Maxwell's equations VS. Lorentz & Coulomb force equations

[Dale]

Try clearing your cache when you reload the page. I think you can do it by doing something like CTRL-F5 or refreshing several times. The page is correct when I look at it now.

Oh, thanks, you are right. After a few refreshes it came out right.

 

[varga]

No. You solve Maxwell's equations to obtain an expression for the E and B.

Yes, I'm glad we established that, but there is also E and B on the right-hand side, what those two stand for? And so also, where did B go from Liénard-Wiechert formula for E?

=============================
1.) All I'm asking is to see those two Maxwell's equations solved for E and B, not their curls, and where I can see the full expression of the E and B terms on the right hand side.


2.) http://en.wikipedia.org/wiki/Maxwell's_equations -- If you look at Wikipedia article you may find that E stands for "electric field" and if you follow the link you will find COULOMB'S LAW, they also say B stands for "magnetic field" and if you follow the link you will find LORENTZ FORCE and BIOT-SAVART LAW. Special Relativity error correction is NOT fundamental part of Maxwell's equations.


3.) So, it turns out this one:

...can be written like this:
https://www.physicsforums.com/latex_images/26/2626683-0.png

But where did B go from the right hand side? Where did curl go from the left hand side? Let's suppose that is still Maxwell's equation, but look if we take Special Relativity and error correction out, what are we left with? Coulomb's law. And again, where did B go? Why is E not interacting with B anymore?


4.) - "In the case of a charged point particle q moving at a constant velocity v, then Maxwell's equations give the following expression for the electric field and magnetic field:

\mathbf{E} = \frac{q}{4\pi \epsilon_0} \frac{1-v^2/c^2}{(1-v^2\sin^2\theta/c^2)^{3/2}}\frac{\mathbf{\hat r}}{r^2}

\mathbf{B} = \mathbf{v} \times \frac{1}{c^2} \mathbf{E}


As if Maxwell's equations were relativistic, but anyway, once we take out Special Relativity and its error correction out we are left with "my" equations:

\mathbf{E} =\frac{q}{4\pi\epsilon_0}\ \frac{\mathbf{\hat r}}{r^2}

\mathbf{B} =\frac{\mu_0 q \mathbf{v}}{4\pi} \times \frac{\mathbf{\hat r}}{r^2}

http://en.wikipedia.org/wiki/Biot–Savart_law
(ref. Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.)


- Is everyone now happy to accept that Coulomb's law and Biot-Savart law can be derived from Maxwell's equations and vice versa? I'm not, where is E and B interaction where did curls go?

 

[PhaseShifter]

In CLASSICAL PHYSICS, the real input for E and B terms on the right hand side of Maxwell's equations is Coulomb's law and Biot-Savart law, is it not?

Try working a simpler problem using those equations.

Model light (an EM wave) with a frequency of 5x10^12 Hz propagating along the x axis, in a vacuum (charge density =0 everywhere).

 

[Born2bwire]

Yes, I'm glad we established that, but there is also E and B on the right-hand side, what those two stand for? And so also, where did B go from Liénard-Wiechert formula for E?

=============================
1.) All I'm asking is to see those two Maxwell's equations solved for E and B, not their curls, and where I can see the full expression of the E and B terms on the right hand side.


2.) http://en.wikipedia.org/wiki/Maxwell's_equations -- If you look at Wikipedia article you may find that E stands for "electric field" and if you follow the link you will find COULOMB'S LAW, they also say B stands for "magnetic field" and if you follow the link you will find LORENTZ FORCE and BIOT-SAVART LAW. Special Relativity error correction is NOT fundamental part of Maxwell's equations.


3.) So, it turns out this one:

...can be written like this:
https://www.physicsforums.com/latex_images/26/2626683-0.png

But where did B go from the right hand side? Where did curl go from the left hand side? Let's suppose that is still Maxwell's equation, but look if we take Special Relativity and error correction out, what are we left with? Coulomb's law. And again, where did B go? Why is E not interacting with B anymore?


4.) - "In the case of a charged point particle q moving at a constant velocity v, then Maxwell's equations give the following expression for the electric field and magnetic field:

\mathbf{E} = \frac{q}{4\pi \epsilon_0} \frac{1-v^2/c^2}{(1-v^2\sin^2\theta/c^2)^{3/2}}\frac{\mathbf{\hat r}}{r^2}

\mathbf{B} = \mathbf{v} \times \frac{1}{c^2} \mathbf{E}


As if Maxwell's equations were relativistic, but anyway, once we take out Special Relativity and its error correction out we are left with "my" equations:

\mathbf{E} =\frac{q}{4\pi\epsilon_0}\ \frac{\mathbf{\hat r}}{r^2}

\mathbf{B} =\frac{\mu_0 q \mathbf{v}}{4\pi} \times \frac{\mathbf{\hat r}}{r^2}

http://en.wikipedia.org/wiki/Biot–Savart_law
(ref. Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.)


- Is everyone now happy to accept that Coulomb's law and Biot-Savart law can be derived from Maxwell's equations and vice versa? I'm not, where is E and B interaction where did curls go?

We already provided you the solved equations. For a point source you get back the fields generated by the Lienard-Wiechart potentials. For an arbitrary charge and current source, you get the Jefimenko equations.

You cannot take out special relativity. This has to be the fourth time I have told you, classical electrodynamics automatically satisfies relativity. You cannot remove it because it is inseparable from the theory. This is easily seen when you solve for the wave equations in a source-free vacuum which predicts the wave speed to be c. Special relativity is derived from Maxwell's equations so there are no corrections.

Those equations are not correct because you have stipulated an acclerating charge.

You cannot derive Maxwell's equations from Coulomb's Law and Biot-Savart Law. Once again, these equations are only valid for statics and they do not preserve special relativity. Using them in your example you lack the retarded time and position and the additional electromagnetic radiation that arises due to the acceleration of the charge.

 

[Repainted]

1.) All I'm asking is to see those two Maxwell's equations solved for E and B, not their curls, and where I can see the full expression of the E and B terms on the right hand side.


2.) http://en.wikipedia.org/wiki/Maxwell's_equations -- If you look at Wikipedia article you may find that E stands for "electric field" and if you follow the link you will find COULOMB'S LAW, they also say B stands for "magnetic field" and if you follow the link you will find LORENTZ FORCE and BIOT-SAVART LAW. Special Relativity error correction is NOT fundamental part of Maxwell's equations.


3.) So, it turns out this one:

...can be written like this:
https://www.physicsforums.com/latex_images/26/2626683-0.png

But where did B go from the right hand side? Where did curl go from the left hand side? Let's suppose that is still Maxwell's equation, but look if we take Special Relativity and error correction out, what are we left with? Coulomb's law. And again, where did B go? Why is E not interacting with B anymore?


4.) - "In the case of a charged point particle q moving at a constant velocity v, then Maxwell's equations give the following expression for the electric field and magnetic field:

\mathbf{E} = \frac{q}{4\pi \epsilon_0} \frac{1-v^2/c^2}{(1-v^2\sin^2\theta/c^2)^{3/2}}\frac{\mathbf{\hat r}}{r^2}

\mathbf{B} = \mathbf{v} \times \frac{1}{c^2} \mathbf{E}


As if Maxwell's equations were relativistic, but anyway, once we take out Special Relativity and its error correction out we are left with "my" equations:

\mathbf{E} =\frac{q}{4\pi\epsilon_0}\ \frac{\mathbf{\hat r}}{r^2}

\mathbf{B} =\frac{\mu_0 q \mathbf{v}}{4\pi} \times \frac{\mathbf{\hat r}}{r^2}

http://en.wikipedia.org/wiki/Biot–Savart_law
(ref. Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.)

1) Already gave that to you. Check the previous post.

2) I have no idea what you're saying. Classical Electrodynamics is the basis for which the Lorentz transforms were developed the first time.

3) When we solved for E, we wanted to find it in terms of the velocity and acceleration of the charge did we not? If we wanted to find the E field in terms of the B-field we simply have Faraday's law and the other term in Ampere's law.

4) You can't derive Maxwell's equations from Coloumb's law and the Biot-Savart law because they deal with stationary charges(or steady currents without E-fields), while Maxwell's equations deal with all kinds of motions of charges. So you can derive 2 from 1, but not 1 from 2.

Yes, I'm glad we established that, but there is also E and B on the right-hand side, what those two stand for? And so also, where did B go from Liénard-Wiechert formula for E?

As for this, when we solve Maxwell's equations, we have to make sure our expressions for E and B hold for ALL FOUR equations. So in other words we're solving 4 partial differential equations simultaneously, and the solutions that we get are solutions of all 4 equations.

 

[Frame Dragger]

Thanks for the confirmation. This is really weird. When I go to the page it displays the wrong equation, but when I go to edit it has the right equation. Do edits have to be "approved" or something before they take effect?

Depending on the status of the page, some do.


@ALL: Please, I know you're all trying to be good, decent people and teach Varga what he needs to know. What you're missing is that I don't believe he inderstands ANY of what you're telling him. He knows what he's said, and where he starts to paste from Wiki he no longer has a working understanding. At some point, it's no longer kind to string him along, and the time has come to send him back to HS/University or tell him to READ the wikipedia article and not just paste it. I think the time has come to recognize that a major barrier exists in communicating basics of E&M to him, and also that he strongly and unwaveringly believes that HE IS RIGHT. You first need to educate him systematically from the groud up, before he'll believe you. If you enjoy someone "challenging" you with first year EM, and going in circles, fair enough. If not... cut him loose already, or start to explain the first principles he only pretends to grasp.

 

[Dale]

varga, at this point I have to agree with Frame Dragger's assessment. Every single question that you have asked in the quoted post below has been answered for you multiple times in this thread. I think you must have some very fundamental lack of knowledge to have failed to grasp the repeated answers. I think that an internet forum is probably not going to be an effective way for you to learn and I would recommend traditional classroom instruction or at a minimum taking a traditional textbook and working the homework problems for yourself.

Yes, I'm glad we established that, but there is also E and B on the right-hand side, what those two stand for?

This comment makes me think that you do not even understand algebra, let alone vector calculus. A solid understanding of vector calculus is a pre-requisite for EM problems.

And so also, where did B go from Liénard-Wiechert formula for E?

This was given already multiple times:
Equation 21 from http://fermi.la.asu.edu/PHY531/larmor/index.html given in post 31
Fourth equation in the "Corresponding values of electric and magnetic fields" subsection from http://en.wikipedia.org/wiki/Liénar...onding_values_of_electric_and_magnetic_fields also in post 31
Second equation in Repainted's post 47

1.) All I'm asking is to see those two Maxwell's equations solved for E and B, not their curls, and where I can see the full expression of the E and B terms on the right hand side.

That was also given multiple times:
Equations 19 and 21 from http://fermi.la.asu.edu/PHY531/larmor/index.html given in post 31
3rd and 4th equations in the "Corresponding values of electric and magnetic fields" subsection from http://en.wikipedia.org/wiki/Liénar...onding_values_of_electric_and_magnetic_fields also in post 31
1st and 2nd equations in Repainted's post 47

2.) http://en.wikipedia.org/wiki/Maxwell's_equations -- If you look at Wikipedia article you may find that E stands for "electric field" and if you follow the link you will find COULOMB'S LAW, they also say B stands for "magnetic field" and if you follow the link you will find LORENTZ FORCE and BIOT-SAVART LAW. Special Relativity error correction is NOT fundamental part of Maxwell's equations.

Yes it is, this has been pointed out by Born2bwire in posts 3 and 54, espen180 in post 5, SpectraCat in post 22, and myself in posts 31 and 50. If that were not enough, in post 9 espen180 even linked to the derivation of the second postulate of relativity from Maxwell's equations.

3.) So, it turns out this one:

...can be written like this:
https://www.physicsforums.com/latex_images/26/2626683-0.png

But where did B go from the right hand side? Where did curl go from the left hand side? Let's suppose that is still Maxwell's equation, but look if we take Special Relativity and error correction out, what are we left with? Coulomb's law. And again, where did B go? Why is E not interacting with B anymore?

E and B are interacting as per the general formulation of Maxwell's equations. This is shown clearly in the derivation which is given on both of the links that I posted, as is the solution for B.

4.) - "In the case of a charged point particle q moving at a constant velocity v, then Maxwell's equations give the following expression for the electric field and magnetic field:

\mathbf{E} = \frac{q}{4\pi \epsilon_0} \frac{1-v^2/c^2}{(1-v^2\sin^2\theta/c^2)^{3/2}}\frac{\mathbf{\hat r}}{r^2}

\mathbf{B} = \mathbf{v} \times \frac{1}{c^2} \mathbf{E}As if Maxwell's equations were relativistic, but anyway, once we take out Special Relativity and its error correction out we are left with "my" equations:

\mathbf{E} =\frac{q}{4\pi\epsilon_0}\ \frac{\mathbf{\hat r}}{r^2}

\mathbf{B} =\frac{\mu_0 q \mathbf{v}}{4\pi} \times \frac{\mathbf{\hat r}}{r^2}

http://en.wikipedia.org/wiki/Biot–Savart_law
(ref. Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.)

Yes, this is exactly what I said (and others elsewhere) in posts 24, 31, 37, and 50: Maxwell's equations reduce to the Coulomb equation for the specific case of a point charge at rest. If the point charge is moving then Maxwell's equations automatically generate a relativistic correction term and an acceleration (radiation) term, neither of which are predicted by Coulomb but both of which are verified experimentally.

- Is everyone now happy to accept that Coulomb's law and Biot-Savart law can be derived from Maxwell's equations and vice versa? I'm not, where is E and B interaction where did curls go?

No, Coulomb's law and Biot-Savart can be derived from Maxwell's equations, but NOT vice versa. See my posts 24 and 28, and those posted by others.

varga, in short, this conversation is going around and around in circles. I will be glad to respond to any new question that you post, but I am not going to re-hash points that have already been thoroughly addressed. I don't know if your failure to grasp the repeated and direct answers to each one of your questions is due to a very basic lack of education or due to a deliberate attempt to ignore information which contradicts your preconceptions. Either way, I don't think that it can be addressed via an internet forum conversation.

 

[varga]

Going in circles? It is not my fault SpectraCat said: - "The Lienard-Weichert potentials are the *inputs* to the Maxwell's equations to allow you to calculate the fields."


So, can someone finally get this straight then?!?
========================================

EQUATION 3.

EQUATION 4.

X.) 'B' on right side of 3. should be replaced with equation 4.?

Y.) 'B' on right side of 3. should be replaced with Biot-Savart law?

Z.) 'B' on right side of 3. should be replaced with Lienard-Wiechert B?

Q.) Depending on situation both Y. and Z. can be used to substitute 'B'?




[EDIT: DaleSpam, I do not see the answer in your reference, please provide citation and copy/paste the particular equation so I know what you mean to be referring to, thank you.]

 

[Dale]

DaleSpam, I do not see the answer in your reference, please provide citation and copy/paste the particular equation so I know what you mean to be referring to, thank you.

What part of

Equation 21 from http://fermi.la.asu.edu/PHY531/larmor/index.html

don't you understand. If you cannot do something as simple as follow a link and find Equation 21 I don't know how you think anyone can possibly help you over the internet. IMO, you are in serious need of classroom instruction; internet instruction is not likely to be successful in your case.

 

[Repainted]

So, can someone finally get this straight then?!?
========================================

EQUATION 3.

EQUATION 4.

X.) 'B' on right side of 3. should be replaced with equation 4.?

Y.) 'B' on right side of 3. should be replaced with Biot-Savart law?

Z.) 'B' on right side of 3. should be replaced with Lienard-Wiechert B?

Q.) Depending on situation both Y. and Z. can be used to substitute 'B'?

X.) Yes. Though I should say its not that it SHOULD be replaced, it just can be replaced. We normally use Vector Identities to simplify things, and not substitute directly. For example, we can take the curl on both sides of 3, then sub in the curl(B) from 4.

Y.) No.

Z.) No.

Q.) Yes. Though I think it would be more precise to say that depending on the situation, the equations in Y.) and Z.) are solutions for B.

 

[varga]

X.) Yes. Though I should say its not that it SHOULD be replaced, it just can be replaced. We normally use Vector Identities to simplify things, and not substitute directly. For example, we can take the curl on both sides of 3, then sub in the curl(B) from 4.

Y.) No.

Z.) No.

Q.) Yes. Though I think it would be more precise to say that depending on the situation, the equations in Y.) and Z.) are solutions for B.

THANK YOU! Finally... thank you, whooo. Let me clarify, Q; do you mean to say if velocity is non-relativistic we can use Biot-Savart and Coulomb's law with good accuracy? Do you mean to say there are situations where Y.) and Z.) are applicable but Maxwell's equations are not? Basically, what do you mean by "depending on the situation, the equations in Y.) and Z.) are solutions for B.", it sounds exclusive of everything else, like nothing else IS solution, but they ARE.


I'll just leave this like that for a while so everyone get a chance to disagree with what you said and hence avoid any confusion and "going in circles". For a start, I can say that I generally agree and I thank you again for making this much more clear.

 

[Repainted]

THANK YOU! Finally... thank you, whooo. Let me clarify, Q; do you mean to say if velocity is non-relativistic we can use Biot-Savart and Coulomb's law with good accuracy?

Thats right, Biot-Savart and Coulomb's law are good approximations for the E-fields of slow moving charges, obviously the faster the charges are moving, the less accurate they become.

Do you mean to say there are situations where Y.) and Z.) are applicable but Maxwell's equations are not? Basically, what do you mean by "depending on the situation, the equations in Y.) and Z.) are solutions for B.", it sounds exclusive of everything else, like nothing else IS solution, but they ARE.

I don't get what you mean. Your first statement contradicts your second. Y.) and Z.) are solutions from Maxwell's equations, so how can Maxwell's equations not be applicable to the situations Y.) and Z.) are?

Maybe I wasn't clear. What I meant was, when we solve Maxwell's equations for B, we get Y.) or Z.) or something else, it depends on the situation.

 

[varga]

What part ofdon't you understand. If you cannot do something as simple as follow a link and find Equation 21 I don't know how you think anyone can possibly help you over the internet. IMO, you are in serious need of classroom instruction; internet instruction is not likely to be successful in your case.

Sorry to anger you, kiddo. Do you understand this particular effect of "retarded time" is not even experimentally confirmed?


http://maxwell.ucdavis.edu/~electro/magnetic_field/images/bptchrg.jpg

Anyway, have I ever told you what I actually do? Do you know what that is? -- B field: The magnitude potential and its density distribution of B field does change during this time proportionally to velocity. Geometrically this potential is toroidal and its magnitude drops off uniformly with the inverse square law in a plane perpendicular to velocity vector, but it decreases as this angle goes from 90 degrees to 0 when it aligns with the velocity vector and where magnetic potential is zero, directly in line behind and in front of the charge. If we trace the magnitude potential around the charge with some arbitrary but constant radius it will describe a "ball squeezed from the front and behind" (doughnut). Therefore, I can say this field does have rotation (curl) defined by the cross product, and that divB != 0.

Has anyone ever told you that is how magnetic field of a moving charge looks like? What do you say is the divergence of that field? -- In any case, we are not going in circles - I accept all these equations, answers - but now, are you ready to pick one of those "correct" ones and solve that basic problem numerically so we can actually compare and see how big this error really is?

 

[Frame Dragger]

Sorry to anger you, kiddo.

*winces* Oh boy... this is going to get so ugly.

EDIT: Varga... do yourself a favour and learn to take criticism with more grace than this.

 

[varga]

Thats right, Biot-Savart and Coulomb's law are good approximations for the E-fields of slow moving charges, obviously the faster the charges are moving, the less accurate they become.

Thank you. All I'm trying to do here is to plot one of those CORRECTED equations as velocity goes up against classical Coulomb and Biot-Savart, all I want to see just how big this error is at 800m/s, 2,900m/s, 18,475m/s...

I don't get what you mean. Your first statement contradicts your second. Y.) and Z.) are solutions from Maxwell's equations, so how can Maxwell's equations not be applicable to the situations Y.) and Z.) are?

Maybe I wasn't clear. What I meant was, when we solve Maxwell's equations for B, we get Y.) or Z.) or something else, it depends on the situation.

Hmmm. That sounds as if Maxwell's equations produce other equations and not actual numerical results. Maybe we should use real words instead of X, Y, Z. Can you please rephrase your original statement and be more specific what situations did you have in mind and what would be suitable equation for each of those particular situations, that should settle it: - "Though I think it would be more precise to say that depending on the situation, the equations in Y.) and Z.) are solutions for B."

 

[ZapperZ]

This thread is closed because there is no hope.

Zz.