# Maxwell's equations VS. Lorentz & Coulomb force equations

I find Maxwell's equations insufficient and superfluous having the Lorentz & Coulomb's force equations. As far as I see magnetic (Lorentz force) and electric (Coulomb's force) interaction is best defined by these two equations themselves, and although Maxwell's equations can describe quite a few electromagnetic interactions, by not having Lorentz force incorporated in any equation nor included as separate one, Maxwell's equations are automatically doomed to fail to describe any interaction due to this very, very important force. Did I miss something?

Why describe electric interaction with any other but with 'Electric force' equation?

Why describe magnetic interaction with any other but with 'Magnetic force' equation?

*** Electric interaction
Step 1: Electric field potential given by Coulomb's law
Step 2: Electric force (acceleration vector) given by Coulomb's law

*** Magnetic interaction
Step 3: Magnetic field potential given by Biot-Savart law
Step 4: Magnetic force (acceleration vector) given by Lorentz force
====================================================

As far as I see that's all what is necessary to solve any em interaction, no?

1. Gauss's law
- This is obviously about Coulomb's law/electric potential, so why would this equation be "more suitable"?

2. Gauss's law for magnetism
- divB = 0, what in the world? Instead of to describe magnetic potentials or force, to put here Biot-Savart law, or Lorentz force equation or Ampere's force law, they included some equation that has result already calculated in advance? No monopoles? That's as useful as stating "there is no other intelligent life in the Universe".

- Is there anything here we can not calculate with the time integral of four steps given above? How is this induction supposed to accurately describe complete interaction if it is oblivious to Lorentz force?

4. Ampère's circuital law
- Is there anything here we can not calculate with the time integral of four steps given above? This is the best candidate to be saying something about Lorentz force, if it only included "Ampere's force law" too.

I suppose the explanation why these equations are in this form is because that is the most suitable for practical application and experimental setups, but still, my greatest concern is how any of that can accurately work without incorporating Lorentz force and Biot-Savart law in the same fashion as Coulomb's law and electric potential/force.

Last edited:

Dale
Mentor
2020 Award
You do need all four of Maxwell's equations and the Lorentz equation. Biot-Savart is only an approximation to Maxwell.

Born2bwire
Gold Member
You do need all four of Maxwell's equations and the Lorentz equation. Biot-Savart is only an approximation to Maxwell.

To reiterate, practically the entirety of all classical electrodynamics is derived from Maxwell's equations and the Lorentz force. You really do not need much more than that. Practically any set of equations that you can reference are derivable from these five equations (Coulomb's Law (only valid for electrostatics), Biot-Savart (only valid for magnetostatics), Lorentz transformations (special relativity is automatically satisfied in Maxwell's equations), etc.). Still, Lorentz force is only really needed in terms of trajectory problems. Maxwell's equations already incorporate charge and current sources. Thus we do not need to go through the Lorentz force as a mediator for inducing current sources from fields for many problems.

You can think of maxwells equations as equations of motion for the field, and the lorentz force as EOM for the currents. If you are looking for a single equation, it is the EM lagrangian, which will get you to all the equations of motion for the field and currents.

- Is there anything here we can not calculate with the time integral of four steps given above? How is this induction supposed to accurately describe complete interaction if it is oblivious to Lorentz force?

4. Ampère's circuital law
- Is there anything here we can not calculate with the time integral of four steps given above? This is the best candidate to be saying something about Lorentz force, if it only included "Ampere's force law" too.

Electromagnetic waves? Ampere's Law and Faraday's Law together give the velocity and E/B ratio in electromagnetic waves. There are no charges or magnetic dipoles present in the wave. The electric and magnetic field sustain eachother. Can you explain that using your time integral 4-step method?

The knowledge that radiowaves, infrared, visible light, x-rays and gamma rays all are the same phenomenon but with different energies, the prediction of the speed of light and probably the theory of relativity are thanks to Maxwell's equations.

Last edited:
SpectraCat
How are you gonna get them accelerated electrons to emit EM radiation using your 4 step program above? Maxwell's equations and the Lorentz equation accurately describe *all* classical physical phenomena involving electricity and magnetism .. any proposed "substitute" would also need to do that, at the very least.

espen180 said:
Electromagnetic waves? Ampere's Law and Faraday's Law together give the velocity and E/B ratio in electromagnetic waves.

I'm glad you mentioned it. What waves? Yeah, I know how text-book description goes, but none of these equations has to do anything with any radiation, these equations are used to describe experimental setups with current carrying wires and permanent magnets, where properties are measured in amperes, volts, ohms and millimeters. How any of this has anything to do with any em WAVES? It is about fields, not waves. But if Maxwell's equations can describe photons, then Lorentz and Coulomb's force equations can too.

There are no charges or magnetic dipoles present in the wave. The electric and magnetic field sustain eachother. Can you explain that using your time integral 4-step method?

Ok, tell me what experiment are you talking about, show the result given by Maxwell's equations and I will demonstrate I can do the same, or better, with the four steps above.

Electric and magnetic fields do not sustain each other. According to Coulomb's law, Lorentz force and Biot-Savart law, electric field is intrinsic property of any single electron (charge), while magnetic fields forms proportionally to velocity, there is no creation of any other fields here, there will always be electric field whose potential magnitude will be independent of everything, and the magnitude potential of the magnetic field of moving charge will vary according to velocity vector.

Magnetic field is EFFECT of charge motion, it is not the CAUSE for it, however since all moving electrons can interact with this magnetic field, what it can do is to cause electron displacement, i.e. it can cause electric CURRENT, but that does not mean it can be the CAUSE or CREATE any new electrostatic potential.

The knowledge that radiowaves, infrared, visible light, x-rays and gamma rays all are the same phenomenon but with different energies, the prediction of the speed of light and probably the theory of relativity are thanks to Maxwell's equations.

Can you point out what Maxwell's equation predicts the speed of light? And again, what Maxwell's equation says anything about any photons (em radiation)?

SpectraCat; said:
How are you gonna get them accelerated electrons to emit EM radiation using your 4 step program above?

I really have no idea how Maxwell's equations do it, what are you referring to?

Maxwell's equations and the Lorentz equation accurately describe *all* classical physical phenomena involving electricity and magnetism .. any proposed "substitute" would also need to do that, at the very least.

I agree that together with the Lorentz force they do describe all. What I'm saying is that Lorentz force in this form "F= q(E + v x B)", where it is integrated with Coulomb's force, can do it alone. One equation VS. four, do you accept the challenge?

Can you point out what Maxwell's equation predicts the speed of light? And again, what Maxwell's equation says anything about any photons (em radiation)?

http://en.wikipedia.org/wiki/Electr...thesis_that_light_is_an_electromagnetic_wave"
Note that it skips a step. Instead of showing the final wave equations using the permeability and permittivity constants the author immediately substitutes them.

Compare with this to see that Maxwell's equations, using constants measures from electrostatics and magnetostatics, really does predict the wave nature and speed of electromagnetic waves.
http://en.wikipedia.org/wiki/Wave_equation#Introduction"

Note that this was before quantum physics and thus the model was later exanded to account for the particle nature of light.

Last edited by a moderator:
Dale
Mentor
2020 Award
I agree that together with the Lorentz force they do describe all. What I'm saying is that Lorentz force in this form "F= q(E + v x B)", where it is integrated with Coulomb's force, can do it alone. One equation VS. four, do you accept the challenge?
The Lorentz force and the Coulomb's force is woefully inadequate:
1) no way to calculate the magnetic field
2) no waves
3) no interaction between magnetic field and electric field
4) infinite speed of a propagating electric field

DaleSpam said:
The Lorentz force and the Coulomb's force is woefully inadequate:
1) no way to calculate the magnetic field
2) no waves
3) no interaction between magnetic field and electric field
4) infinite speed of a propagating electric field

1) no way to calculate the magnetic field

Step 3.

F= q(E + v x B); E here refers to Coulomb's Law, and B to Biot-Savart law, or at least I'll define it like that, so that basically unrolls to these four:

*** Electric interaction
Step 1: electric field potential given by Coulomb's law
Step 2: Electric force (acceleration vector) given by Coulomb's law

*** Magnetic interaction
Step 3: Magnetic field potential given by Biot-Savart law
Step 4: Magnetic force (acceleration vector) given by Lorentz force
====================================================

- How Maxwell's equations calculate magnetic field?

2) no waves

- I'll make them just like they did. What do you think E and B stand for in Maxwell's equations?

3) no interaction between magnetic field and electric field

- Magnetic and electric fields DO NOT interact.

4) infinite speed of a propagating electric field

- What is propagation speed of E and B fields in Maxwell's equations? What is the expression for E and B?

http://en.wikipedia.org/wiki/Electr...thesis_that_light_is_an_electromagnetic_wave"
Note that it skips a step. Instead of showing the final wave equations using the permeability and permittivity constants the author immediately substitutes them.

Thanks, that's very interesting. Do you think you could explain the process or point someplace I can find answer to these questions:

- "To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum and charge free space, these equations are: Taking the curl of the curl equations gives: - What in the world curl of E and B fields, and even the curl-of-the-curl, is supposed to have with the speed of light? What are they trying to do here, how did anyone ever figure out they should combine curls to get any information about any speed?

- How can we talk about any velocity if we are not calculating FORCES, how can field potentials and their shape tell us anything about any velocity without calculating the FORCE and ACCELERATION first? I need to understand what and WHY they did so I can do the same thing with "my" equations.

Compare with this to see that Maxwell's equations, using constants measures from electrostatics and magnetostatics, really does predict the wave nature and speed of electromagnetic waves. Is this it, the beginning of it? As a time integral my equations are already in similar form like this, can you help me make a 'wave equation' out of it?

#### Attachments

Last edited by a moderator:
Matterwave
Gold Member
3) no interaction between magnetic field and electric field

- Magnetic and electric fields DO NOT interact.

I'm just going to address this one point. If magnetic and electric fields don't interact, then how can you get propagation of an EM wave? In an electro-magnetic wave, the changing electric fields induce a changing magnetic field and vice-versa. That's how you get to propagate that wave out.

How would you explain all of EM radiation if you assert that electric fields and magnetic fields don't interact? I would like to know this.

Born2bwire
Gold Member
- How can we talk about any velocity if we are not calculating FORCES, how can field potentials and their shape tell us anything about any velocity without calculating the FORCE and ACCELERATION first? I need to understand what and WHY they did so I can do the same thing with "my" equations.

Electromagnetic waves do not need sources to propagate. There are no forces or accelerations, there are no charges or currents. Is this it, the beginning of it? As a time integral my equations are already in similar form like this, can you help me make a 'wave equation' out of it?

We are the ones that are telling you that it will not work. Why should we attempt to work out assertions that are unanimously being detracted?

Again, you are using static equations for time-varying situations. This will not work.

Dale
Mentor
2020 Award
1) no way to calculate the magnetic field

Step 3.

F= q(E + v x B); E here refers to Coulomb's Law, and B to Biot-Savart law,
That's fine, you can certainly use Biot-Savart to calculate a B field, but you cannot derive Biot-Savart from Coulomb and Lorentz force, so this is a third equation.

- I'll make them just like they did.
I would very much like to see that. Kindly post your derivation of the wave equation from only Coulomb, Lorentz force, and Biot-Savart.

- Magnetic and electric fields DO NOT interact.

- What is propagation speed of E and B fields in Maxwell's equations?
c, which agrees with experiment. Infinity does not.

Matterwave said:
I'm just going to address this one point. If magnetic and electric fields don't interact, then how can you get propagation of an EM wave? In an electro-magnetic wave, the changing electric fields induce a changing magnetic field and vice-versa. That's how you get to propagate that wave out.

Photon is made of how many electric and magnetic fields, you say?

Please show me that equation where I can see how E and B interact.

How would you explain all of EM radiation if you assert that electric fields and magnetic fields don't interact? I would like to know this.

I'm using the same E and B fields as Maxwell, only I do calculations per point particle, so why do you think I would not to arrive to the same result?

Born2bwire said:
Electromagnetic waves do not need sources to propagate. There are no forces or accelerations, there are no charges or currents.

- What is the relation between the speed of light and the curl of the curl of E and B field?

- Can E and B interact, do we add magnetic and electric vectors, do we superimpose them, or do electric only interact with electric and magnetic with magnetic fields?

Again, you are using static equations for time-varying situations. This will not work.

How about we take on some practical experiment and actually see what will work and what will not work?

DaleSpam said:
I would very much like to see that. Kindly post your derivation of the wave equation from only Coulomb, Lorentz force, and Biot-Savart.

Me too, but without any help, I'll need time. Do you think Maxwell's equations were not derived from Coulomb's law and Biot-Savart law?

No, they do not. I explained 'induction' above. Please, show the equation you believe describes this interaction of E and B field.

c, which agrees with experiment. Infinity does not.

What is expression for E and B in Maxwell's equations?

kcdodd said:
You can think of maxwells equations as equations of motion for the field, and the lorentz force as EOM for the currents.

Yes, kind of like that, but the other way around.
This the whole point behind my arguments, thanks for that.

1.) Maxwell's equation are about em field potentials - Coulomb's law and Biot-Savart law, but approximated in relation to currents and charge densities.

2.) Lorentz force equation is about em fields and forces - Coulomb's law, Biot-Savart law, Coulomb and Lorentz force, but in relation to point charges, no approximations.

These two deal with the same E and B fields, all the same constants are there, all the relations, divergence, curl, flux or whatever is there. There is nothing in 1. that is not in 2, but there are things in 2. that are not in 1. Were approximations for charge densities and current potentials in Maxwell's equations derived from the point particle equations or was it the other way around?

Last edited:
Born2bwire
Gold Member
How about we take on some practical experiment and actually see what will work and what will not work?

Sure, here are two simple problems for you to work out.

1. What are the total fields produced by an electron that is moving in a circle of constant radius R and at some constant speed v? We will let v vary as we see fit. This can easily be done by assuming a constant B field applied normally to the plane of oscillation.

2. What are the total fields at some point (X, 0, 0) as a function of time produced by an electron located at the origin that starts at rest from t=-\infty to t = -0 and starts oscillating along the z-axis. The oscillations can be modeled as a harmonic oscillator of magnitude A and angular frequency \omega.

Photon is made of how many electric and magnetic fields, you say?

Using Coloumb and Lorentz, it is a poorly posed question to ask "how many" E or B fields. There is only a total E and B field which is a superposition of all of them. However, Maxwell can calculate the individual fields in the superposition. Coloumb/Lorentz cannot. In the case of a photon, you can say that there is a single E and B field which source each other.

Please show me that equation where I can see how E and B interact.

I'm using the same E and B fields as Maxwell, only I do calculations per point particle, so why do you think I would not to arrive to the same result?

There are no particles in the wave, hence the E field is the source of the B field and vice versa.

- What is the relation between the speed of light and the curl of the curl of E and B field?

The wave equation is a solution of Maxwell's equations. It shows that EM waves can only exist if they have speed c.

- Can E and B interact, do we add magnetic and electric vectors, do we superimpose them, or do electric only interact with electric and magnetic with magnetic fields?

I don't understand the question. E fields add together, as do B-fields. The time derivative of one field (ex. E) is related to the other (ex. B).

Me too, but without any help, I'll need time. Do you think Maxwell's equations were not derived from Coulomb's law and Biot-Savart law?

They weren't. They were derived experimentally.

No, they do not. I explained 'induction' above. Please, show the equation you believe describes this interaction of E and B field.

What is expression for E and B in Maxwell's equations?

B: Mag. Gauss and Ampere

1.) Maxwell's equation are about em field potentials - Coulomb's law and Biot-Savart law, but approximated in relation to currents and charge densities.

2.) Lorentz force equation is about em fields and forces - Coulomb's law, Biot-Savart law, Coulomb and Lorentz force, but in relation to point charges, no approximations.

Coloumb and Lorentz define E and B fields. Maxwell describes them and explains their sources.

There is similar thread where everyone agreed about some of the main points I'm arguing here:

OP:
I am trying to bridge a gap between Maxwell equations and Lorentz force. I know that they are not independent and in theory, one could be derived from the other but I cannot see that.

More physics oriented people prefer the Lorentz force because it describes the effect of B and E as a final force on the particle, which is good.

Born2bwire:
- Oh, one nice way you can derive the Lorentz force ...

marcusl:
- Maxwell's equations may be derived in the same way, starting from Coulomb's law...

atyy:
- The Lorentz force cannot be derived from Maxwell's equations.

atyy
- I was thinking that the Lorentz force law can't be derived from Maxwell's equations in the same way that charge conservation can...

samalkhaiat:
- No, it is not possible to derive Newton's 2nd law (i.e., Lorentz force) from Maxwell equations. If such cosistent derivation is possible, then Maxwell or Lorentz would have done it.
=====================

So, why can't we agree they can be derived one from the other, for a start?

Sure, here are two simple problems for you to work out.

1. What are the total fields produced by an electron that is moving in a circle of constant radius R and at some constant speed v? We will let v vary as we see fit. This can easily be done by assuming a constant B field applied normally to the plane of oscillation.

2. What are the total fields at some point (X, 0, 0) as a function of time produced by an electron located at the origin that starts at rest from t=-\infty to t = -0 and starts oscillating along the z-axis. The oscillations can be modeled as a harmonic oscillator of magnitude A and angular frequency \omega.

Ok, but you are supposed to show the solution with Maxwell's equations. I really need to understand better what are these equations actually doing and we need the result so we can compare.

TEST EXAMPLE:
- Electron is moving in a line accelerating from 100m/s to 500m/s what is E and B equal to numerically when its velocity is 300m/s, and please say with few words how E and B change over time.

espen180 said:
However, Maxwell can calculate the individual fields in the superposition. Coloumb/Lorentz cannot. In the case of a photon, you can say that there is a single E and B field which source each other.

What is the value of E and B if an electron is moving at 400m/s?

Ok, two electrons move with the velocity of 900.000m/s next to each other in the same direction and separated by 0.05mm. What are the values of their E and B fields. What is their separation distance after 10 seconds?

B: Mag. Gauss and Ampere

E stands for 'electric field', what do you replace the variable E with?
B stands for 'magnetic field', what do you replace the variable B with?

Coloumb and Lorentz define E and B fields. Maxwell describes them and explains their sources.

"Define" and "describe" has pretty much the same meaning, except that 'define' is superior to 'describe'. But at least you know we are talking about the same fields and same properties.

SpectraCat
So, why can't we agree they can be derived one from the other, for a start?

Did you see the post in that thread that gives references for where you can find out how to derive the Maxwell equations from a relativistic treatment of Coulomb's law? Why not just check those out, since they seem to be what you are looking for.

I doubt that you will ever convince the rest of us that the Maxwell equations are "insufficient and superfluous" or whatever it was you said in your OP, and I am almost certain that you will not replace them with your treatment, whatever it is. What will most likely happen if you go through the derivation by yourself is that you will come up with precisely the same equations as Maxwell, thereby reinventing the wheel. Not that it won't be an instructive exercise worth doing ...

Also, it seems that Maxwell was aware of the Lorentz force, and included it in some form as one of his original 8 equations. The Lorentz force cannot be derived directly from Maxwell's equations alone, although it does follow after some reasonable assumptions are made (see http://arxiv.org/abs/physics/0206022) However, this certainly does not mean that Maxwell's equations can be derived from the Lorentz force, which is what you seem to be claiming.

1) What is the value of E and B if an electron is moving at 400m/s?

2) Ok, two electrons move with the velocity of 900.000m/s next to each other in the same direction and separated by 0.05mm. What are the values of their E and B fields. What is their separation distance after 10 seconds?

3)
E stands for 'electric field', what do you replace the variable E with?
B stands for 'magnetic field', what do you replace the variable B with?

4)
"Define" and "describe" has pretty much the same meaning, except that 'define' is superior to 'describe'. But at least you know we are talking about the same fields and same properties.

1) Gauss's Law gives the E-field and Ampere's Law Gives the B-field.

2) Troublesome integral, so I'm not gonna bother with it. I'm guessing your argument is that Maxwell doesn't tell anything about forces, only fields. Even if this is true, the fields are defined from forces, so the argument doesn't work.

3) I don't understand the question.

4) I disagree. For example, I can define space-time curvature as the source of gravitational attraction, but I still don't know how to work with it. Einstein's equation describes space-time curvature and lets me do stuff. The same applies to E and B-fields.

Dale
Mentor
2020 Award
Do you think Maxwell's equations were not derived from Coulomb's law and Biot-Savart law?
No. In fact, it is quite the other way around. Coulomb's law and the Biot-Savart law can be derived from Maxwell's equations in the electrostatic and magnetostatic limits respectively. Coulomb and Biot-Savart are special cases of the more general Maxwell, and it is simply not mathematically possible to start with only a special case and derive a generalization, you always have to go the other direction.

DaleSpam said:
No. In fact, it is quite the other way around. Coulomb's law and the Biot-Savart law can be derived from Maxwell's equations in the electrostatic and magnetostatic limits respectively. Coulomb and Biot-Savart are special cases of the more general Maxwell, and it is simply not mathematically possible to start with only a special case and derive a generalization, you always have to go the other direction.

Let me make sure I understand you correctly. Biot-Savart and Coulomb's law, equations for POINT PARTICLES, are special case of charge distributions and average current density used in Maxwell's equations? I suppose then 'amperes' is what describes general case, while individual charge and velocity is special case, and so the 'current' in a wire is general case and electric potential of a single charge given by the Coulomb's law is special case?

Why are you trying to disagree without even thinking, what is your agenda? Do you have something against Lorentz force or have you fallen in love with Maxwell's equations? I see absolutely no reason why would anyone take any sides in this argument. -- What is the full expression for the E and B terms found on the right side of Maxwell's equations?