 laws of Maxwell?

1. Apr 10, 2010

### nonequilibrium

I'm a bit confused about how the Lorentz Force (as a law) stands in relation to the laws of Maxwell (independent of each other? dependent?). There are two concrete examples I can think of where they interfere with each other:

1: Lorentz Force & Faraday's Law of Induction

Imagine a rectangular conducting loop (no current) hanging in a magnetic field (in such a way that the magnetic flux through the loop is at a maximum, aka the magnetic field lines are parallel to the surface vector of the loop). One of the 4 rods of the loop is moveable and can be dragged away in such a way that the surface area of the loop is increased. Now let this rod move at a constant speed v, then the change of the magnetic flux through the loop is = B*L*v with B the external magnetic field and L the length of the moving rod. Faraday's law of induction predicts a current induced by a voltage equal in size to the changing flux.

Now you can predict the exact same current by not using Faraday's Law of Induction and instead nothing more than the Lorentz Force by noting that because the rod is moving away at a constant speed v, the charges inside the rod experience a Lorentz Force and start moving, which constitutes a current flow. (It gaves the same numerical result, yet a different interpretation)

2: Lorentz Force & Ampère's Law

Imagine a magnet standing vertically, south pole downward, north pole upward. Now hold a circular loop a few inches above the north pole. Viewed from above, there is a clockwise current in the loop. Ampère's Law predicts that the loop acts like a little magnet with its north pole aimed downward and its south pole aimed upward. You can conclude the loop will be pushed away from the magnet, because the two north poles are directed at each other.

Again, ignore Ampère's Law and simply calculate the Lorentz Force that the magnet exercises on the current (F = B*I*L). You'll have to do this for an infinitesimal piece and then integrate it and you will see that the resulting force is aimed directly upward. Conclusion: same result as with Ampère's law.

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This is all very confusing, because practically they seem to be very different things, but in both cases they predict the same experimental result. How do you even know they are the same result? Maybe they are different phenomenons and have to be ADDED to each other. How can you know? And is it logical that sometimes the Lorentz Force can predict what different Mawellian laws say? It's like the Lorentz Force is derived from multiple laws of Maxwell, but then only for a specific case (because the Lorentz Force seems to be equivalent in these cases, but we know we can't always use it instead of the laws of Maxwell)

Very curious and grateful to all helpers,
mr. vodka

2. Apr 11, 2010

### Born2bwire

The Lorentz force is a supplementary law to Maxwell's equations. Together, these five equations are adequate enough to solve practically the vast majority of electrodynamic problems. I would point out though that the Ampere's Law and Faraday's Law in Maxwell's equations are not the Ampere's Force Law and Faraday's Force Law as you have described here. Maxwell's equations does not directly calculate the force. Though I expect you could get a force calculation from a purely field standpoint by using the Maxwell stress tensor.

You can derive the various force laws, Coulomb's law, Ampere's force law, Faraday's force law, etc., by combining the Lorentz force with the fields predicted by Maxwell's equations for the given sources.

3. Apr 11, 2010

### Repainted

Erm... I'm pretty sure the thread starter isn't talking about Ampere's Force Law and Faraday's Force Law(personally, I've never heard of a Faraday's Force Law), but IS talking about the ones in Maxwell's set of equations.

What his question is, if I were to attempt to reiterate, is that the phenomena of electromagnetic induction (induced emf due to a changing magnetic flux) can be described completely using the Lorentz Force law (at least for a coil of wire), as he tried to illustrate in the first portion of his post. Hence, why do we need Faraday's law when both methods produce the same result?

This has also puzzled me. More responses would be great.

4. Apr 11, 2010

### nonequilibrium

But Born2bwire, if Lorentz' Force is supplementary to the laws of Maxwell, how come (in certain situations...) Lorentz Force can describe an effect that also a law of Maxwell does. In my first example, you can calculate the current exactly by either Faraday's Law of Induction or the Lorentz Force. This is very confusing. One could argue, for a fact, that there is no reason to assume the Lorentz Force and Faraday's Law are predicting the same phenomenon (as they're independent equations (?)) so you could argue that the actual current is twice as much because you have to add the two phenomenons. I'm pretty sure this is actually not the case, but then my question is: how do you even know that (without verifying it experimentally)?

Also please don't forget my 2nd example where Ampère's Law and the Lorentz Force seem to predict the same phenomenon (as the explanation that might work for my first example maybe won't go up for my second).

Thanks for both replies.
I hope more will follow :)

mr. vodka

5. Apr 11, 2010

### jean-ro88

Hello!

First of all I want to emphasize that Maxwell's equations are the basics of the classical electromagnetic theory and that Lorentz's Force is just a piece of this theory ( it's not a supplementary law ). Maxwell's equations deal with the connections between electromagnetic fields, charges and currents. Lorentz's Force shows us how a charged particle behaves in a magnetic field and therefore is a tool for describing the movement and the trajectory of that particle. I will give an example and I hope you will understand the origins of Lorentz's Force.
Let's take the first example with the rectangular loop and let's assume that the vertical rods are not conductors. When the upper rod moves up with a certain velocity ( v ) the charged particles within the rod will also move up with that velocity ( let's presume they were stationary at the begginig ) and because they are in a magnetic field they will feel a Lorentz force that will move them. If the field is towards us ( coming out of the loop ) then the positive charged particles will move towards the right edge of the rod and the negative charged ones towards the left edge ( well, it's just the electrons who are in fact moving but neither way, there will be a polarization ). Ok, so how long does this work? The new polarized bar acts just like a capacitor and creates an electric field inside it that will give birth to an electric force that will balance Lorentz's Force. So after a while all particles will be in balance. Now here's the trick: what if we study the whole phenomenon from a different point of view. Let's say we are on the moving rod and therefore, in this case, there is no velocity for us and there is no Lorentz Force. But the charges are stationary for us. The electric field created by the polarized bar still gives birth to an electric force so they should be moving. Yet, they are standing there, at a constant distance from us. How do we explain this? The explanation is this: for us there is another electric field in the bar and the origin of this field is in Faraday's law of induction. How do we connect it to Lorentz's Force? Simple:

$$\vec{E'}$$ = - $$\vec{v}$$ $$\times$$ $$\vec{B}$$

As you can see, the result are the same but the way we see them are different and depend on every system ( the observer ). In fact, magnetism is a relativistic effect of electricity: it depend on the observer. For an electron that is in the middle of the bar there is no magnetic field. There are just 2 electric fields: one given by the induction and one given by the charged particles at the edges of the bar. So, Lorentz's Force somehow derives from this aspect. It's just a result of Maxwell's equations and we use it to find the movement of the charges ( using Newton's laws ).
I hope you can see my point. I await your messages.

Last edited: Apr 11, 2010
6. Apr 11, 2010

### nonequilibrium

That's a really interesting experiment. Electromagnetism makes inertial frames of reference so terribly entertaining to play with!

A little remark:
"For an electron that is in the middle of the bar there is no magnetic field."
There is still a magnetic field, but I get what you mean.

I'm not sure what your explanation is, exactly. It seems more like an elaboration on my first example? For example, in the first reference frame, where the bar has a speed, you can solve the potential difference by the Lorentz force (as you did) or Faraday's Law, which was the reason I posted this thread. There is no distinction there. But is very interesting how you displayed that with a small change in observation, Faraday's Law will still work but Lorentz' force will not. And thank you for letting me know the Lorentz law is deduced from the 4 laws of Maxwell.

It does seem tempting to say "Faraday's Law of Induction is more general than Lorentz' Force" but this cannot be correct, cause there obviously are cases (single charges) where Faraday's Law will be no good. So it's hard to know when you have to use both laws if you want to know what happens in a certain experiment and it seems like there is no way to know if they are talking about the same phenomenon (as in our examples) or not. Maybe the sole cure for this is to go and see how exactly Lorentz' Force is derived from the laws and then in each case think about that deduction to see if they will predict the same phenomenon or not.

Can I presume Ampère's Law is also used in its deduction, as my 2nd example (see my original post) shows they sometimes also tell you the same expected phenomenon.

I must say that of what I've seen of the whole of physics so far, electromagnetism seems to be the most majestical of all -- all the quantum weirdness aside.

7. Apr 11, 2010

### Repainted

Are you sure? I've always thought that the Lorentz Force Law was needed in addition to Maxwell's Equations and cannot be derived from them. Maxwell's equations tell you what the fields are (if you solve them for every point in space and time), and the Lorentz Force Law tells you how the fields affect charges (through the force), is that not so?

8. Apr 12, 2010

### SpectraCat

The Lorentz force is one of Maxwell's original equations. For historical reasons of which I do not know the details, the number of laws was "standardized" to four, and the Lorentz force was dropped, and later "rediscovered" by Lorentz in a slightly different form. However, you are correct that it is generally considered a necessary and separate equation to the standard 4 Maxwell equations. I do not think it can be derived from Maxwell's equations alone, without some other postulates ... at least I have never seen such a derivation.

9. Apr 13, 2010

### jean-ro88

Hello everybody!

I've read you messages. I must admit that I didn't use the most appropiate words in this phrase:
Lorentz Force is strongly connected to Maxwell's Equations but you need to take in account the reference frame in order to obtain it ( just like I said in my example ). My teacher always told me that Maxwell's equations are fundamental and that they govern all the electromagnetic phenomena ( if we consider the classical approach ). Furthermore, he also emphasized that Lorentz's Force is very useful but showed me that it is strongly connected to the frame of reference. Of course, there is a difference because we are talking about fields, currents and charges in Maxwell's equations and about forces in the other case ( just as I said in my previous message ). Anyway, I see them connected to each other and that's why I gave that example. I could be wrong though...
By the way, I didn't know that Maxwell discovered this force. I always thought it was Lorentz's discovery.

I think it was dropped because of esthetical reasons too. Maxwell's equations are indeed beautiful! Lorentz's Force would have broken their symmetry.

Oh, and thanks for the remark mr. vodka:
I've been thinking about this. I've consulted some of my books and I came upon this subject. Some of them admit that there is no magnetic field in that reference frame ( the electron's reference frame ). Then I imagined another experiment:
Let's say we have 2 positive charges ( far away from each other so that they do not interact ) that are moving up with different velocities ( v1 and v2, v2 > v1 ). After a period of time they enter a region where there's a constant magnetic field ( let's assume is coming out of the screen :) ) and an electric field ( from the right edge to the left edge ). Now suppose that the first charge is in balance and moves up with the same velocity. Therefore v1 = E/B. Let's take 2 reference frames: one stationary and one that is moving with the first charge. The first observer will see that one charge is moving up with the same velocity ( just as I said before ) while the other charge will have an aditional a circular trajectory besides it's first one. The second observer will see the first charge stationary and will conclude that there are 2 electric fields that balance each other. But he will see the other charge move in a circle. If we admit that there is no magnetic field in the second frame of reference then we cannot explain the movement. So I think you are right about that! Still, I was wondering about the density of electromagnetic energy: if the magnetic field is the same in all inertial reference frames while the electric field changes then the total electromagnetic density changes. Is this alright?

10. Apr 13, 2010

### Antiphon

Lorentz force equation is independent of maxwell's equations. Maxwells eqations relate charges and fields. The definition of mechanical force exerted on a charge by a unit field is a separate postulate.

11. Apr 14, 2010

### Repainted

Hey guys, I think we're missing the main question here, which is to put it rather crudely, can Faraday's Law of Induction be derived from the Lorentz Force Law(or maybe the other way around)?

The short answer is probably no, but look at it this way.

First of all, the original integral form of Faraday's Law came from the relation between the induced E-Field in a coil of wire and the rate of change magnetic flux through the coil of wire. Now is it possible, through some form of weird calculus, to reproduce Faraday's Law using the Lorentz Force Law, by letting the equation of the coil of wire(its a equation of a curve in space I suppose?) by some function (probably a parametric one since its a curve), where all points on that wire are moving at some certain velocity given by another (vector) function v(x,y,z,t), and the magnetic field all around in space by another (vector) function B(x,y,z,t).

Now using the force on any charge in the wire is equal to F = q(v x B), we have F/q = v x B. So if we do the line (work) integral of F/q around the whole wire, would we be able to, through some strange calculus, end up with an equation which says that the work done per unit charge(which is the line integral of F/q around the whole closed coil of wire) is equal to the negative rate of change of magnetic flux(through the coil of wire) with respect to time?

You see what I'm trying to do here? Both the Lorentz Force Law and Faraday's Law can describe the same things for say, a square coil, a circular coil, rotating or moving in a magnetic field. However, what if the shape of the coil is not so simply, and its motion in the B-field is not a simple rotation or translation as well? At times like these, we always use Faraday's Law because its just silly to attempt to do it the Lorentz Force way, but is it actually possible to show that the two methods are equivalent for ANY strange coil shape undergoing ANY strange motion while the B-field in space (and therefore through the coil) is changing defined by ANY function?

12. Apr 14, 2010

### nonequilibrium

Okay, a few responses to some people:

jean: you got me thinking more about my statement... Cause in a reference frame where the magnetic field is fixed but a charge is moving, there is a Lorentz force, but in the reference frame initially moving with the same speed as the charge, the charge is stationary. But the magnetic field is now moving. Does it make sense to say a magnetic field is moving if it is uniform across the universe? In some way or another, there must originate a force that pushes the charge.

More about your own example: if a certain parallel capacitor forms an electric field E, can you say that all charges moving relative to those capacitor-plates feel the same E? It may seem like a stupid question, because it's so often used (in a velocity selector as you describe) but is it so self-evident, as moving charges feel other electric fields?

Repainted: Don't forget my second example in my original post! That linked Lorentz' Force to Ampère's Law. So if you were able to deduce Lorentz' Force from Faraday's Law, that would mean Faraday's Law and Ampère's Law aren't independent, and the 4 Maxwellian Laws are independent (correct?). So Lorentz' & Faraday's Law cannot be equivalent.

Then there might be the option that Faraday's Law follows out of Lorentz' Law, but that is doubtful, because Lorentz' Law says nothing about a magnetic field that varies strengthwise in time (correct?) and Faraday's Law can.

How is it possible that Lorentz' Law, independent of the Maxwellian equations, can in some cases replace at least two Maxwellian Laws (Ampère's and Faraday's)?

I suppose the only explanation for this can be: Lorentz' Law is deduced from at least Ampère's Law, Faraday's Law AND something outside of the Maxwellian Laws. Would anybody agree?

13. Apr 14, 2010

### Repainted

This.

I'm pretty sure for the original integral form of Faraday's Law, its not about the magnetic field strength varying with time, but rather the magnetic FLUX through the coil (varying with time). And that shouldn't be a problem in calculation using only the Lorentz Force Law.

So in other words, maybe I'll break it up into steps for the calculation.

1. Find an expression for flux through the coil by doing a surface integral for the magnetic field. Just regular Math here.
2. Take the time derivative of the flux.
3. Find an expression for the line (work) integral for F/q = v x B.
4. Show 2 and 3 are equivalent.

You see what I did there? Using the Lorentz Force Law alone, it seems possible to deduce the integral form of Faraday's Law, and from there you can deduce the differential form using Stokes Theorem and all. Similar methods for Ampere's Law?

I also know I'm probably wrong since plenty of people must have thought of this before me and yet the notion still stands that Maxwell's equations can't be deduced from anything. So... is there any assumption that I made unknowingly in the 4 steps above? Or if it just plain wrong, in what way is it so?

The Lorentz Force Law IMO isn't exactly a law since stems from the way we define E-fields and B-fields (force per unit charge (or a test charge) and force per unit length per unit current placed at right angles). I mean, its pretty obvious what the force would be given the fields and how they're defined. So it doesn't really need to be derived doesn't it?

Last edited: Apr 14, 2010
14. Apr 16, 2010

### netheril96

Neither the Faraday's law of induction,which states that the electromotive force is proportional to the changing rate of magnetic flux,nor Ampere's law,which tells you the force on a current exerted by a magnetic field,part of Maxwell's equations.

The EMF in a loop which is moving is indeed caused by Lorentz force.What Maxwell's equation tells is the EMF in a loop in a time-varying magnetic field is caused by a vortex electric field.

Lorentz force are independent of Maxwell's equations.The former deals with how electromagnetic field affects charges,while the latter deals with how charges generate fields.

15. Apr 16, 2010

### elect_eng

You are definitely making some good points. I have an opinion here, but since I can't prove it conclusively, I'll just give some references for further reading. Recently, Ben Niehoff pointed out a good paper by Frank Munley. Although this paper addresses the resolution of some paradoxes related to Faraday's Law, it does show examples of obtaining the same answers either with Faraday's Law, or the Lorentz force law.

http://www.hep.princeton.edu/~mcdonald/examples/EM/munley_ajp_72_1478_04.pdf [Broken]

Also, the book "Foundations of Classical Electrodynamics", by F.W. Hehl and Y. N. Ovukhov provides an approach that makes it easier to resolve some of these questions. It offers a very clear axiomatic based treatment of electromagnetics in a modern formulation that includes general relativity. In this treatment, FL is a direct consequence of very basic experimentally consistent and accepted facts.

1. Conservation of electric charge
2. Existence of the Lorentz force
3. Conservation of Magnetic Flux

Other axioms are considered as well, but it seems these three are the most basic and understandable and perhaps are a sufficient foundation for Maxwell's Equations in the context of this thread.

Last edited by a moderator: May 4, 2017