Maxwell's equations VS. Lorentz & Coulomb force equations

[Dale]

Then you convince me, by solving one simple example. All I'm asking here is for explanation how to apply those equations, how to get some real numbers and analyze what do numbers mean.Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s.

a.) Solve for B and E when its velocity is 700m/s.

b.) Describe briefly how E and B change during that time.

SpectraCat is right, there is a reason that entire courses are devoted to this subject. However, the specific problem of the Maxwell's equation fields and potentials due to an arbitrarily moving point charge has been solved. The solution is called the Lienard-Wiechert potential. Here is the Wikipedia page:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential
but I prefer this page from Arizona State University because it shows the notation in more detail:
http://fermi.la.asu.edu/PHY531/larmor/index.html

As you can see the Lienard-Weichert field reduces to the Coulomb field for a stationary charge, but for moving charges they are different with a relativistic correction to the Coulomb field term and a radiation term for an accelerating charge. This difference is experimentally measurable and agrees with Maxwell and disagrees with Coulomb.

 

[Born2bwire]

SpectraCat is right, there is a reason that entire courses are devoted to this subject. However, the specific problem of the Maxwell's equation fields and potentials due to an arbitrarily moving point charge has been solved. The solution is called the Lienard-Wiechert potential. Here is the Wikipedia page:
http://en.wikipedia.org/wiki/Liénard–Wiechert_potential
but I prefer this page from Arizona State University because it shows the notation in more detail:
http://fermi.la.asu.edu/PHY531/larmor/index.html

As you can see the Lienard-Weichert field reduces to the Coulomb field for a stationary charge, but for moving charges they are different with a relativistic correction to the Coulomb field term and a radiation term for an accelerating charge. This difference is experimentally measurable and agrees with Maxwell and disagrees with Coulomb.

Oh nice. I only knew of Jackson that had the field equations for the Lienared-Wiechert potentials.

 

[PhilDSP]

By the way, Maxwell did include the equivalent of the Lorentz force law in his set of fundamental equations (he got there much earlier than Lorentz) But after Maxwell died, a committee of mostly British physicists restructured Maxwell's model and removed that and the linkage to scalar and vector potentials (as well as other things they thought muddled the simplicity of the 4 equation structure)

We know now that potential is more fundamental than EM fields (as the sole prominent objector to removing them, Lord Kelvin, suspected over a hundred years ago)

 

[varga]

I agree with this 100%. The thing is, it takes most of us a considerable amount of time to learn all of that the first time we see this stuff (usually in college). There are formal courses that take (at least) a year that show folks how to use Maxwell's equations to solve problems in physics. You should expect to take a similar amount of time, and frankly, it's nothing short of staggeringly arrogant that you have declared them "insufficient and superfluous", without at least HAVING THIS MUCH TRAINING TO BEGIN WITH!

Sorry if I was not clear, I do believe to have adequate education and knowledge of the subject, I also did check Wikipedia and Google, as I always do, and I even took my textbooks out, to confirm what I could. It is with this knowledge and understanding that I still decided to try Maxwell's equations vs Lorentz & Coulomb's law.

Get yourself some books and start reading, or enroll in a course...

Yep, I did all that, but here we are having this conversation about it and we have different opinions. I made my arguments and heard yours, now it is the time to prove the point with some numbers.

Why should I (or anyone) do your homework for you? If you can't solve such basic problems, then you shouldn't worry too much about whether or not we really need Maxwell's equations, or trying to derive substitutes. Learn about Maxwell's equations first, then criticize them.

Basic problem, ok. Why? Maybe because it would take less time to solve than to write what you did. / To prove your point. To have conversation. To find out the truth. To exchange information. To try to teach me a lesson. To have fun and learn... all these reasons as why you and me joined this forum, eh? That's why we're here, right?

 

[varga]

SpectraCat is right, there is a reason that entire courses are devoted to this subject.

Ok, but all I need is full expression of all the terms in any equation and calculator to see what result it gives with whatever arbitrary parameters, or I can use some actual experimental measurements and hence confirm the numbers and validity or accuracy of the equation. I do not need to know even any math to do that, just one example to see what is what. I can not solve this example myself because my argument is that it can not be done and that at best it will give the same result as what I get via Lorentz force equation.

However, the specific problem of the Maxwell's equation fields and potentials due to an arbitrarily moving point charge has been solved. The solution is called the Lienard-Wiechert potential. Here is the Wikipedia page:
http://en.wikipedia.org/wiki/Li%C3%A...hert_potential
but I prefer this page from Arizona State University because it shows the notation in more detail:
http://fermi.la.asu.edu/PHY531/larmor/index.html

As you can see the Lienard-Weichert field reduces to the Coulomb field for a stationary charge, but for moving charges they are different with a relativistic correction to the Coulomb field term and a radiation term for an accelerating charge.

Thank you, but that would prove my point as that is not one of the four equations that we call "Maxwell's equations". What error are we talking about anyway? The one due to limited speed of propagation of gravity and em field potentials? If so, then what that has to do with this example where there is only one charge? Even with two or more, how large would that error be with closely spaced charges if this effect is not even concern for interplanetary distances where gravity field of the Sun is supposed to "lag" eight *minutes* and yet we can not measure it? I again object to the involvement of special relativity, I'm talking about the velocity of ~700m/s, but nevertheless I'm very interested to compare actual results and see how large this error correction really is.

This difference is experimentally measurable and agrees with Maxwell and disagrees with Coulomb.

Ok, what experiments? Can you point some papers or articles? I'd like to know what is the difference in prediction, how much error there is in Coulomb's law and how error changes as the velocity increases, at what speed it really becomes measurable and important factor, stuff like that.


p.s. What is "Larmor radiation", is that the same, similar or completely different to "synchrotron radiation"?

 

[SpectraCat]

Ok, but all I need is full expression of all the terms in any equation and calculator to see what result it gives with whatever arbitrary parameters, or I can use some actual experimental measurements and hence confirm the numbers and validity or accuracy of the equation. I do not need to know even any math to do that, just one example to see what is what. I can not solve this example myself because my argument is that it can not be done and that at best it will give the same result as what I get via Lorentz force equation.

So let's start there .. what do you get for the Lorentz force equation?

Thank you, but that would prove my point as that is not one of the four equations that we call "Maxwell's equations".

Ummm ... no. Maxwell's equations are *general* ... this is the solution for a specific case, which is directly relevant to your question. The E & B fields are *derived from* the charge density and current density *using* Maxwell's equations. The Lienard-Weichert potentials are the *inputs* to the Maxwell's equations to allow you to calculate the fields.

What error are we talking about anyway? The one due to limited speed of propagation of gravity and em field potentials? If so, then what that has to do with this example where there is only one charge? Even with two or more, how large would that error be with closely spaced charges if this effect is not even concern for interplanetary distances where gravity field of the Sun is supposed to "lag" eight *minutes* and yet we can not measure it? I again object to the involvement of special relativity, I'm talking about the velocity of ~700m/s, but nevertheless I'm very interested to compare actual results and see how large this error correction really is.

No, read it again .. there are two corrections, one for the relativistic effects to the Coulomb field, and the other for *emission of radiation* ... you know, that phenomenon that you didn't think existed for accelerated charges? Note how the Maxwell's equations derive the existence of such a phenomenon (radiating EM waves) from the basic, simple inputs, in a manner that is consistent with experiment. That is a nice example of why the rest of us do not find them "insufficient" or "superfluous".

 

[Dale]

Thank you, but that would prove my point as that is not one of the four equations that we call "Maxwell's equations".

Yes, it is. The Lienard Weichert potentials are derived directly from Maxwells equations for the special case of a point charge (as I said earlier, a delta-function) distribution as you requested. The derivation is pretty simple and straightforward, anyone can verify it for themselves. Simply substitute in q \, \delta(\mathbf{r}-\mathbf{r0}(t)) for the charge distribution and q \, \mathbf{r0'}(t) \, \delta(\mathbf{r}-\mathbf{r0}(t)) for the current distribution and simplify.

What error are we talking about anyway?

Did you even read the links in detail? Do you see how the fields reduce to the Coulomb field for a point charge at rest? The error is then the other terms which are non-zero for a moving or accelerating charge.

 

[espen180]

Sorry if I was not clear, I do believe to have adequate education and knowledge of the subject, I also did check Wikipedia and Google, as I always do, and I even took my textbooks out, to confirm what I could. It is with this knowledge and understanding that I still decided to try Maxwell's equations vs Lorentz & Coulomb's law.

Yep, I did all that, but here we are having this conversation about it and we have different opinions. I made my arguments and heard yours, now it is the time to prove the point with some numbers.

You claim to have an understanding of Maxwell's equations. If so, why are you using the differential form to do calculations on point charges? You should know to use the integral form.

 

[Frame Dragger]

I warned you folks about him...

 

[Dale]

Hi Everyone,

Can someone check this expression for me from a reputable textbook? (my introductory text didn't go this far) I believe that the Wikipedia formula is wrong:
\vec{E}(\vec{x},t) = q\left(\frac{\vec{n}-\vec{\beta}}{\gamma^2(1-\vec{\beta}\cdot\vec{n})^3R^2}\right)_{\rm{ret}} + \frac{q}{c}\left(\frac{\vec{n}\times[(\vec{n}\times\vec{\beta})\times\vec{\dot{\beta}}]}{(1-\vec{\dot{\beta}}\cdot\vec{n})^3R}\right)_{\rm{ret}}<br />

But if my memory and understanding are right then I think the correct formula is:
\vec{E}(\vec{x},t) = q\left(\frac{\vec{n}-\vec{\beta}}{\gamma^2(1-\vec{\beta}\cdot\vec{n})^3R^2}\right)_{\rm{ret}} + \frac{q}{c}\left(\frac{\vec{n}\times[(\vec{n}-\vec{\beta})\times\vec{\dot{\beta}}]}{(1-\vec{\dot{\beta}}\cdot\vec{n})^3R}\right)_{\rm{ret}}

This is an important difference since the first form results in no radiation for an accelerating point charge which is momentarily at rest.

 

[varga]

So let's start there .. what do you get for the Lorentz force equation?

Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s.

a.) Solve for E and B when its velocity is 700m/s.

b.) Describe briefly how E and B change during that time.

According to above equations...

E field: The magnitude potential and its density distribution of E field does not change during this time. Geometrically this potential is spherical and its magnitude drops off uniformly with the inverse square law. If we trace the magnitude potential around the charge with some arbitrary but constant radius it will describe a "ball" or sphere. Therefore, I can say this field actually has no rotation (curl) and that divE = 0.

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 'right-hand rule', and that divB != 0.


Your turn, what do you get from Maxwell's equations?

Ummm ... no. Maxwell's equations are *general* ... this is the solution for a specific case, which is directly relevant to your question. The E & B fields are *derived from* the charge density and current density *using* Maxwell's equations. The Lienard-Weichert potentials are the *inputs* to the Maxwell's equations to allow you to calculate the fields.

I asked 10 times what E and B terms stand for on the right hand side of those equation. Anyway, I can use those same expressions to substitute E and B in Lorentz force equation, but why would I if error is 398 places behind the decimal point?

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?

No, read it again .. there are two corrections, one for the relativistic effects to the Coulomb field, and the other for *emission of radiation* ... you know, that phenomenon that you didn't think existed for accelerated charges?

No radiation, only retarded time.

Wikipedia article explicitly states no quantum effects are taken into account

Note how the Maxwell's equations derive the existence of such a phenomenon (radiating EM waves) from the basic, simple inputs, in a manner that is consistent with experiment. That is a nice example of why the rest of us do not find them "insufficient" or "superfluous".

I hear you, but I don't see anyone is able to demonstrate any of that.

 

[Born2bwire]

Hi Everyone,

Can someone check this expression for me from a reputable textbook? (my introductory text didn't go this far) I believe that the Wikipedia formula is wrong:
\vec{E}(\vec{x},t) = q\left(\frac{\vec{n}-\vec{\beta}}{\gamma^2(1-\vec{\beta}\cdot\vec{n})^3R^2}\right)_{\rm{ret}} + \frac{q}{c}\left(\frac{\vec{n}\times[(\vec{n}\times\vec{\beta})\times\vec{\dot{\beta}}]}{(1-\vec{\dot{\beta}}\cdot\vec{n})^3R}\right)_{\rm{ret}}<br />

But if my memory and understanding are right then I think the correct formula is:
\vec{E}(\vec{x},t) = q\left(\frac{\vec{n}-\vec{\beta}}{\gamma^2(1-\vec{\beta}\cdot\vec{n})^3R^2}\right)_{\rm{ret}} + \frac{q}{c}\left(\frac{\vec{n}\times[(\vec{n}-\vec{\beta})\times\vec{\dot{\beta}}]}{(1-\vec{\dot{\beta}}\cdot\vec{n})^3R}\right)_{\rm{ret}}

This is an important difference since the first form results in no radiation for an accelerating point charge which is momentarily at rest.

Yeah, your equation is correct. That must have been a recent edition because I distinctly remember being put off by the fact that the article gave the scalar and vector potentials and not the field equations.

EDIT: Did you just fix that? I went over there to correct it but when I loaded up the edit page it had the corrected equations... sneaky...

Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s.

a.) Solve for E and B when its velocity is 700m/s.

DaleSpam just gave you the equations.

\mathbf{E}(\mathbf{r},t) = \frac{q}{4 \pi \epsilon_0} \frac{R}{\left( \mathbf{R}\cdot\mathbf{u}\right)^3} \left[ (c^2-v^2\right)\mathbf{u}+\mathbf{R}\times\left(\mathbf{u}\times\mathbf{a}\right) \right]

where

\mathbf{R}= \mathbf{r}-\mathbf{w}(t_r)
\dot{\mathbf{w}}(t_r) = \mathbf{v}

the vector w gives the postion of the particle at the retarded time t_r.

Anyway, if you do know electromagnetics as you assert then you should already be more than familiar with Maxwell's Equations, synchrotron radiation, Lienard-Wiechart potentials and etc. These are all topics in undergraduate EM physics courses. There really should be nothing to discuss.

 

[varga]

Yes, it is. The Lienard Weichert potentials are derived directly from Maxwells equations for the special case of a point charge (as I said earlier, a delta-function) distribution as you requested.

That is special relativity. Why use special relativity correction? Why not use Coulomb's law and Biot-Savart law like we are supposed to in classical electromagnetism?

Why complicate the terms of INPUT, we are looking at relation between Lorentz force and Maxwell's equations, we can use the same input terms for E and B in both, that does not help us, it only complicates unnecessarily. How large do you think this error could possibly be in our simple example?

And if Liénard-Wiechert potentials define E and B only as an INPUT for the right hand side of Maxwell's equations, then what does that mean? The values of E and B given by Liénard-Wiechert potentials are not "complete", so they need to be further processed?

Did you even read the links in detail? Do you see how the fields reduce to the Coulomb field for a point charge at rest? The error is then the other terms which are non-zero for a moving or accelerating charge.

You cut off the rest of that paragraph where I explained what I mean. "Retarded time" is an effect related primarily to DISTANCE between TWO interacting fields, the greater the distance the more "delay", more "error"...

1.) What retarded time has to do with this case and only one charge, where is the distance, distance to what? "OBSERVER"?

2.) How large do you expect error correction to be if we can not even measure the whole 8 minutes delay of Sun's gravity field?

 

[varga]

You claim to have an understanding of Maxwell's equations. If so, why are you using the differential form to do calculations on point charges? You should know to use the integral form.

My understanding may be wrong, please use whichever form suits you, and you do not need to use Special Relativity correction.

Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s. Solve for E and B when its velocity is 700m/s.

 

[varga]

Hi Everyone,
Can someone check this expression for me from a reputable textbook?
This is an important difference since the first form results in no radiation for an accelerating point charge which is momentarily at rest.

I surely would like to see numerical result of those equations applied to our example, but I still do not see how that fits back in Maxwell's equations, especially since both sets are supposed to deal with time-varying aspect of it. Are you sure those terms are input to Maxwell's equations and not something we use INSTEAD?


Now, let's stick to classical electromagnetism and go from the beginning. How are Maxwell's equations, including those E and B on the right side, defined in CLASSICAL ELECTROMAGNETISM, can you write that down please?

 

[varga]

DaleSpam just gave you the equations.

\mathbf{E}(\mathbf{r},t) = \frac{q}{4 \pi \epsilon_0} \frac{R}{\left( \mathbf{R}\cdot\mathbf{u}\right)^3} \left[ (c^2-v^2\right)\mathbf{u}+\mathbf{R}\times\left(\mathbf{u}\times\mathbf{a}\right) \right]

where

\mathbf{R}= \mathbf{r}-\mathbf{w}(t_r)
\dot{\mathbf{w}}(t_r) = \mathbf{v}

the vector w gives the postion of the particle at the retarded time t_r.

So does that gives us the full information about E field, or is that just an INPUT for the right hand side of Maxwell's equation to calculate B field? What do you think this error correction would be in our example, 534 places behind the decimal point, or larger?

Anyway, if you do know electromagnetics as you assert then you should already be more than familiar with Maxwell's Equations, synchrotron radiation, Lienard-Wiechart potentials and etc. These are all topics in undergraduate EM physics courses. There really should be nothing to discuss.

Yes, there is nothing to discuss anymore. Now it is a matter of whether any of you can actually apply these equations as you are asserting they can be applied, and you do not even need to struggle with SR and retarded time, or any corrections, just print down the damn equation that can solve this most simple example, in classical form, without relativistic corrections, like undergraduates would:

Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s.
- Solve for E and B when its velocity is 700m/s.

 

[Repainted]

I asked 10 times what E and B terms stand for on the right hand side of those equation. Anyway, I can use those same expressions to substitute E and B in Lorentz force equation, but why would I if error is 398 places behind the decimal point?

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?

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

Q: Electron accelerates along X-axis in 10 seconds from 200m/s to 900m/s.
- Solve for E and B when its velocity is 700m/s.

These were taken from the Wikipedia link that you obviously didn't read(and have been posted twice in the thread).

I think you might want to know what the the symbols mean(again from Wikipedia):

is the charges velocity divided by c and

is the vector position of the charge. The 'ret' emphasises that we are considering only the retarded solutions.
γ is the Lorentz factor and

is a unit vector pointing from the retarded position of the charge to the observer.

These were derived from Maxwell's equations, they aren't to be input into Maxwell's equations or anything. Input them into the Lorentz force law(which is basically just the definition of the E and B fields) and you get the force on a charged particle due to your moving electron.

 

[Dale]

Yeah, your equation is correct. That must have been a recent edition because I distinctly remember being put off by the fact that the article gave the scalar and vector potentials and not the field equations.

EDIT: Did you just fix that? I went over there to correct it but when I loaded up the edit page it had the corrected equations... sneaky...

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?

 

[Born2bwire]

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?

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.

 

[Dale]

That is special relativity. Why use special relativity correction? Why not use Coulomb's law and Biot-Savart law like we are supposed to in classical electromagnetism?

Because classical electromagnetism is relativistic (Maxwell's equations are Lorentz invariant). Historically, that was the whole motivation for developing special relativity. Also, as I have pointed out before, Coulomb's law simply does not apply in the situation of a moving point charge, it is an approximation to Maxwell's equations that assumes stationary charges (electrostatic).

Why complicate the terms of INPUT, we are looking at relation between Lorentz force and Maxwell's equations, we can use the same input terms for E and B in both, that does not help us, it only complicates unnecessarily.

I don't understand what you mean by "terms of INPUT". You are the one who specified a point charge. That is the only input, delta functions at an arbitrary location representing charge and current distribution due to a point charge. The Lienard-Wiechert fields are the solution of Maxwell's equations (the output result) using delta function charge and current distributions as the input representing an arbitrarily moving point charge.

I don't know how I can possibly make this any clearer. For the question you posed the answer is the Lienard-Wiechert field, which you can see is different from inappropriately applying Coulomb's law to an electrodynamic situation.

You cut off the rest of that paragraph where I explained what I mean. "Retarded time" is an effect related primarily to DISTANCE between TWO interacting fields, the greater the distance the more "delay", more "error"...

1.) What retarded time has to do with this case and only one charge, where is the distance, distance to what? "OBSERVER"?

2.) How large do you expect error correction to be if we can not even measure the whole 8 minutes delay of Sun's gravity field?

The retarded time is the distance between the source (point charge) and the position in space where you are calculating the field (divided by c). So the distance between the charge and each point in space if you are calculating the field everywhere.

I don't know what you think any of this has to do with gravity. We are talking about electromagnetism, not gravitation. It is a completely irrelevant comment. Please stick to one topic.

Now, let's stick to classical electromagnetism and go from the beginning. How are Maxwell's equations, including those E and B on the right side, defined in CLASSICAL ELECTROMAGNETISM, can you write that down please?

Did you forget how to spell google?
http://en.wikipedia.org/wiki/Maxwell's_equations#General_formulation

 

[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.