Maxwell's equations: meaning, derivation and applicability

[ambros]

Hello, I'm a computer programmer and although I do have education in classical physics including electrical courses and as well as practical experience with assembly and testing of UV radiation equipment, their power supplies and overall wiring, I'm surprised to realize I have no idea what is the purpose, derivation (experimental origin) and the meaning of Maxwell's equations as I can not recall to have used any of them for anything.

 

 

This realization happened when I was supposed to simulate - model and visualize in 3D - the simplest electromagnetic interaction there is, between electric and magnetic fields from only two electrons/positrons, including the tracing of their individual shape/intensity. Everyone referred me to Maxwell's equations, but - to cut the story short - after much trouble, search and experimenting I had to conclude Maxwell's equations are simply useless for this. They seemed far too specific in relation to experimental setups with wires and permanent magnets, and so they seemed to lack much of the information required to solve any such general case with individual free charges.

 

I found a solution via Coulomb's law, Biot-Savart law and Lorentz force, I also found I have complete information of these EM field potentials in those equations, including their field "density" or magnitude distribution and geometry, and as well as description of force vectors and not only field vectors, I found that these "older" equations incorporate far more general and complete information about electromagnetic fields and forces.

 

 

Well, that is my surprising conclusion anyway, so of course I'm looking for some explanation and hopefully someone here is very familiar with Maxwell's equations so can explain my misunderstandings and say something about their actual implications - what they mean, what they describe, how was it derived and how to apply them to practical situations.

 

 

 

TO BE MORE SPECIFIC:

 

1. Gauss's law: divE= p/e0

 

- When do we use this equation? How to get rid of the divergence operator so to solve for just E, and would that be a vector or scalar quantity? Divergence of E field according to Coulomb's law is zero, it has uniform magnitude gradient dropping off with inverse square law, does that not mean divE=0?

 

 

 

2. Gauss's law for magnetism: divB= 0

 

- According to Biot-Savart law which actually describes this magnetic field potential for point charges, *not wires*, this field is toroidal, its magnitude falls off with inverse square law in perpendicular plane to velocity vector (current direction), but it also falls with the angle according to vector cross product, so at the end it looks like doughnut (toroid/torus) and not like a "ball" of an electric field. This actually means that divergence of this particular magnetic field 'due to moving charge' (this is not intrinsic magnetic dipole moment), has non zero divergence and non zero rotation (curl).

 

Yes, if you take an infinite wire then divB=0, but that does not say anything about individual fields, it is very specific case that does not reveal anything about how individual magnetic fields look in front and behind that 90 degree plane, it is very crude approximation and hence lacks information. -- Let's say divB=0, then what is just B equal to? I do not see any information about B field here, so where and when do we ever use this equation?

 

 

 

3. Maxwell–Faraday equation: rotE= - dB/dt

 

- According to Coulomb's law E field has no rotation (curl), it is more of a "radial" kind of thing, so what in the world can this mean if we get rid of the curl operator and solve for just E? How can 'curl of E' tell us anything if 'curl of E' is always supposed to be constant and zero?

 

What does "dB" refer to?

a.) to second equation: rotE= - (divB= 0)/dt ?

b.) to fourth equation: rotE= - (rotB= J + dE/dt)/dt ?

c.) to Biot-Savart law: rotE= - (B= k qv x 1/r^2)/dt ?

d.) something else?

 

 

 

4. Ampère's circuital law: rotB= J + dE/dt

 

- What do we get when we get rid of the curl, how to do it, and what just B then equals to? What J equals to? What "dE" refers to, 1st equation, 3rd equation, Coulomb's law? Are these equations for just one field or do they require at least two like Coulomb's law has Q1 and Q2 and Newton's gravity has M1 and M2?

 

Does rotB, J and dE refer to the fields (potential difference) of one and the same particle, or rotB refers to one field and dE to separate another field? And also, 3rd and 4th equations appear to be kind of 'circular definition' and self-referencing, but hopefully answers to previous question will explain this.

 

 

 

In addition, what is the full and exact meaning of "changing electric field causes... B" or "varying magnetic field produces... E". How can E or B field vary if you look at only one charge (electron) or two? Can E potential of individual charges actually change and can there be a creation of any new magnetic or electric potential (new fields)?

 

 

 

Thank you.

Edited March 19, 2010 by ambros

[darkenlighten]

TO BE MORE SPECIFIC:

 

1. Gauss's law: divE= p/e0

 

- When do we use this equation? How to get rid of the divergence operator so to solve for just E, and would that be a vector or scalar quantity? Divergence of E field according to Coulomb's law is zero, it has uniform magnitude gradient dropping off with inverse square law, does that not mean divE=0?

 

So for this, you would use this when you are looking at something with a charge to find the electric field around it. You want to apply stokes theorem to get it into integral form. From there you would take a gaussian surface around the object enclosed in order to get the electric field at a distance away from the object.

 

So ∇·E = Q/εo ⇒ ∫E·nda = Q(enclosed)/εo

 

If you have more questions about this I can expand.

 

 

2. Gauss's law for magnetism: divB= 0

 

- According to Biot-Savart law which actually describes this magnetic field potential for point charges, *not wires*, this field is toroidal, its magnitude falls off with inverse square law in perpendicular plane to velocity vector (current direction), but it also falls with the angle according to vector cross product, so at the end it looks like doughnut (toroid/torus) and not like a "ball" of an electric field. This actually means that divergence of this particular magnetic field 'due to moving charge' (this is not intrinsic magnetic dipole moment), has non zero divergence and non zero rotation (curl).

 

Yes, if you take an infinite wire then divB=0, but that does not say anything about individual fields, it is very specific case that does not reveal anything about how individual magnetic fields look in front and behind that 90 degree plane, it is very crude approximation and hence lacks information. -- Let's say divB=0, then what is just B equal to? I do not see any information about B field here, so where and when do we ever use this equation?

 

 

Okay so here you can find the magnetic field using amperes law which is similar to guass's law, which is ∇ x B = μoI, or get the integral form. so you would, for a wire, apply an amperian loop around the wire and take the line integral and set it equal to μoI, where I is the current enclosed.

 

∇ x B = μoI ⇒ ∫B·dl = μoI(enclosed)

 

Once again ask further if you want more info.

 

 

3. Maxwell–Faraday equation: rotE= - dB/dt

 

- According to Coulomb's law E field has no rotation (curl), it is more of a "radial" kind of thing, so what in the world can this mean if we get rid of the curl operator and solve for just E? How can 'curl of E' tell us anything if 'curl of E' is always supposed to be constant and zero?

 

What does "dB" refer to?

a.) to second equation: rotE= - (divB= 0)/dt ?

b.) to fourth equation: rotE= - (rotB= J + dE/dt)/dt ?

c.) to Biot-Savart law: rotE= - (B= k qv x 1/r^2)/dt ?

d.) something else?

 

Here dB/dt is the derivative of the magnetic field, or the change in magnetic field. You can find B from Ampere's Law. So this is an induced electric field from a change in the magnetic field, once again apply stokes theorem to get it in integral form.

 

4. Ampère's circuital law: rotB= J + dE/dt

 

- What do we get when we get rid of the curl, how to do it, and what just B then equals to? What J equals to? What "dE" refers to, 1st equation, 3rd equation, Coulomb's law? Are these equations for just one field or do they require at least two like Coulomb's law has Q1 and Q2 and Newton's gravity has M1 and M2?

 

Does rotB, J and dE refer to the fields (potential difference) of one and the same particle, or rotB refers to one field and dE to separate another field? And also, 3rd and 4th equations appear to be kind of 'circular definition' and self-referencing, but hopefully answers to previous question will explain this.

 

In addition, what is the full and exact meaning of "changing electric field causes... B" or "varying magnetic field produces... E". How can E or B field vary if you look at only one charge (electron) or two? Can E potential of individual charges actually change and can there be a creation of any new magnetic or electric potential (new fields)?

 

Thank you.

 

Lastly I already gave Amperes law above and the dE/dt is maxwells correction and isnt always needed, especially if there is not a change in the electric field. and J is the volume charge, so I = ∫J·da

 

There for you can clearly see that these will work for wires and most likely for what you were trying to do. If you want me to show you how to do specific scenario please ask. Like how you would get a magnetic field around a wire with current I or something like that.

[ambros]

Ok, thank you.

 

So for this, you would use this when you are looking at something with a charge to find the electric field around it. You want to apply stokes theorem to get it into integral form. From there you would take a gaussian surface around the object enclosed in order to get the electric field at a distance away from the object.

 

So ∇·E = Q/εo ⇒ ∫E·nda = Q(enclosed)/εo

 

If you have more questions about this I can expand.

 

We will eventually need to stick to differential form due to our time stepping algorithm being time integral in itself, beside equations are more illustrative in differential form. Ok, so even after you "cooked" it I still do not see what is just E equal to, that's some 'dot product' there, I do not really see how can we apply either form, and why, oh why just not use Coulomb's law. EXAMPLE: - electron is moving along x-axis at 25m/s, what is the magnitude of its electric field potential in arbitrary direction at distance 'r'.

 

Coulomb's law: E = k * q/r^2

 

1st equation - Gauss's law: E = ?

 

 

Okay so here you can find the magnetic field using amperes law which is similar to guass's law, which is ∇ x B = μoI, or get the integral form. so you would, for a wire, apply an amperian loop around the wire and take the line integral and set it equal to μoI, where I is the current enclosed.

 

∇ x B = μoI ⇒ ∫B·dl = μoI(enclosed)

 

Once again ask further if you want more info.

 

Do you think you can describe the full 3D shape of a magnetic field of a single moving charge with Gauss's law for magnetism? Can we please stick to differential form, it's not there just for decoration, right? Anyway, that is 'dot product' of B and some stuff, but what is just B equal to? EXAMPLE: - electron is moving along x-axis at 25m/s, what is the magnitude of its magnetic field potential in arbitrary direction at distance 'r'.

 

Biot-Savart law: B = k * qv x 1/r^2

 

2nd equation - Gauss's law for magnetism: B = ?

 

 

Here dB/dt is the derivative of the magnetic field, or the change in magnetic field. You can find B from Ampere's Law. So this is an induced electric field from a change in the magnetic field, once again apply stokes theorem to get it in integral form.

 

Lastly I already gave Amperes law above and the dE/dt is maxwells correction and isnt always needed, especially if there is not a change in the electric field. and J is the volume charge, so I = ∫J·da

 

Are you actually saying these equations can not be used before we apply stokes theorem, gaussian surfaces and get them in integral form first? So, the 3rd equation defines dB/dt with 4th equation, and then 4th equation defines dE/dt with 3rd equation, is that not self-referencing and circular definition? EXAMPLE: - electron is moving along x-axis at 25m/s, what is the magnitude of its electric and magnetic field potential in arbitrary direction at distance 'r'.

 

Coulomb's law: E = k * q/r^2

Biot-Savart law: B = k * qv x 1/r^2

 

3rd equation - Maxwell–Faraday: E = ?

4th equation - Ampère's circuital law: B = ?

[Radical Edward]

Maxwell's equations are the most fundamental equations in that you can use them to describe any classical electromagnetic system.

 

that doesn't make them the most practical to use however, because often there is quite a bit of work involved in actually deriving stuff from the equations. For complex systems, such as Mie scattering, thin film optics, fibre optics, Supercritical Fluorescence, evanescent wave calculations it is often critical that you start from Maxwell's equations (or very close). For example for thin film calculations you need the wave equation which is derived directly, then you remove the time dependence and simulate the system based on boundary conditions dictated my Maxwell's eqns. Often you need to integrate them with a number of other mathematical theorems to describe systems. I use the dyadic Green functions a lot in my work for example.

 

The real issue is determining the appropriateness of using particular levels of description. You could describe a magnifying glass from first principles using maxwell's equations if you wanted to, but you would be stupid to, since geometrical approximations work perfectly well.

 

This page here shows you how to get coulomb's law from maxwell's first equation:

 

http://people.bu.edu/wwildman/WeirdWildWeb/courses/sl/resources/physics/em/em01.htm

 

So if Coulombs law works best for you, then use coulomb's law - there's no need to re-invent the wheel.

[darkenlighten]

Okay so point like charges, it is usually best to use Coloumb's Law and the Biot-Savart Law, but for everything else it is probably not true. Guass's Law and Ampere's law make things so much more simpler when it comes to everything else.

 

And yes you can use the laws in differential form, but you would need to know the E or B before hand. Here is an example:

 

 

So for inside it would be different because the charge enclosed would be different. And Guass and Ampere's Law call for symmetry.

 

The Biot-Savart Law only holds for steady currents, so one electron moving will not be accurately depicted by the Biot-Savart Law, nor can you really with Ampere's Law.

 

Are you actually saying these equations can not be used before we apply stokes theorem, gaussian surfaces and get them in integral form first? So, the 3rd equation defines dB/dt with 4th equation, and then 4th equation defines dE/dt with 3rd equation, is that not self-referencing and circular definition? EXAMPLE: - electron is moving along x-axis at 25m/s, what is the magnitude of its electric and magnetic field potential in arbitrary direction at distance 'r'.

 

I am saying you cannot solve for E or B without applying Stoke's Theorem. You can use it in differential form if you already have E or B but that will obviously not solve for either. You might be able to use it to find E and B but I cannot think of anyway off hand. And the dB/dt and dE/dt are external known fields one way or another.

 

Also for your moving electron and the E field:

 

[ambros]

So if Coulombs law works best for you' date=' then use coulomb's law - there's no need to re-invent the wheel.[/quote']

 

As far as the original problem goes it is solved, but it uncovered great many things for me, so what is left now is curiosity because this solution implies Coulomb's and Biot-Savart law tell different and more complete story than Maxwell's equations and yet they are supposed to talk about the same E and B fields.

 

 

There are two kinds of fields, "radial" like gravity and electric fields, and we have "rotational", like vortexes, whirlpools or magnetic fields. Uniform and constant "radial" field potentials have zero divergence and zero rotation (curl), it's a uniform magnitude distribution and inverse square law which defines topology and geometry of an electric field, not the other way around. You can not define any numerical result if all your equation says is -"IT IS ROUND", that's poor description to obtain some real numbers, especially if the field-lines or force-lines are not actually round nor curved.

 

The quantity of electric potential of a single electron is discrete and the smallest quantified one, it does not vary or change but only its magnitude is "felt" less as the distance increases in whatever direction. The only way E can change is when measured from arbitrary reference frame or when there is relative displacement and change in the distance 'r', which automatically then represents the difference in electric potential, but there is no way B field can 'create' or 'cause' E field, it can only interact with the B field of *another charge(s)*, displace them and hence create electric current or motion of electrons, resulting in a change of RELATIVE electric (and magnetic) potential of TWO or more charges, and B field can not change or create any new electric potential in any other way. E field, however, "creates" B field due to its spatial motion so the magnitude (range) of B field increases with the velocity of E field, this is not quite what Maxwell's equations say, except for 4th equation, that one seem not very wrong, but it seem to be just one-way approximation of Lorentz force and Biot-Savart law. 3rd equation can be *interpreted* *generally* so it is "correct", but it's far from being NUMERICAL solution or exact definition. The very fact it is an integral (infinite wire approximation) make these equations loose degrees of freedom.

 

This actually is self-referencing definition in any case, but it's fine, it's called 'recursive algorithm', that's where output of the previous derivative becomes input for the next one within integral called "em induction", technically it's a fractal, which is what I find very interesting.

 

 

This page here shows you how to get coulomb's law from maxwell's first equation:

 

http://en.wikipedia.org/wiki/Gauss%27s_law -- This page says: -"Gauss's law can be derived from Coulomb's law..", which is to be expected as Coulomb's law describes this field fully, it actually has information about both curl and divergence, and flux and gradient.. everything. Then, that page also says: -"Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E." -- With this, I believe Wikipedia confirmed my suspicion and completely answered all my questions regarding this 1st equation - Gauss's law - it's really useless until we "cook" it and end up with Coulomb's law, by some magic.

 

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Merged post follows:

Consecutive posts merged

 

 

Okay so point like charges' date=' it is usually best to use Coloumb's Law and the Biot-Savart Law, but for everything else it is probably not true. Guass's Law and Ampere's law make things so much more simpler when it comes to everything else.

[/quote']

 

http://en.wikipedia.org/wiki/Ampere

 

1.) - "The Ampere is the SI unit of electric current, the ampere is a measure of the amount of electric charge passing a point per unit time."

 

2.) - "DEFINITION: Ampère's force law states that there is an attractive force between two parallel wires carrying an electric current. This force is used in the formal definition of the ampere... a current of one ampere is one coulomb of charge going past a given point per second"

 

It is amazing to see this unit is actually defined with Biot-Savart Law and Lorentz force (Ampere's force law), but in any case I do not see any problem of converting Amperes to individual moving electrons, while the other way around is like modeling individual molecules of water by measuring the flow in liters per second. I can model any electric current in any wire, finite or not, by "thin rows" of electrons going really fast, or with "thick bunched rows" of many electrons drifting slowly. There is no scenario that can not be solved with Coulomb's, Biot-Savart law and Lorentz force.

 

 

So, I should be able to put two electrons next to each other 1 meter apart and give them velocity of "1 Ampere" in the same direction, this would effectively model the experimental setup for the unit of 'Ampere' and two its parallel wires. However we do not even need to integrate with Biot-Savart and Coulomb's law, this can be solved in one step, in a single time instant and if we now employ Lorentz force equation and see if interaction between the two virtual magnetic fields of these two "wires" matches "attractive force of 2 × 10–7 newtons per metre of length". If it does, then I believe that makes a full circle as far as correctness and validity goes.

 

 

And yes you can use the laws in differential form, but you would need to know the E or B before hand.

 

Ok, so this means that 1st and 2nd equation are about individual field potentials, while 3rd and 4th equations are about at least TWO spatially separated fields, this should be about forces (em interaction, induction), but I'm afraid that can't work without Lorentz force equation.

 

 

The Biot-Savart Law only holds for steady currents, so one electron moving will not be accurately depicted by the Biot-Savart Law, nor can you really with Ampere's Law.

 

With point charges I do not see any problem to that or anything else.

 

 

I don't know what to say about your pictures, it helped me find those derivations in Wikipedia and understand it better eventually, thank you, but you ended up twice with some sort of Coulomb's law and have not given one solution for B field, which I guess would look like Biot-Savart law anyway. -- Thanks again, and I do appreciate your help and everyone's involvement, even though we inevitably must disagree, unless I'm terribly wrong, which seems unlikely, to me.

Edited March 21, 2010 by ambros
Consecutive posts merged.

[darkenlighten]

By definition of the Biot-Savart Law, you need steady currents in order for it to be correct. What you have for a single charge is incorrect. You need steady currents and a single charge moving is not a steady current, unless it is in circular motion like an electron "orbit" and then you need the current and not qv.

 

And I didn't solve for a B field because it is the exact same method as solving for an E field using Guass's Law, but with Amperian Loops (line integrals).

 

It doesn't have to be infinite wires, that was just the easiest example, it can solve finite solutions.

 

And of course it will come out the same as Coloumb's and Biot-Savart Law, otherwise it would be incorrect...And Maxwell's Equations were derived from Coloumb's and Biot-Savart Laws, but they make finding the E and B field so much easier.

 

What is this about there is no way a B field can create an E field??? That is wrong, it has to be changing, but a changing B field WILL create an E field, that is what the Maxwell-Faraday Equation states.

 

Yea I'm going to have to disagree with you also and be stern that you are wrong. At least in your method of thinking about this...

[Bignose]

And yes you can use the laws in differential form, but you would need to know the E or B before hand. Here is an example:

 

No, E and B are all coupled together and you solve them all together. You will have to use some sophisticated computational techniques, but it can be done.

 

The simplest idea is some sort of sequential solver. That is, you start with an initial guess for B0 and E0. Then, treating E0 as a constant, you solve for B, and call that solution B1. Then, treating B1 as a constant, you solve for E1. And repeat, probably a great number of times. There are all sorts of constraints on how large of a jump between iterations you can let your solution take for the simulation to remain constant, what kind of method you use to solve, etc. etc., that are beyond the scope of a forum post. There is a huge body of work on the computational solution of nonlinear equations, however. I am most familiar with the methods to solve fluid mechanics equations, but the high-level view is the same. Both fluid mechanics and Maxwell's equations are conservation equations, coupled together and nonlinear. If the situation does not allow simplifying assumptions, then you may have to solve the entire system computationally.

[ambros]

By definition of the Biot-Savart Law' date=' you need steady currents in order for it to be correct. What you have for a single charge is incorrect. You need steady currents and a single charge moving is not a steady current, unless it is in circular motion like an electron "orbit" and then you need the current and not qv.

[/quote']

 

I'm talking about Biot-Savart for point particles. Ok, I understand why you disagree, but I can demonstrate, I think. Do you want to come up with some scenario, or would you prefer to see if I can indeed reconstruct the virtual experiment for the unit of "Ampere" and reproduce the results with two parallel virtual wires made only of one electron each?

 

 

Circles, spirals, orbits, loops... I have not really tried it, but the point is that we can "freeze the time" in virtual reality (or with pen and paper) and if you look at any single point (wire or not) you can reduce it into single charge with the velocity vector pointing in the direction of the wire and with the velocity magnitude to mach given amperes or voltage/ohms, this effectively models the wire and electric current at that point, and as we can do this for any arbitrary point it means we can obtain virtual "measurements" when, where and as much as we please.

 

If you can define "unsteady current" quantitatively then it is all the same, it is easy to model things like turning switch off/on every half second, or send more or less charges down the virtual wire, or send them with different velocities at certain intervals, whatever.

 

 

What is this about there is no way a B field can create an E field??? That is wrong, it has to be changing, but a changing B field WILL create an E field, that is what the Maxwell-Faraday Equation states.

 

I do not think your phrasing actually represents what Maxwell's equations are saying, let me correct:

 

WRONG:

- Changing B field WILL create an E field

 

CORRECT:

- Changing B field MAY create a difference in E potential - induce current

(moving magnet around *straight* wire will hardly induce any current)

 

 

Fields can not be created, that's like saying electrons get created. Charge, electron and electric field is one and the same thing if you look at point particles, there is no electron that is not electric charge which is not electric field, in these equations simply "q". Electron is fundamental particle, the smallest amount of charge (electric field) that can exist, it can not change, disappear or be created in these experiments.

 

 

Coulomb's law: E = k * q/r^2

Biot-Savart law: B = k * qv x 1/r^2

 

Anyhow, what I'm saying is wrong as much as these two equations are wrong, I'm just saying what I read from them. Do you see any relation of E field with the B field in Coulomb's law? That's what I said, but it is still not generally in disagreement with 3rd Maxwell's equation as I explained above B can indeed create electric potential by displacing other electrons and hence creating electric current. What I'm saying is that B field can not create any new E field/electron/charge.

 

 

rotE= - dB/dt

 

Imagine these are two electrons, we are trying to figure out how E of one electron changes as it passes through the dB of the other electron. Do you think this magnetic field can change distribution or the amount of electric charge single electron carry? Do you not agree potential of an electric field is "radial" and that it has no rotation, so do you really think the change in magnitude potential in this magnetic field can indeed make any changes about *rotation* (curl) of the electric field? But most obviously, if electric fields have constant and zero rotation than this equation simply "does not compute".

[darkenlighten]

I'm talking about Biot-Savart for point particles. Ok, I understand why you disagree, but I can demonstrate, I think. Do you want to come up with some scenario, or would you prefer to see if I can indeed reconstruct the virtual experiment for the unit of "Ampere" and reproduce the results with two parallel virtual wires made only of one electron each?

 

And so am I...

 

I do not think your phrasing actually represents what Maxwell's equations are saying, let me correct:

 

WRONG:

- Changing B field WILL create an E field

 

CORRECT:

- Changing B field MAY create a difference in E potential - induce current

(moving magnet around *straight* wire will hardly induce any current)

 

Fields can not be created, that's like saying electrons get created. Charge, electron and electric field is one and the same thing if you look at point particles, there is no electron that is not electric charge which is not electric field, in these equations simply "q". Electron is fundamental particle, the smallest amount of charge (electric field) that can exist, it can not change, disappear or be created in these experiments.

 

No, a changing Magnetic Field will induce an Electric Field. And no the charge, electron and electric fields are not one in the same when it comes to point particles. They exist because of each other but not one in the same.

 

Coulomb's law: E = k * q/r^2

Biot-Savart law: B = k * qv x 1/r^2

 

Anyhow, what I'm saying is wrong as much as these two equations are wrong, I'm just saying what I read from them. Do you see any relation of E field with the B field in Coulomb's law? That's what I said, but it is still not generally in disagreement with 3rd Maxwell's equation as I explained above B can indeed create electric potential by displacing other electrons and hence creating electric current. What I'm saying is that B field can not create any new E field/electron/charge.

 

I'm glad you said this, take a look:

 

This is taken from the "Introduction to Electrodynamics" (Third Edition) by David J. Griffiths

 

 

rotE= - dB/dt

 

Imagine these are two electrons, we are trying to figure out how E of one electron changes as it passes through the dB of the other electron. Do you think this magnetic field can change distribution or the amount of electric charge single electron carry? Do you not agree potential of an electric field is "radial" and that it has no rotation, so do you really think the change in magnitude potential in this magnetic field can indeed make any changes about *rotation* (curl) of the electric field? But most obviously, if electric fields have constant and zero rotation than this equation simply "does not compute".

 

So you are asking if a changing Magnetic Field of an electron would change another electrons charge??? Because it would only change the E field around the electron, not the properties of the electron itself. This goes back to what you were trying to say about it being one in the same. To change the E Field around an electron does not constitute a change in the electron's properties.

 

Hopefully you are starting to understand something of what I am saying.

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Merged post follows:

Consecutive posts merged

Note that B = (μo/4π) * (q v x r) / r^2 is an alright approximation for v^2 << c^2, but is nonetheless not the exact answer. I am not sure how accurate you wanted to be with this.

 

And I'm just going to add in here to make it clear, for point like particles, Coloumb's Law and Biot-Savart Law is probably going to be the easiest way to get the E and B field. I just want to make sure we were not ever arguing that.

[ambros]

No, a changing Magnetic Field will induce an Electric Field.

 

So you are asking if a changing Magnetic Field of an electron would change another electrons charge??? Because it would only change the E field around the electron, not the properties of the electron itself. This goes back to what you were trying to say about it being one in the same. To change the E Field around an electron does not constitute a change in the electron's properties.

 

Hopefully you are starting to understand something of what I am saying.

 

Your phrasing is wrong, which makes what you say ambiguous. Current (motion) is what is induced, E fields are there before and after with their fields unchanged. What changes is relative position and hence RELATIVE electric potential, not the potential of their own electric fields. E field is not "around" electron, it's what electron is and what does not change, it's what defines its charge and all the rest of its electric properties.

 

 

And no the charge, electron and electric fields are not one in the same when it comes to point particles. They exist because of each other but not one in the same.

 

The "q" symbol in these equations I can call "charge", "electron" or "electric field", and you should know exactly what is it I'm talking about, it can not be anything else because that's what electron is, there is nothing more to it.

 

 

I'm glad you said this, take a look:

 

This is taken from the "Introduction to Electrodynamics" (Third Edition) by David J. Griffiths

 

What part of "simply wrong" did convince you?

 

Please pick some example so I can demonstrate.

 

 

Note that B = (μo/4π) * (q v x r) / r^2 is an alright approximation for v^2 << c^2, but is nonetheless not the exact answer. I am not sure how accurate you wanted to be with this.

 

I have analyzed that, "retarded time" error can not be experimentally confirmed, which directly implies this correction is absolutely unnecessary as we would not be able to verify the difference anyway, but if there is such experiment please let me know. The other thing is that we can always look at these fields from their reference frame, where they are stationary, right?

 

 

And I'm just going to add in here to make it clear, for point like particles, Coloumb's Law and Biot-Savart Law is probably going to be the easiest way to get the E and B field. I just want to make sure we were not ever arguing that.

 

Yes, but I'm saying it is the only way, not just easiest. Maxwell's equations are about WIRES, because of that they lack one whole dimension, behind and in front of the charge.

[darkenlighten]

Your phrasing is wrong, which makes what you say ambiguous. Current (motion) is what is induced, E fields are there before and after with their fields unchanged. What changes is relative position and hence RELATIVE electric potential, not the potential of their own electric fields. E field is not "around" electron, it's what electron is and what does not change, it's what defines its charge and all the rest of its electric properties.

 

No. Electrons have a charge and electric field, but it is not its charge or electric field, they are not one in the same, see http://www.scienceforums.net/forum/showthread.php?t=46717

 

The "q" symbol in these equations I can call "charge", "electron" or "electric field", and you should know exactly what is it I'm talking about, it can not be anything else because that's what electron is, there is nothing more to it.

 

If you say this you are incorrect, q is not an electron, nor is it an electric field. I don't know why you would insist on this.

 

What part of "simply wrong" did convince you?

 

Please pick some example so I can demonstrate.

 

I have analyzed that, "retarded time" error can not be experimentally confirmed, which directly implies this correction is absolutely unnecessary as we would not be able to verify the difference anyway, but if there is such experiment please let me know. The other thing is that we can always look at these fields from their reference frame, where they are stationary, right?

 

B = (μo/4π) * (q v x r) / r^2 is a fine enough approximation, its just not completely accurate.

 

Yes, but I'm saying it is the only way, not just easiest. Maxwell's equations are about WIRES, because of that they lack one whole dimension, behind and in front of the charge.

 

Maxwell's equations are not about just wires, they can evaluate the E and B field of a point like charge, a surface, and a volume of any type. I'm sorry you are wrong on this. And I'll state again, Maxwell's equations are derived from Coloumb's and Biot-Savart Law.

 

I think for you to really understand what is going on you would need to take an Electrodynamics course. Griffiths "Introduction to Electrodynamics" is a really good book if you can get it or buy it.

[Bignose]

And I'll state again, Maxwell's equations are derived from Coloumb's and Biot-Savart Law.

 

No, it is the other way around. Coloumb's Law and Biot-Savart Law are special cases of Maxwell's equations.

 

Maxwell's equations are based on Ampere's and Faraday's laws. Maxwell's laws at their basis are just conservation laws that apply to anything that can be conserved. Mass, energy, momentum, angular momentum, charge, magnetic field, etc. etc. Then, using Ampere's Law and Faraday's laws as constitutive equations (like Newton's law of viscosity goes into deriving the Navier-Stokes equations of fluid mechanics), you get Maxwell's equations.

 

This is a decent write up: http://www.engr.uconn.edu/~lanbo/DeriveMaxwell.pdf

 

Maxwell's equations are not about just wires, they can evaluate the E and B field of a point like charge, a surface, and a volume of any type. I'm sorry you are wrong on this.

 

I completely agree that Maxwell's equations are not just in wires, however. They are full three dimensional field equations. They predict resultant magnetic field and electric fields in any geometry at all. That is why they are so powerful, they are not constrained to just simple shapes.

[ambros]

Ok, I'll leave semantics alone and concentrate on more practical issues.

 

 

1.) Biot-Savart is a fine enough approximation, its just not completely accurate.

 

- I say it has maximum accuracy in regards to any real world experimental measurements you can find. Please bring on some practical scenario so we can plug in some numbers and I will try to demonstrate.

 

 

2.) Maxwell's equations can evaluate the E and B field of a point like charge.

 

- Please demonstrate: electron is moving along x-axis at 25m/s, what is the magnitude of its electric and magnetic field potential in arbitrary direction at distance 'r'. No need to do derivation, just say what equations did you start with and write down what is the final expression for E and B.

 

 

 

I completely agree that Maxwell's equations are not just in wires, however. They are full three dimensional field equations. They predict resultant magnetic field and electric fields in any geometry at all. That is why they are so powerful, they are not constrained to just simple shapes.

 

Can you demonstrate by solving the problem above?

 

Would you say gravity field has curl? If you forget about Maxwell's equations and consider everything else you know about electric fields, electrons and their charge - would you say electric fields have any rotation (curl) and can this curl ever change?

[Radical Edward]

"Gauss's law can be derived from Coulomb's law.."

 

don't quote mine....

 

wikipedia says:

 

Gauss's law can be used to derive Coulomb's law,[2] and vice versa.

[darkenlighten]

Ok, I'll leave semantics alone and concentrate on more practical issues.

 

 

1.) Biot-Savart is a fine enough approximation, its just not completely accurate.

 

- I say it has maximum accuracy in regards to any real world experimental measurements you can find. Please bring on some practical scenario so we can plug in some numbers and I will try to demonstrate.

 

 

Here is the derivation:

 

2.) Maxwell's equations can evaluate the E and B field of a point like charge.

 

- Please demonstrate: electron is moving along x-axis at 25m/s, what is the magnitude of its electric and magnetic field potential in arbitrary direction at distance 'r'. No need to do derivation, just say what equations did you start with and write down what is the final expression for E and B.

 

Can you demonstrate by solving the problem above?

 

Would you say gravity field has curl? If you forget about Maxwell's equations and consider everything else you know about electric fields, electrons and their charge - would you say electric fields have any rotation (curl) and can this curl ever change?

 

I already solved for the E-Field above using Gauss's Law. To solve for B field is the same method, and for approximation it will be the same results we have already discussed.

[Mr Skeptic]

This page here shows you how to get coulomb's law from maxwell's first equation:

 

http://people.bu.edu/wwildman/WeirdWildWeb/courses/sl/resources/physics/em/em01.htm

 

Technically, from maxwell's first equation and knowledge that the field is spherically symmetric.

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There is no scenario that can not be solved with Coulomb's, Biot-Savart law and Lorentz force.

 

Well I don't think you can get electromagnetic waves from those.

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Fields can not be created, that's like saying electrons get created. Charge, electron and electric field is one and the same thing if you look at point particles, there is no electron that is not electric charge which is not electric field, in these equations simply "q". Electron is fundamental particle, the smallest amount of charge (electric field) that can exist, it can not change, disappear or be created in these experiments.

 

You're talking here of conservation of charge, not of electric field. Yes, Maxwell's equations do say charge is conserved, but field is not charge. From Maxwell's first equation, if you put a box around something and measure the overall field coming out of the box (ie the divergence) this will give you the charge inside the box. However, an electron-positron pair can be created, satisfies conservation of charge, and yet has a field. The overall charge involved is zero, but the electric field is not the charge. An electromagnetic wave has both magnetic and electric fields but no charge whatsoever.

[ambros]

Here is the derivation:

 

What is that for? I am offering you a demonstration' date=' so to resolve this argument with practical scenario and real world measurements. You were supposed to give some experimental setup that can be solved with Maxwell's equations and where you believe Coulomb and Biot-Savart laws are just approximations so I can prove otherwise.

 

 

I already solved for the E-Field above using Gauss's Law.

 

You ended up with Coulomb's law, which Wikipedia and other people here confirmed is not really possible. It also makes no sense to have a set of equations and all you can do with them is derivation to get some other formulas that by some magic incorporate completely different relations. The question here is about applicability of Maxwell's equations and I am yet to see any direct numerical result come out of them at all.

 

 

To solve for B field is the same method, and for approximation it will be the same results we have already discussed.

 

What approximation are you talking about? The only approximation in the whole story comes from wires, integrals and the unit of Ampere. It is no approximation if you know positions and velocity vectors, approximation is when you say: -"around 6.242 × 10^18 electrons passing a given point each second constitutes one ampere."

 

And it gets worse. You see, amperes do not really tell you the VELOCITY of electrons, which is the most important variable in regards to the magnitude of B field. This definition, and so automatically Maxwell's equations too, are oblivious to different properties of different materials. The same current (amperes) WILL NOT produce the B field of the same magnitude in different conductors, because the speed and amount of moving charges will be different. -- Are you really trying to tell me:

 

divB=0 --> B= k*qv x 1/r^2 ?

 

Where is magnetic constant, charge and *velocity* in Gauss's law for magnetism? Most obviously, do you not see these two equations are opposite and contradict one another? Gauss says: "divB=0", but Biot-Savart describes: divB!=0, so whom do you chose to believe - the first equation that is approximated with line integrals and amperes, that can not even produce any numerical results on its own, or equation that actually gives results and is used in practice as it can describe this field with not just wires and electric currents, but also per point charge (maximum resolution).

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Well I don't think you can get electromagnetic waves from those.

 

I see you are skeptic' date=' Mr Skeptic. Which is good, especially if applied without prejudice. Interesting thing however is that I do not get anything else but waves, curls and spirals. For example, from a random positron-electron "soup" these charges will couple in electric dipoles oscillating around each other and actually gain directional velocity as a combined entity, while their oscillation plane may end up polarized vertically, horizontally or can twirl around as in twisted wire or double helix, so what I just described really is a transverse electromagnetic wave.

 

To reproduce this all you need is the Coulomb's law, Biot-Savart law, Lorentz force and some program that can integrate this motion and draw it on the screen using Newton's laws of motion, i.e. kinematics equation. It is similar to modeling planetary orbits with Newton's law of gravity and the result would be similar if it was not for magnetic fields and Lorentz force, which is what turns orbits into directional oscillations, spirals, waves and such.

 

 

I can even get the speed of light out of these equations, it is built-in value in magnetic and electric constant. Speaking of which, this is the only way Maxwell's equations can do it too, there is no other numerical value in those equations but these constants, so if they can produce any *numbers* from combining *symbols* 'cur', div', E' and 'B', that's only because these numbers were there before, it's a farce because those are experimental numbers.

 

 

However, an electron-positron pair can be created, satisfies conservation of charge, and yet has a field. The overall charge involved is zero, but the electric field is not the charge.

 

Hey, that's what I'm talking about too, what a coincidence. Ok, I submit "charge" and "field" have different units, but it is technically the same as when you measure electric field potential you actually measure its electric charge, no measurable filed potential = no net charge.

 

There is no case where you can measure electric charge that you are not actually measuring electric field potential, and there is no case when you measure electric potential and can say there is no net charge, so if photon consist of electric fields, then it means it has electric charge, but is another story why do we measure that charge to be zero.

 

It is really a paradox to have electric fields and not have electric charge, unless of course we are talking about superposition and net or combined amount of charge aka electric field potential.

 

 

 

An electromagnetic wave has both magnetic and electric fields but no charge whatsoever.

 

In the positron-electron wave described above there will be two charges and their electric and magnetic fields, their net charge will be close to zero, depending on where and when you measure it and how close you can get and how fast you can make the measurement and how well you can isolate a single photon... it's most likely to give zero value simply due to rapid changes of negative and positive amplitudes.

[Bignose]

Speaking of which, this is the only way Maxwell's equations can do it too, there is no other numerical value in those equations but these constants, so if they can produce any *numbers* from combining *symbols* 'cur', div', E' and 'B', that's only because these numbers were there before, it's a farce because those are experimental numbers.

 

The equations themselves are incomplete without the appropriate boundary and initial conditions that allow them to be solved (i.e. give the values of E and B).

 

Please don't take this to be rude, but in your first post you say "I do have education in classical physics" and you have forgotten about about initial and boundary conditions?

 

Look, here's a simple example. Velocity = dx/dt. Sure, you can integrate this, given a function for velocity. Let's take a really simple example, V=1. You can integrate this, x(t) = 1 + x(0). The value x(t) is completely unknowable without knowing what its value at time 0 was, x(0).

 

It is exactly the same for Maxwell's equations. You have to have the appropriate boundary and initial conditions to compute the values. The initial and boundary conditions are just as important and necessary as the equations to solve.

 

In a slightly different vein here, I am confused about where this thread is going or was meant to go. It started off innocently enough asking about how to solve some equations, but it seems to have turned into some sort of anti-Maxwell's equations rant -- using words like "farce" when describing the equations. Which I don't get at all, because Maxwell's equations are among the most verified equations known to mankind. The computer you are using wouldn't work unless Maxwell's equations didn't work, as just one of many, many examples. The problem as I see it, is a lack of understanding of the vector operations, and a lack of how to apply them, specifically applying them via Maxwell's equations. A good advanced book on the topic may go a long way to clearing this up. From the table of contents, Maxwell's Equations and the Principles of Electromagnetism by Fitzpatrick seems like an excellent place to start. If you aren't familiar with vector equations, Shey's Div, Grad, Curl and All That is extremely excellent.

 

But, if I might ask a favor, if an Anti-Maxwell rant is your agenda, could you just lay it all out on the table now instead of trying to be coy about it?

Edited March 23, 2010 by Bignose

[darkenlighten]

What is that for? I am offering you a demonstration, so to resolve this argument with practical scenario and real world measurements. You were supposed to give some experimental setup that can be solved with Maxwell's equations and where you believe Coulomb and Biot-Savart laws are just approximations so I can prove otherwise.

 

Edit: That was to show you why B = (μo/4π) * (q v x r) / r^2 is just an approximation for a moving charge if you look at the math I have given you. I'm starting to lose confidence in your math though since you are still not getting it. I mean you should be able to plug in the numbers and or even just look and see how the exact E and B Fields derived here will be different than the approximation. I'm not trying to argue with you, I'm just trying to show you why I am right.

 

You ended up with Coulomb's law, which Wikipedia and other people here confirmed is not really possible. It also makes no sense to have a set of equations and all you can do with them is derivation to get some other formulas that by some magic incorporate completely different relations. The question here is about applicability of Maxwell's equations and I am yet to see any direct numerical result come out of them at all.

 

Stop reading Wikipedia, because that is just wrong. Of course you will end up with the same field! Otherwise Gauss's Law would be wrong and well of course it's not. And its not my some magic, its math...Do you not understand that Gauss's Law will give you the same E field that Coulomb's Law will, that's the idea.

 

What approximation are you talking about? The only approximation in the whole story comes from wires, integrals and the unit of Ampere. It is no approximation if you know positions and velocity vectors, approximation is when you say: -"around 6.242 × 10^18 electrons passing a given point each second constitutes one ampere."

 

The approximation I am talking is the above one that you ignored since you believed it had no relevance, so please look again. And I never gave an approximation, the equation I solved for the wire is the EXACT field you would experience for that scenario. Do I really need to solve other scenarios for you to understand its not just for wires. You do need symmetry in the problem, but any that is Spherically, Cylindrically or Planar Symmetrical can be easily solved by Gauss's and Ampere's Law, which are apart of Maxwell's equations.

 

But the approximation is really an aside and is not going to make or break everything else we are talking about.

 

And it gets worse. You see, amperes do not really tell you the VELOCITY of electrons, which is the most important variable in regards to the magnitude of B field. This definition, and so automatically Maxwell's equations too, are oblivious to different properties of different materials. The same current (amperes) WILL NOT produce the B field of the same magnitude in different conductors, because the speed and amount of moving charges will be different. -- Are you really trying to tell me:

 

divB=0 --> B= k*qv x 1/r^2 ?

 

Where is magnetic constant, charge and *velocity* in Gauss's law for magnetism? Most obviously, do you not see these two equations are opposite and contradict one another? Gauss says: "divB=0", but Biot-Savart describes: divB!=0, so whom do you chose to believe - the first equation that is approximated with line integrals and amperes, that can not even produce any numerical results on its own, or equation that actually gives results and is used in practice as it can describe this field with not just wires and electric currents, but also per point charge (maximum resolution).

 

The equation is not ∇ · B = 0 in order to solve for the B field it is ∇ x B = μoI ⇒ ∫B·dl = μoI(enclosed) and you can easily derive the same resulting B field with this method. And as you can see ∇ · B = 0 is always zero, because the divergence of a curl is always zero.

Edited March 23, 2010 by darkenlighten

[Mr Skeptic]

I see you are skeptic, Mr Skeptic. Which is good, especially if applied without prejudice. Interesting thing however is that I do not get anything else but waves, curls and spirals. For example, from a random positron-electron "soup" these charges will couple in electric dipoles oscillating around each other and actually gain directional velocity as a combined entity, while their oscillation plane may end up polarized vertically, horizontally or can twirl around as in twisted wire or double helix, so what I just described really is a transverse electromagnetic wave.

 

To reproduce this all you need is the Coulomb's law, Biot-Savart law, Lorentz force and some program that can integrate this motion and draw it on the screen using Newton's laws of motion, i.e. kinematics equation. It is similar to modeling planetary orbits with Newton's law of gravity and the result would be similar if it was not for magnetic fields and Lorentz force, which is what turns orbits into directional oscillations, spirals, waves and such.

 

 

I can even get the speed of light out of these equations, it is built-in value in magnetic and electric constant. Speaking of which, this is the only way Maxwell's equations can do it too, there is no other numerical value in those equations but these constants, so if they can produce any *numbers* from combining *symbols* 'cur', div', E' and 'B', that's only because these numbers were there before, it's a farce because those are experimental numbers.

 

I considered this before, as the "medium" for an electromagnetic wave and with massless charged particle-antiparticle pairs. However I never went so far as to calculate it. I'd like to know how you got your wave to go at the speed of light however since some of the energy would inevitably have to end up as potential energy for the separated charges. It would seem to me that this would give a lag depending on the density of the particle-antiparticle pairs, and any mass they might have.

 

As for finding the speed of light, before finding it as a solution to Maxwell's equations, people had no idea light was an electromagnetic wave (and I don't think they had a concept of an electromagnetic wave either). It wasn't the actual speed so much as that it was there, that was impressive.

 

Hey, that's what I'm talking about too, what a coincidence. Ok, I submit "charge" and "field" have different units, but it is technically the same as when you measure electric field potential you actually measure its electric charge, no measurable filed potential = no net charge.

 

There is no case where you can measure electric charge that you are not actually measuring electric field potential, and there is no case when you measure electric potential and can say there is no net charge, so if photon consist of electric fields, then it means it has electric charge, but is another story why do we measure that charge to be zero.

 

It is really a paradox to have electric fields and not have electric charge, unless of course we are talking about superposition and net or combined amount of charge aka electric field potential.

 

You can certainly have fields with no net electric charge (dipoles for example). There's also electromagnetic waves of course, which don't require your particle-antiparticle soup to make sense. The reason I considered a particle-antiparticle soup in the first place was to avoid some action at a distance sort of things, but then I also end up with the field of an electron having to be made of said soup and that really gets annoying.

 

In the positron-electron wave described above there will be two charges and their electric and magnetic fields, their net charge will be close to zero, depending on where and when you measure it and how close you can get and how fast you can make the measurement and how well you can isolate a single photon... it's most likely to give zero value simply due to rapid changes of negative and positive amplitudes.

 

Oh and just wondering, how do you propose we get magnetic fields without magnetic charges?

[ambros]

Do I really need to solve other scenarios for you to understand its not just for wires.

 

You did not solve anything with any Maxwell's equation' date=' that was Coulomb's equation which can not be derived from Gauss's law, but I forgive you, so to concentrate on more peculiar part, B field.

 

The equation is not ∇ · B = 0 in order to solve for the B field it is ∇ x B = μoI ⇒ ∫B·dl = μoI(enclosed) and you can easily derive the same resulting B field with this method.

 

Why not to use 2nd equation - Gauss's law for magnetism? Again, we can not use integrals as we are solving for point charges in a single time instant so we can "look" at it from all sides, in full 3D, and line integrals can not do it, ok? That half-baked equation you offered can not even be applied to the problem, and it's still not solved for B®.

 

rotB= J + dE/dt --> ∫B·dl = μoI(enclosed)? Why are you giving me equation for loops? Even if it was correct equation it would not be applicable in that form, that is not solution for B®, but some integral of some dot product, as I said the 1st time, what in the world you imagine you can do with that?

 

 

And as you can see ∇ · B = 0 is always zero, because the divergence of a curl is always zero.

 

My friend, you are being "two-dimensional", integral has sucked one whole dimension out of those equations, with them you can not see even a little bit in front nor behind, this is what you see if you look at single charge in single point in time with Maxwell's equations (left):

 

 

http://maxwell.ucdavis.edu/~electro/magnetic_field/ans2.html

...and this (right) is what Biot-Savart for point charges and I see.

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In a slightly different vein here, I am confused about where this thread is going or was meant to go. It started off innocently enough asking about how to solve some equations, but it seems to have turned into some sort of anti-Maxwell's equations rant -- using words like "farce" when describing the equations.

 

I did not mean to assert that is a fact, I am trying to condense my meanings in as less words as I can, so rather I'm illustrating how I see it from my point of view, it was meant as a bit of joke, provocation, and rant indeed, sorry.

 

Do gravity fields have curl?

 

 

But, if I might ask a favor, if an Anti-Maxwell rant is your agenda, could you just lay it all out on the table now instead of trying to be coy about it?

 

Biot-Savart and Coulomb's law define quite different properties of E and B fields, and together with Lorentz force they describe different, though somewhat similar, interaction, i.e. induction, mechanism. -- Am I right, am I wrong? What would it mean? What do you suggest should be done in order to find the answer, how to approach this issue?

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I'd like to know how you got your wave to go at the speed of light however since some of the energy would inevitably have to end up as potential energy for the separated charges.

 

The greatest challenge for me in all that was to scale the numbers and avoid problems with decimal precision' date=' to slow it down and make the whole thing animate smoothly while constrained with the hardware performances - in short, I thrown all the units out, scaled decimal places, kept just relations, so I really have no idea in what time and size scale I ended up with.

 

Help me solve "the scale problem" *backwards* (if possible), find out the velocity in normal units and see if those electron-positron waves turn out to move with the speed of light... and if so, then I'll share the prize with you, for rediscovering the photon and solving the wave-particle duality paradox.

 

 

As for finding the speed of light, before finding it as a solution to Maxwell's equations, people had no idea light was an electromagnetic wave (and I don't think they had a concept of an electromagnetic wave either). It wasn't the actual speed so much as that it was there, that was impressive.

 

There is a lot I'm still yet to learn about it, but I think it was there the whole time, those are experimentally measured numbers that came first with Coulomb's law and Biot-Savart law... elasticity and viscosity of the "vacuum", as they saw it back then, I believe.

 

How can we combine symbols without any initial condition, without any numbers and yet get some number out of it? How many fields was he considering, how many fields are inducing themselves in his wave equation or 'speed of light' derivation - one, two fields? What is the physical meaning of those 'boundary conditions', positron-electron? Maybe that's it, are these 'boundary conditions' the same thing as 'initial conditions', like relative position and velocity of the fields as with Newton's laws and kinematics?

 

 

Oh and just wondering, how do you propose we get magnetic fields without magnetic charges?

 

As always, as described by Biot-Savart law - magnetic field is an effect of motion of an electric field and its magnitude is proportional to the velocity of an electric filed. If we can measure this magnetic field potential we can say we are measuring magnetic charge (analog measurement). I'm not sure if I understand the question, are you talking about "magnetic monopoles"?

 

 

You get magnetic fields just like you get vorticity in fluids - due to the motion, of the wingtip in that particular photo above. So, we should not be surprised Biot-Savart law applies to these situations in aerodynamics as well. -- 100$ question: two parallel bullets("wires") - two bullets are shot in the same time and they move in the same direction close to each other through some gas, considering vorticity effect as shown on the picture above - will these two bullets attract, repel or neither?

Edited March 23, 2010 by ambros
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[Klaynos]

http://www.ansoft.com/products/hf/hfss/

 

Solve numerically EM structures using only maxwell's equations.

[ambros]

http://www.ansoft.com/products/hf/hfss/

 

Solve numerically EM structures using only maxwell's equations.

 

That's quite impressive. I will submit and bow to Maxwell's equations if they can do that, but I bet 50pts that was done with Coulomb's law and Biot-Savart law. I could not find mention of any equations in product sheets. Do you know how it was done and where it is used?

[Klaynos]

That's quite impressive. I will submit and bow to Maxwell's equations if they can do that, but I bet 50pts that was done with Coulomb's law and Biot-Savart law. I could not find mention of any equations in product sheets. Do you know how it was done and where it is used?

 

I can promise you that this uses maxwell's equations and only maxwell's equations.

 

It is a finite element method model, it splits models into small boxes and solves maxwell's equations inside the boxes using the boundary conditions set by the surrounding boxes.

 

It is used for many applications from antenna design for radar systems to novel EM structures at optical frequencies.