Talk:Biot–Savart law/Archive 1

Archive 1

Listed on Wikipedia:Votes for deletion Feb 20 to Feb 26 2004, redirected. Discussion:

• Biot-Savart law - text has nothing to do with the Biot-Savart law]] Fuzheado 23:18, 20 Feb 2004 (UTC)
  • I've redirected it to Biot-Savart's Law. Maximus Rex 23:42, 20 Feb 2004 (UTC)
  • I switched them the other way, since the preferred usage is tends to be without the apostrophe. -- Decumanus 08:57, 21 Feb 2004 (UTC)
  • List on redirects for deletion as copy/paste move which needs to be fixed. Anthony DiPierro 16:22, 21 Feb 2004 (UTC)

-- Graham  :) 21:08, 26 Feb 2004 (UTC)

---

In fact, the use of the apostrophe is clearly wrong. It implies that there was a single person named 'Biot-Savart', whereas the two names in fact belong to two different people. I'm going to fix all pages that link to the bad spelling. --Smack 00:40, 15 Nov 2004 (UTC)

Laplace's Law

I was looking for this version of Laplace's law: http://hyperphysics.phy-astr.gsu.edu/Hbase/ptens.html#lap Can someone either fix the redirection, maybe add a disambiguation, or explain where this should correctly fit?

first sentence; more general form?

The Biot-Savart law describes the magnetic field set up by a steady current density.

This is true, but as later noted in the article Biot-Savart is used extensively in aerodynamics. In fact it has been the lynchpin of all vortex models of flows around bodies for the past 70 years. Given its prominence in aerodynamics, shouldn't the first sentence of this article be changed to more generally describe application of Biot-Savart?

I made an attempt to modify the intro as you suggested. Feel free to improve it, expand it, or revert and start over. -- Metacomet 01:38, 27 December 2005 (UTC)

Just thought someone should know, I searched on Laplace Law, and got the following link: http://en.wikipedia.org/wiki/Laplace%27s_Law however, the article i got was Biot-Savart Law... I see no obvious connection between them. //Wikipedia reader ;)

Really, there are two different uses of the name "Biot-Savart law." One is the strict use in the E&M sense of finding the B field from a current, the other is the basic math of inverting a curl. Inverting the curl is all you are doing in finding B from J, and it's also all you are doing to find v given the vorticity in a fluid. You could also use it to find A (the vector potential) from B, up to gauge, or whatever. I think it makes more sense to separate out the core mathematical concept in the Biot-Savart law, somewhere on this pagePetwil 06:01, 8 September 2006 (UTC)

pronunciation

how to pronounce "Biot-Savart"?

-- As they are French, probably the correct is without trailing "t", but this is only a guess. The rest I suppose is pronounced phonetically. -- Mtodorov 69 14:24, 4 May 2006 (UTC)

I tried doing phonetic but I can't get the unicode to print right. Bee-oh, followed by sa like sa in sand but a bit more like "ah" as in "ah I see", art like the English word art but without the t, and with the r pronounced like a French r - fricative, swallowed r on the roof of your mouth, not the front of your mouth. But that won't do for the entry, eh? I dunno how to do the French phonetic unicode.... :)Petwil 06:24, 8 September 2006 (UTC)

Factors of 2

...are easily lost in this subject. I've 'corrected' to my understanding (Batchelor, 'Fluid Dynamics', eqn 2.6.4) - if you think I'm wrong, I'll need a reference. Linuxlad 13:20, 22 January 2007 (UTC)

geneneral statement first?

Could we see a general statement of this law in terms of vector analysis first, and then its applications to electromagnetics and to aerodynamics afterwards? Michael Hardy 02:12, 13 March 2007 (UTC)

The Introduction

While the application to aerodynamics is very interesting, it is not something that people expect to read in an introductory paragraph about the Biot-Savart law. A full section on the aerodynamics parallel and applications is nevertheless most welcome, but we should not overlook the fact that the law was first conceieved of in conjunction with electromagnetism and it is with electromagnetism that it is primarily associated.

The issue of the Biot-Savart Law being the inverse of the curl operator may be a matter of technical interest to mathematicians but it is hardly suitable material to include in the introductory paragraph. It's a bit like saying that a quantity is the product of two quotients. David Tombe 16th April 2007 (125.24.135.73 09:55, 16 April 2007 (UTC))

These two para is wrong-headed. The Biot-Savart law is standardly taught to 2nd/3rd year aero-engineers and has been around in that use for over 100 years. And it IS at the last just a formal vector field relationship! Bob aka Linuxlad (talk) 09:52, 13 October 2009 (UTC)

Coordinate Frame Origin

The Biot-Savart law contains the inverse square law of distance.

If we consider electromagnetic radiation deep in space, where do we fix the origin of the coordinate frame within which the inverse square term of the Biot-Savart law is measured? David Tombe 17th April 2007 (125.24.192.94 16:21, 17 April 2007 (UTC))

Lorentz Transformation

The Lorentz transformation acts on the full electromagnetic field tensor to produce the Biot-Savart law. See http://hepth.hanyang.ac.kr/~kst/lect/relativity/x850.htm This tensor already contains the magnetic vector potential term A. If we remove A from the equation, we cannot obtain the Biot-Savart law. Therefore it is not true to say that the Biot-Savart law can be obtained by applying the Lorentz transformation to Coulomb's law. We need to have the full set of Maxwell's equations to begin with. (86.155.139.178 21:47, 9 July 2007 (UTC))

Template:Electromagnetism vs Template:Electromagnetism2

I have thought for a while that the electromagnetism template is too long. I feel it gives a better overview of the subject if all of the main topics can be seen together. I created a new template and gave an explanation on the old (i.e. current) template talk page, however I don't think many people are watching that page.

I have modified this article to demonstrate the new template and I would appreciate people's thoughts on it: constructive criticism, arguments for or against the change, suggestions for different layouts, etc.

To see an example of a similar template style, check out Template:Thermodynamic_equations. This example expands the sublist associated with the main topic article currently being viewed, then has a separate template for each main topic once you are viewing articles within that topic. My personal preference (at least for electromagnetism) would be to remain with just one template and expand the main topic sublist for all articles associated with that topic.--DJIndica 16:46, 6 November 2007 (UTC)

Undefined constant

A constant $K_{m}$ begins to appear half way down page without any definition nor word of explanation. —Preceding unsigned comment added by 138.40.94.135 (talk) 18:14, 15 January 2008 (UTC)

Showing that Ampère's circuital law is the curl of the Biot-Savart law

Take the curl of,

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

This expands into four terms under the product rule. The two v terms vanish since v is a vector and not a vector field. The two terms left are v(div E) and (v.grad)E.

The former is equal to ρv which equals J. the latter is the convective term which we ignore at stationary points in space.

Hence curl B = J. This is Ampère's circuital law.George Smyth XI (talk) 06:18, 5 April 2008 (UTC)

If you're suggesting that the "derivation" of Ampere from Biot-Savart be put in, it seems reasonable and topical enough. It would probably be better to use a derivation that starts from the common form of the Biot-Savart law, instead of starting from the unconventional form in terms of E. See Jackson p178-9 for this derivation, for example. We can also put in the "derivation" of Gauss's law for magnetism while we're at it, see Jackson p179. Maybe we can use show/hide boxes to not clutter up the article with vector manipulations? --Steve (talk) 17:33, 5 April 2008 (UTC)

Steve, you asked for a citation for the above expression. That is hardly necessary. It follows directly from the previous section. Anyway, here is web link which backs it up. it's at about equation (19). http://hepth.hanyang.ac.kr/~kst/lect/relativity/x850.htm George Smyth XI (talk) 08:02, 6 April 2008 (UTC)

Well everything I've seen, including that link, indicate that this is the formula for the magnetic field of a point charge moving at constant velocity (changing neither direction nor speed). The section itself makes it sound more general than that, so I rewrote and resectioned accordingly. I'd still like to see a citation for any expression in that section being called "the Biot-Savart law", as opposed to "the formula for the magnetic field of a point charge moving at constant velocity" :-) --Steve (talk) 18:04, 6 April 2008 (UTC)

Steve, I'm happy enough if you put in the derivation of Ampère's circuital law from the Biot-Savart law. Try it for a few days. If it appears too cluttered then you can always side link it.

When you look at that derivation, you will notice that the inverse square law aspect of Biot-Savart plays no role in obtaining the current density term. George Smyth XI (talk) 09:58, 7 April 2008 (UTC)

Steve, Now that you have put in the derivation, I don't actually think that you need to offer the 'hide' option. I think it should form a full section in its own right. It shows that the Biot-Savart law is a solution to the differential form of Ampère's circuital law within the context of current understanding at textbook level. It is a well presented piece of information.

One point however worth noting is that as the curl of B is undefined at the origin, we cannot say that div B is zero everywhere. We have a dilemma which cannot be resolved within the context of current textbook level knowledge. I do however believe that it can be resolved, but that we will have to modify our understanding of the physical meaning of the terms in the Biot-Savart law. I don't think that the inverse square law applies on the large scale. George Smyth XI (talk) 11:47, 10 April 2008 (UTC)

Well, for a continuous current distribution, where J is finite at every point, the integrals all converge, B is finite everywhere, and its divergence is zero everywhere. That's absolutely mathematically rigorous, and certainly includes the "origin". So I guess what you're saying is that as you successively reduce the thickness of a finite-thickness wire, the divergence of B is zero everywhere, zero everywhere, zero everywhere, and then you finally get to zero thickness and the divergence is...undefined? Certainly physicists have never had a problem saying that the divergence is still zero in the limit. As for mathematicians, I'm sure that there are various formal, completely-mathematically-rigorous ways to take the divergence in the zero-thickness limit, and I'm quite sure that the result would be zero. In any case, this section includes the proof, straight out of two textbooks, that the Biot-Savart law is consistent (yes, at every point) with Gauss's law for magnetism, so I'm restoring the section title and description to include that reliable-source-verifiable fact.

Hmm, I'm partial to the show/hide box. It's important that any reader understand that these three laws are consistent. It's not important that they be able to follow the proof. By expanding the box, we make readers less likely to be able to read the important parts, since they might get put off by the vector manipulations. --Steve (talk) 16:40, 10 April 2008 (UTC)

Steve, the divergence cannot be zero at the origin of in an inverse position function. That applies to the E in the Coulomb force and it applies to any attempts to write B in that same form.

It seems to me that you are keen to highlight this effect in relation to E but to sweep it under the carpet in relation to B. George Smyth XI (talk) 03:49, 11 April 2008 (UTC)

George, here's the equation, straight out of a textbook:

$\mathbf {B} (\mathbf {r} )=\nabla \times \left({\frac {\mu _{0}}{4\pi }}\int d^{3}r'{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)$

If J is a continuous, finite current distribution, then the integral converges and we have here an explicit expression for B as the curl of a finite, well-defined vector field. The divergence of a curl is zero at every point. I don't know what point you're defining as the "origin", but the divergence of B is certainly zero there too. If you define the Biot-Savart law as "an inverse position function", then your claim that "the divergence cannot be zero at the origin of an inverse position function" is demonstrably false.

But I don't know why I'm arguing with you about this when you don't even believe the Biot-Savart law is true. Like I've said, if there's even a slight chance that you're right and every physicist in the last century is wrong about this, what are you doing wasting your time editing Wikipedia when you could be winning your Nobel Prize??? --Steve (talk) 06:11, 11 April 2008 (UTC)

Steve, and what's the curl of B when the denominator is equal to zero? George Smyth XI (talk) 10:11, 11 April 2008 (UTC)

Well, pick an r, any r. Here's the vector field in question:

$\left({\frac {\mu _{0}}{4\pi }}\int d^{3}r'{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\right)$

This contains an integral over r'. The integrand will go to infinity as r' approaches r, but the integral is finite. (Remember, three-dimensional integrals that go as (r')^-1 do not diverge near the origin.) That's why this vector field is a finite, well-defined vector field at each point r (assuming that J is a finite-valued function that goes to zero at infinity). There's no problem in computing the curl of this finite, well-defined vector field, at the point r. And its curl, as proven in the textbooks and outlined in the article, is B, which again is a finite, well-defined vector field at every point. And since B is a finite, well-defined vector field, which is the curl of a different finite, well-defined vector field, its divergence is zero at the point r. Happy now? Anything I can spell out in more detail? --Steve (talk) 16:09, 11 April 2008 (UTC)

Steve, Nobody is talking about integral forms. We are talking about the Biot-Savart law,

$d\mathbf {B} ={\frac {\mu _{0}}{4\pi }}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$

The inverse square law implies a discontinuity. The divergence cannot be zero at the origin. Hence the physical significance of the Biot-Savart law is not fully understood. There is probably a physical explanation that will make the Biot-Savart law correct. But that physical explanation does not exist in modern textbooks. Meanwhile, there is something seriously wrong with the Biot-Savart law. It contradicts the idea that div B = 0 everywhere. George Smyth XI (talk) 00:46, 12 April 2008 (UTC)

The form you gaves does not implies a discontinuity at the origin, it implies a singularity, meaning we have to take the limit as r goes to zero to know what the behaviour of that function is (because in physics, behaviour at singularities are given by limits). dB doesn't even have to be finite, because we are interested in the divergence of B, and not dB itself. To find the divergence of B, you use the reasoning that Steve presented earlier, and while you integrate over a region where the integrand has a singularity, it doesn't matter because B is well-behaved and everything comes out right. Headbomb (talk) 17:47, 12 April 2008 (UTC)

Hmm. I was talking about the version of the Biot-Savart law:

$\mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int d^{3}r'\mathbf {J} (\mathbf {r} ')\times {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}$

Try re-reading the above conversation now that we're hopefully on the same page. I was explaining to you the textbook proof that this law, assuming J is finite, yields a non-singular B whose divergence is well-defined everywhere, and equal to zero everywhere. Are we in agreement on this well-known fact? If so, then we can start talking about other versions, such as "differential", or with I instead of J.

Again, you're saying that the Biot-Savart law gives a magnetic field whose divergence is not zero everywhere, right? If so, you're not just talking about the "physical significance" of the Biot-Savart law, you're saying that either the Biot-Savart law is false, or Gauss's law for magnetism is false. You're on to something Nobel-prizewinning either way. Have you talked to any professional physicists about this yet? If not, what are you waiting for? --Steve (talk) 17:29, 12 April 2008 (UTC)

Steve and Headbomb, Considering the integral version only clouds the issue. The integral version is a summation of,

$d\mathbf {B} ={\frac {\mu _{0}}{4\pi }}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$

around an entire current loop. So let's just look at the core Biot-Savart law of the main article. There is an inverse square law term in it, hence there is a singularity at the origin, and hence neither the curl nor the divergence are defined at the origin.

You have tried to avoid the issue by taking the curl after you have done the summation (Integration).

The problem ultimately lies in the fact that no satisfactory physical model yet exists to back up the Biot-Savart law. If we had such a physical model, then we could better see the mechanical significance to terms such as B.

I'm sure that ultimately you will find that div B = 0 everywhere, but only when viewed on the large scale. The microscopic version as used in relativity,

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

avoids this problem. It's curl leads to Ampère's circuital law in a simple fashion, without having to use delta functions, just as the curl of the Lorentz force law leads to Faraday's law.

There is clearly a problem with the Biot-Savart law and this echoes right through to the issue of displacement current, in so much as the very existence of displacement current disproves the Biot-Savart law. The Biot-Savart law instantly breaks down in the dynamic state. In truth, you will find that no such law, exactly in the form of the Biot-Savart law, holds on the large scale. Div B = 0 holds on the large scale, but not the Biot-Savart law.

My ultimate point is that the term 'Gauss's law for magnetism', while technically correct, and used with increasing frequency in modern literature, is not in fact the best term to apply to div B = 0, since it misses the whole point of that equation. The older terminology 'No magnetic monopoles' is a much more informed name for div B = 0. George Smyth XI (talk) 08:33, 13 April 2008 (UTC)

dB has a singularity, not B. B is the integral of dB over a region, and there is absolutely nothing wrong with integrating over a region containing a singularity. There is no problem with the Biot-Savart law as written in differential form. The curl exists everywhere, the divergence exists everywhere and B exist everywhere, even if dB has a singularity at the origin. What's so hard to get about this?Headbomb (talk) 08:51, 13 April 2008 (UTC)

Headbomb, it's no different to the Coulomb force. The Coulomb force has zero divergence except at the origin. If we use an inverse square law in the Biot-Savart law, we are in the same boat. The only hope of resolving the dilemma is to understand the microscopic structure of Faraday's tubes of force and see where these sources actually are on the microscopic scale. George Smyth XI (talk) 11:14, 13 April 2008 (UTC)

George, we're interested in what the divergence of B is. Not dB, but B. And B is given by an integral. So we should absolutely be talking about the integrated form. In Coulomb's law, if you use the integrated form, you can compute the divergence, and you get the charge density just like you expect. For this law, you can do the same thing, use the integral form, compute the divergence, and you get zero at every point. (The difference ultimately comes from the fact that the Biot-Savart law has a cross-product in it, while Coulomb's law does not.)

So let's talk about the integrated form. Do you agree that if you use the integrated Biot-Savart law, (which is what, in real life, you do use), then a continuous current distribution will give you a B which is defined everywhere and divergenceless everywhere? If not, which part of this argument do you not follow? --Steve (talk) 16:15, 13 April 2008 (UTC)

Steve, the integral B is merely a summation of many smaller elements of B. Those smaller dB's are contributory elements to B. Each of those elements has got a source. Hence the final B has actually got many sources. You cannot sweep those many sources under the carpet by doing an integral.

The final B may well be solenoidal on one scale. But the Biot-Savart law is telling us that there are sources on another microscopic scale. Hence the Biot-Savart law cannot hold true on the same scale that div B = 0 holds on.

I know that Biot-Savart has got a cross product whereas Coulomb doesn't. That cross product enables it to be written in the form of a curl. But that curl is not defined at the origin and therefore neither is the divergence of that curl defined at the origin. You cannot paper over this reality by taking the integral (summation) of many dB elements.

You'll get the answer when you discover the focused picture of Faraday's tubes of force. The B line runs through them and it is solenoidal. But you then need to look at what the tubes themselves look like up close.George Smyth XI (talk) 16:59, 13 April 2008 (UTC)

IT.DOESN'T.MATTER. that dB has a singularity in it. B is the result of each dB true, and there is one of those (of infinitesimal width) that has a singularity in it, true. But when you integrate, it really doesn't matter because you get a finite well-defined B, and THAT is the important quantity. It would be as if you would be arguing that there is a problem with evaluating the derivative of Cos(a*x) when a=0 because Cos(a*x)+C, the integral of -sin(a*x)/a over a region x, is undefined at when a=0.Headbomb (talk) 18:37, 13 April 2008 (UTC)

George, you say "The final B may well be solenoidal on one scale". The "final B" is the only B; dB is unmeasurable even in principle. In any case, the "final B" is a vector field; it's a well-defined vector at each point, and we can compute the divergence of this field. This divergence either is, or isn't, zero at any given point. There's no room for "different scales"; there's just a single, final field, B, whose divergence either is or isn't zero.

Incidentally, I'm curious what your feelings are about this equation:

$\int _{-1}^{1}|x|^{-1/2}dx=4$

Do you think this is a reasonable and correct equation to write down, or is it nonsense since the integrand goes to infinity at zero? I'm just trying to figure out exactly where you're coming from. --Steve (talk) 18:33, 13 April 2008 (UTC)

Steve and Headbomb, The dB means a small change in an already existing B. The dB equation is where the physics lies and it involves a source.

You are both trying to cloud the issue by looking at a definite Riemann integral around a closed path. Certainly if we obtain a total value of B by summing over all the small elements of a closed loop, then we will obtain a final definite number. But we cannot take the curl of that number. The curl will already have been taken before the integral is performed.

It is wrong to write the curl outside the definite integral. The theorem that allowed the original cross product expression to be written as a curl comes first.

Essentially, in your integral version, we have an infinite number of sources all around the wire. But if we perform a definite integral for a B that lies beyond the wire, then of course we will get a finite result. We will get a number. But that doesn't get rid of the fact that B has a source under the terms of the Biot-Savart law.

The point is that the form of the Biot-Savart law may be correct on a scale that involves a physical model that is not yet realized in the textbooks.

But as it stands at the moment, there is a big problem because we are left with a dilemma. As it stands at the moment, the Biot-Savart law points to the fact that the equation div B = 0 breaks down at the origin.

The microscopic version should give you a clue to the solution. It links the sources directly with the Coulomb force. The Coulomb force underlies the Biot-Savart law at microscopic level. Hence Faraday's tubes of force while solenoidal on the large scale, contain sources on the microscopic scale.

Compare the situation to a loop of string. It is solenoidal. But the string is full of sources due to the molecules that it is made of on the microscopic scale.

All this is already in the published literature. But not in the kind of journals that you would consider to be acceptable.

But my main purpose here as far as wikipedia is concerned, was to point out that as regards Maxwell's equations, div B = 0 is not about Gauss' law. It is about solenoidality, and hence that equation is better described as 'No magneic monopoles' as was done in the higher quality textbooks of days of old. George Smyth XI (talk) 08:06, 14 April 2008 (UTC)

It is NOT wrong to take the curl outside of the integral. In fact we must take the curl outside of the integral because we are interested in the curl of B (where B is defined by the integral), not in the integral of the curl of the integrand.

In other words,

${\vec {\nabla }}\times {\vec {B}}={\vec {\nabla }}\times {\frac {\mu }{4\pi }}\int {\frac {Id{\vec {l}}\times {\vec {r}}}{r^{3}}}$

and not

${\vec {\nabla }}\times {\vec {B}}={\frac {\mu }{4\pi }}\int {\vec {\nabla }}\times {\frac {Id{\vec {l}}\times {\vec {r}}}{r^{3}}}$

Same remarks applies for the divergence. Headbomb (talk) 16:45, 14 April 2008 (UTC)

Headbomb, for an indefinite integral, either way is correct. The curl is distributive. But we are looking at a summation. We are looking at a Riemann 'definite' integral. We are looking at a final number. We cannot take the curl of a final number. We are looking at the summation of curl terms. The curl must come inside the definite integral.

Basically, if the divergence of a function is zero everywhere, then the function is solenoidal. But if there is a singularity, then the divergence cannot be zero at that singularity. In other words, if B is the curl of a 1/r function, then it must have sources.

So the Biot-Savart law presents a riddle. George Smyth XI (talk) 17:20, 14 April 2008 (UTC)

If either way are equivalent, and that the curl is always defined in one of the ways (taking the curl of the integral), then it is also always defined in the other way (integral of the curl), so there's no problem. If you're saying that they are not equivalent, then the the distinction I made applies, and there is no problem with the Biot-Savart law because the curl of the integral is always defined. And we are looking at a final vector, not a number (which is scalar) so there's no problem there. And the B field does not contain singularities even if dB does, so there's no problem there either. So there is no riddle because B, its curl and its divergence are all well-defined and well-behaved. Headbomb (talk) 17:35, 14 April 2008 (UTC)

Headbomb, the physics begins with the theorem that converts the cross product into a curl. The summation comes afterwards, and so it is essential that the curl is inside the definite integral. In fact, I'm beginning to think now that that is also the case even for the indefinite integral, but that is not important in this argument. We are dealing with a definite integral and the curl must come inside it.

The definite integral results in a vector as you say. Yes. 4 west. You can't take the curl of it. The final result is not a vector field. The final result is a value for B at a particular point.

You are trying to bundle up a myriad of sources inside a packet and pretend that they don't exist.

The Biot-Savart law is about dB. That is a term referring to a small change in B. If dB is inverse square law dependent, then so is B.

Clearly, the Biot-Savart law points to sources. Your challenge is to explain where these sources fit into the solenoidal B lines. George Smyth XI (talk) 04:44, 15 April 2008 (UTC)

The result is B at a particular point eh? That could almost be written as ${\vec {B}}({\vec {r}})$... My my, that really does look like a vector field...

The Biot-Savart law may be about dB, who cares? We're not interested in dB, the curl of dB, the divergence of dB, the integral of the curl of dB or the integral of the divergence of dB. We are interested in B, the curl of B and the divergence of B. You calculate B first, then you take the divergence or the curl.Headbomb (talk) 12:20, 15 April 2008 (UTC)

And if dB is dependent on 1/r^2 implies that B also depends on 1/r^2, that means that all the magnetic fields in the world have spherical symmetry!

Face it George, you've said something wrong, and you're were arguing for it because it made sense in your head. Nothing wrong with that. But then you got proud. Proud to have found something wrong with a very well known equation - you were "smarter" than the rest of the world. We showed you wrong, but you don't want to be wrong, because that would mean you're just a regular Joe, so you're arguing tooth and nails for it, reach at straws and now make totally incoherent statements you would never have made before this debate because you don't think things through anymore. You're more concerned about us being wrong than making sense. Take a break, it'll clear your mind.Headbomb (talk) 12:20, 15 April 2008 (UTC)

Headbomb, You can't just unilaterally declare yourself to have won the argument. The definite (Riemann) integral that you are interested in is a summation of many B terms, all with their own source. The final result is a numerical value. It is not a vector field. The terms that sum together to give that value all have sources. Hence there is a problem with the Biot-Savart law as it stands at present, because it fails to explain where these sources lie in the solenoidal B lines.George Smyth XI (talk) 13:37, 15 April 2008 (UTC)

Here's the expression:

$\mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int d^{3}r'\mathbf {J} (\mathbf {r} ')\times {\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}$

At each value of r, you have a value of B, which is a vector. In other words, B is a vector at each point of space, a.k.a. a "vector field". Agreed? If not, how do you define the term "vector field"? --Steve (talk) 15:54, 15 April 2008 (UTC)

Steve, For a given value of r, you have a value of B. Hence B has a source. At any point in space, the total value of B is the sum of all the contributory sources summed around the current loop. That value is given by the summation (definite integral). It doesn't get rid of the sources. Hence there is a problem as regards reconciling the Biot-Savart law with the idea that div B = 0 everywhere.George Smyth XI (talk) 07:12, 16 April 2008 (UTC)

The Biot-Savart law represents the contribution to the B field made by an infinitesimal section of wire located at the origin. Mathemathically this is

$d{\vec {B}}={\frac {\mu }{4\pi }}{\frac {I({\vec {0}})d{\vec {l}}\times {\vec {r}}}{r^{3}}}$

To clarify things, let's place the source at the r_l

$d{\vec {B}}={\frac {\mu }{4\pi }}{\frac {I({\vec {r_{l}}})d{\vec {l}}\times ({\vec {r}}-{\vec {r_{l}}})}{(r-r_{l})^{3}}}$

The B field is the sum of the contribution of every section of wire. So the contribution to the B field made by a section of a wire going along the path l, from A to B is the integral over dl from a to b.

${\vec {B}}=\int _{a}^{b}d{\vec {B}}=\int _{a}^{b}{\frac {\mu }{4\pi }}{\frac {I({\vec {r_{l}}})d{\vec {l}}\times ({\vec {r}}-{\vec {r_{l}}})}{(r-r_{l})^{3}}}$

Integrating over space coordinates of the wire (r_l removes all the dependencies on dl, and r_l/, leaving us with something that depends exclusively on r, and that does not depend on r in the same way that dB did.Headbomb (talk) 18:57, 15 April 2008 (UTC)

Headbomb, so long as you have a 1/r^2 dependency in the Biot-Savart law, you can never get rid of the reality that on some level, there will be sources.

At the moment you are trying to cloud the issue by using the definite (Riemann) integral around a closed current loop. But you wouldn't be able to do so if I were to use the equation for the magnetic field generated by a single moving point charge. George Smyth XI (talk) 07:09, 16 April 2008 (UTC)

The equation for the magnetic field generated by a single moving point charge is not the Biot-Savart law, so I hardly see how that could be relevent. I assume that you're not so uncomfortable with the mathematics of integration that talking about the Biot-Savart law in the context of an integral over sources isn't going to "cloud any issues". After all, this is how the Biot-Savart law is actually used in all actual calculations, and even the way it's most often defined. Again, we're interested in the divergence of the actual value of B; and the actual value of B, as measured by a compass or whatever, is what you get after integrating.

Anyway, you seem to agree that "For a given value of r, you have a value of B, [which] is the sum of all the contributory sources summed around the current loop. That value is given by the summation (definite integral).". It sounds like you agree that for a given value of r, you have a "final" value of B (or, as I would say, an "actual" value of B). Yet you still assert that B "is not a vector field"?? Can you please explain how, if B(r) is a vector for each point r, B does not constitute a "vector field"?? Here's the definition of "vector field".

By the way, just to be sure, you do understand the roles of r versus r', right? You pick any point r where you're interested in the value of B, and then you integrate over r' to find the value of B at r (holding r fixed for the duration of the integration). Then, if you want, you can pick a different point and compute the field B there too. In this way, you can compute B at any value of r. The "sources" in question are not related to r, but r'. Also, remember that B is a property of the location in space. It exists and is well-defined at each point, even if there's no test charge present to experience that field, and certainly regardless of the test charge's velocity or anything else. I feel like this may also be part of the confusion, since for example you seem to adhere to Maxwell's no-longer-in-use definition of E as "electromagnetic force per charge" (even if the charge is in motion), which is wrong according to modern definitions...and maybe you have some analogous outdated definition that you're thinking of in the case of B. Hope that helps, --Steve (talk) 18:36, 16 April 2008 (UTC)

Steve, there is a value for B due to a small element of the current loop. There is also a total value for B due to the summation of all such elements around the loop.

Each contributory element of B has an associated source due to the inverse square law term.

That's all I'm saying. It doesn't matter if you can obtain a finite value for the total B at a given point. There are still sources for B based on the inherent inverse square law terms in all the constituent elements.

The overall B field is solenoidal. So there are matters which need to be explained. You can't explain this using the definite integral. By drawing attention to the definite integral you are merely zipping the sources up inside a bag and pretending that they don't exist.

There are sources to be accounted for inside the solenoidal B lines. The B lines themselves are solenoidal, but have you enquired into the microscopic structure of Faraday's tubes of force to learn more about the sources?

The Biot-Savart law points us in the right direction, but when the full physical picture emerges, it will be realized that the Biot-Savart law is limited in its extent. George Smyth XI (talk) 01:54, 17 April 2008 (UTC)

Hmm, it does seem to me like you have in your head some definition for B which, like your definition of E, is not the definition used by modern physicists. In modern physics, B is a vector field, i.e. a single, well-defined vector at each point of space and time. You can't have "a value of B due to a small element of the current loop [and] a total value of B". Each point in space and time has one and only one value for B. The thing called "dB" is not a "value for B", it's merely a mathematical formality, a "term that contributes to the actual magnetic field B". dB is certainly not "the magnetic field", but (the total) B is, and it's usually "the magnetic field" that we're interested in talking about.

If you agree that the final, actual value of B can be a finite vector field, then you'll agree that you can compute the divergence of that field, by numerically or analytically calculating the various derivatives of the field B at each point and adding them in the appropriate way. If you've followed the argument given in this article, and in textbooks, and above, then you'll agree that when you do this, and compute the divergence of the final, actual B, you'll get zero at every point. Do you agree with this statement? I'm not making any claim (at this point) about how this statement is best interpreted physically. I'm just saying, you acknowledge that you can obtain a finite final vector field for the total B, so you'll agree that there's nothing stopping you from calculating the divergence of this field, and it's simple enough to mathematically prove that the divergence you calculate will be zero at every point. --Steve (talk) 03:00, 17 April 2008 (UTC)

Each of the elements that sums to give the total value of B is a vector field. The Biot-Savart law as presented overleaf is a vector field. Each of those Biot-Savart elements has zero divergence except at the origin.

The final number and direction for B, which you achieve after doing the Riemann integral is not a vector field.

The value of B is the summation of the value of dB over all the elements. George Smyth XI (talk) 10:08, 17 April 2008 (UTC)

How is the sum of many vector fields not a vector field?Headbomb (talk) 12:17, 17 April 2008 (UTC)

Headbomb, the sum of many vector fields is indeed a vector field. But when you evaluate a vector field at a point in space, you cannot then take the curl of that value.

In this particular case, the vector field for the total value of B is a summation of many curl terms. We can evaluate it at any point in space by performing the Riemann integral.

But what you want to do is sum the parts inside the curl, evaluate, and then take the curl. You cannot do that.

The total magnetic field B is a summation of many elements each with a source.

Therefore there are problems with reconciling the Biot-Savart law and the idea that the divergence of B is zero everywhere. George Smyth XI (talk) 05:21, 18 April 2008 (UTC)

And we don't take the curl of the value of the field at a certain point either. We take the curl at a certain point in the field. The curl does not depend on what value the field is at the point, but rather on how the field varies at that point. Hence the need to find the expression for the field (ie, find B(r) by integrating dB over r_l (Steve used r'), then take the curl with respect to r, $({\vec {\nabla }}\times {\vec {B}}({\vec {r}}))$.

Headbomb, You've just contradicted yourself. You have agreed that we don't take the curl of the value of the field at a certain point. But if we take the curl of a definite integral, which is what you want to do, then that is exactly what we will be doing. We will be taking the curl of the value of the field at a certain point.

For evaluation purposes, we must sum over the curl terms first. Each of those curl terms has a source. Hence we have a problem reconciling the Biot-Savart law with the idea that B has got no sources.George Smyth XI (talk) 07:19, 20 April 2008 (UTC)

Look, I use the terms you use because I want you to understand. If you keep switching what those terms mean, then I can't help you. You used "value" in the sense of value at a specific point, or at least that's the only way the sentence But when you evaluate a vector field at a point in space, you cannot then take the curl of that value. for example the B-field at position r₁ has the value of B(r₁), which could be something like B(r₁). You cannot take the curl of B(r₁), because "B(r₁)" does not give you any information about how the field varies, which is required for taking the curl. So you take the curl of a field, at a specific point, and not the curl of what the field is at a specific point. Mathematically,

$({\vec {\nabla }}\times {\vec {B}}({\vec {r}}))|_{r_{1}}$

and not

$({\vec {\nabla }}\times {\vec {B}}({\vec {r}})|_{r_{1}})$

Headbomb (talk) 12:34, 20 April 2008 (UTC)

Headbomb, you've lost me now. Do you wish to sum all the individual curl terms and then obtain a value? Or do you wish to sum all the terms inside the curl expression, take a value, and then take a curl of that value? Which is it?

If it is the former, then every single one of those curl terms that contributes towards the final value at a point, will have a source.

If it is the latter, then it is a nonsense. George Smyth XI (talk) 12:49, 20 April 2008 (UTC)

Neither. You integrate dB over r' to get B(r), then you take the curl of B(r) with respect to r. Headbomb (talk) 12:56, 20 April 2008 (UTC)

Headbomb, If we do that then we will be taking the curl of a definite integral. The definite integral is a numerical value and it doesn't have a curl. We must keep the curl inside the integral sign. In that case, each contributory element to the final value of B will have a source. Hence the Biot-Savart law has got problems reconciling itself with div B = 0. George Smyth XI (talk) 08:39, 21 April 2008 (UTC)

No, we'll be taken the curl of a vector field. dB depends on two things, r and r'. You integrate over r', and get something that depends on r. Then you take the curl with respect to r. There's no problems. It's the exact same thing as if you'd be intergrating 12xy^2 with respect to x on a region (say -1 to 2) to get (18y^2) and then diffenriate with respect to y (36y).Headbomb (talk) 12:12, 21 April 2008 (UTC)

Headbomb, There is only one variable. That variable is the distance from the point in space in question and the point on the current loop.

Theoretically, a definite integral would be tracing out a closed loop in space with the variable representing the distance to a fixed point on the wire. But in practice, in this case, the point in space is fixed and the distance r varies as we go around the closed electric circuit. George Smyth XI (talk) 12:55, 21 April 2008 (UTC)

"...the distance from the point in space in question (r) and the point on the current loop. (r')"

Looks like two variables to me. When you want to know what the B field is at "the point in space in question" (B(r)) you integrate over "the point on the current loop" (r').Headbomb (talk) 14:47, 21 April 2008 (UTC)

Headbomb, No. There is one variable. That variable is the distance between the point in space and the point on the current loop. The point on the current loop is a source. You cannot cover up that reality no matter how hard you try to confuse the issue with definite integrals. George Smyth XI (talk) 06:39, 22 April 2008 (UTC)

"...the distance between the point in space and the point on the current loop (r-r_l)".

I dunno about you, but the distance between two point depends on where those two points are. Which means it's two variables.Headbomb (talk · contribs) 00:06, 23 April 2008 (UTC)

Headbomb, The Biot-Savart law gives the expression for B at a point in space. The one single varibale in question is the distance from that point to a point on the current loop. To get a final value of B at the point in question, we do a definite integral around the whole loop. That involves one variable. George Smyth XI (talk) 07:35, 23 April 2008 (UTC)

George, it might help if you'd step back for a second and think about how you define terms like "curl" and "vector field". The way I define "vector field" in this context is that a vector field is a function from 3D Euclidean space to 3D Euclidean space. The way I define curl in this context is that "curl" is a certain well-known function whose input is a differentiable vector field and whose output is a vector field, and whose precise definition can be found in any vector calc textbook. Now I know that your definitions are different, since you stated that JRSpriggs needs to "expand [his] understanding of vector fields to include moving points". So, what are the precise, mathematical definitions that you're using for these two terms? And is there any scientist in the last 50 years who uses the same definitions as you? And if not, could you please try to reword your statements in such a way as to be consistent with the terminology that everyone else uses? It would help everyone, including yourself, if you didn't use a term to indicate some concept when everyone else uses the same term to mean something else. --Steve (talk) 22:22, 22 April 2008 (UTC)

Steve, my definition of curl is the same as yours. My definition of a vector field extends beyond the conventional one to include moving points, where the motion of a particle at that point changes the value of the function in question.

In this situation (Biot-Savart law), we don't need to concern ourselves with my extended definition of vector field.

To take a curl, we need a vector field. That means a function which defines a quantity in terms of a position variable. We can only take a curl of a function in its general unevaluated state.

Therefore, in the curl version of Biot-Savart, we must first evaluate each curl term and then sum them all together. We do not sum the expressions inside the curl, obtain a final value and then take the curl.

But let's not lose track of what this was all about. It stemmed from the controversy over whether div B = 0 implies a solenoidal field or an inverse square law field.

We agreed that it is solenoidal providing that we are explicit about the fact that div B = 0 everywhere.

I then drew attention to the fact that the Biot-Savart law leaves a number of questions unanswered because it could be taken either way, hence appearing to cause a dilemma. I was not intending to put my views on this matter unto the main page. George Smyth XI (talk) 07:35, 23 April 2008 (UTC)

My definition of vector fields does not include "moving points", and as such, I don't know how to take the curl of those. I take "a vector field consisting of a moving point" to mean a function from the real line to 3D Euclidean space r(t), and a function from the real line to 3D Euclidean space V(t). Is that right? If so, can you please give an explicit formula which gives the curl of the "vector field" defined by r(t) and V(t)? Maybe it's a bit off-topic, but I'm very curious.

Also, can you please give a precise mathematical distinction between an "evaluated" and "unevaluated" vector field, and explain why it's possible to take the curl of one but not the other? My definition of the curl of a vector field is such that you can take the curl of any differentiable vector field, whether it's known numerically, symbolically, or whatever. Your distinction between "evaluated" and "unevaluated" isn't something I've seen in any textbook or learned in any course, and I frankly can't make any sense of it. Again, if you look up the standard definition of curl, in any book, you'll find a definition that works for any function from 3D Euclidean space to 3D Euclidean space, not just "unevaluated functions", whatever that means.

More generally, George, you have many ideas about classical electromagnetism which manifestly contradict the way that every professional physicist understands the subject. I'm very happy that you're not trying to push your unique point-of-view on this page, but you have been on other pages, and I see it worthwhile to follow through here, on the off-chance that you might actually, for once, understand that you might be wrong when you make statements that explicitly disagree with the entire physics community, rather than the physics community being composed of charlatans. :-) Really what you should be doing is stopping contributing to Wikipedia's electromagnetism articles until you can rectify your understanding with the rest of the world's. This is what we mean by No Original Research. If you live near any university, you should arrange a meeting with a professor, who can either explain to you why you're wrong (this is easier in person with a blackboard than it is online), or co-publish with you a Nobel-prize-worthy, completely-revolutionary paper on electromagnetism. If you're so confident that you're right, then that should be your priority...Nobel prizes are more important than Wikipedia. --Steve (talk) 15:47, 23 April 2008 (UTC)

Steve, a vector field is a point function that is a vector. Generally speaking we can differentiate a function because it has a variable. In the simplest case y =2x, then dy/dx = 2.

But If I draw the graph of y = 2x and then decide that I want the area under that graph between x = 3 and x = 5, I will perform a definite integral. I will then end up with a number.

I cannot then differentiate that number. Likewise with the Biot-Savart law. It caters for the contribution towards a B field at a point due to one element of current. To get the term for the total value of B at a point, we must sum all the dB terms. We're agreed that those dB terms can be written in a curl format, due to a vector identity theorem. Hence we can sum all those curl terms and we will have an expression for the total value of B. We can even do a Riemann integral and obtain a number.

But we cannot do the Riemann integral on the term inside the curl and then take the curl of the resulting number afterwards.

That was your original point. You argued that by obtaining such a Riemann integral, we got rid of any concerns about sources. You argued that the divergence of a curl is always zero, and so it is. But there are still sources inside that expression. So we have a dilemma.

The Biot-Savart law has an inverse square law term which means that B can be div zero on two mutually contradictory counts. In other words, modern electromagnetism is not yet complete.

My point of course was that in Maxwell's equations, we should be emphasizing the solenoidal aspect of div B = 0 since it comes from curl A = B. Hence, the name 'Gauss's law for magnetism' would not have been my preferred choice.

On your other point about taking the curl of vXB, just expand it using the vector identity for curl of a cross product. The two v terms will vanish because they are not vector fields. You will be left with the convective term that adds to the partial term to make a total derivative Faraday's law. That derivation is in a January 1984 paper in some American journal.

What original reserach was I pushing on the other pages? Was it the equivalence of Faraday's law and the Lorentz force?George Smyth XI (talk) 16:14, 23 April 2008 (UTC)

Part 1 Quoting: "Steve, a vector field is a point function that is a vector. Generally speaking we can differentiate a function because it has a variable. In the simplest case y =2x, then dy/dx = 2."

Who taught you vector calculus (or calculus for what matters)? The reason why we can differentiate a function is not because it has a variable. Functions always have variables, and yet some of them are not differentiable (see for example Weierstrass functions which are real and continuous, yet differentiable nowhere, or d|x|/dx at x=0.).

Vector fields are not point functions (what's a point function is anyway?) that are vectors, vector fields are mapping of vectors. A.k.a. their is a vector associated with each point in space. An exemple of a 1 dimensional vector field (also called a scalar field) would a vector such as ${\vec {A}}(x)=x{\hat {x}}$, an exemple of a 3 dimensional vector field would be a vector such as ${\vec {B}}=x{\hat {x}}+y^{3}{\hat {y}}+\sin ^{z}(xyz){\hat {z}}$. N-dimension vector fields are fields of the form

${\vec {V}}(a_{1},a_{2},...,a_{n})=f_{1}(a_{1},a_{2},...,a_{n}){\hat {e_{1}}}+f_{2}(a_{1},a_{2},...,a_{n}){\hat {e_{2}}}+...+f_{n}(a_{1},a_{2},...,a_{n}){\hat {e_{n}}}$,

or more succinctly, ${\vec {V}}({\vec {r}})=f_{1}({\vec {r}}){\hat {e_{1}}}+f_{2}({\vec {r}}){\hat {e_{2}}}+...+f_{n}({\vec {r}}){\hat {e_{n}}}$.

Part 2 Quoting: ":: But If I draw the graph of y = 2x and then decide that I want the area under that graph between x = 3 and x = 5, I will perform a definite integral. I will then end up with a number. ::I cannot then differentiate that number."

Yes you can. Numbers are constant, so if you differentiate it, you get 0. For example, d(32)/dx=0.

Now if you integrate xy with respect to x over x=3 and x=5, you end up with

$\int _{3}^{5}xydx={\frac {25}{2}}y-{\frac {9}{2}}y=8y$

If we differentiate with respect to x, we get 0. If we differentiate with respect to z, we get 0. But if we differentiate with respect to y, we get 8. It is the exact same thing with the Biot-Savart law.

The contribution to the B field at location r made by an element of wire carrying current I is given by the Biot-savart law. If the source is at the origin, the Biot-Savart law takes the form:

$d{\vec {B}}={\frac {\mu }{4\pi }}{\frac {Id{\vec {l}}\times {\vec {r}}}{r^{3}}}$

This form is rather inconvenient if you want to find the B field given by a whole wire, because a wire cannot be contained at the origin. And so to clear up things, let's write Biot-Savart in more convenient form, where it'll give us the contribution to the B field at location r made by an element of wire at location r_l carrying current I.

$d{\vec {B}}={\frac {\mu }{4\pi }}{\frac {I({\vec {r_{l}}})d{\vec {l}}\times ({\vec {r}}-{\vec {r_{l}}})}{|{\vec {r}}-{\vec {r_{l}}}|^{3}}}$

With that form, we can find the field made by a wire going along a path. The B-field at position r is then given by summing over each element of wire, (integral over r_l coordinates):

${\vec {B}}({\vec {r}})=\int {\frac {\mu }{4\pi }}{\frac {I({\vec {r_{l}}})d{\vec {l}}\times ({\vec {r}}-{\vec {r_{l}}})}{|{\vec {r}}-{\vec {r_{l}}}|^{3}}}dV_{l}$

For a solenoid with N turns, this gives the familiar expression ${\vec {B({\vec {r}})}}=\mu NI{\hat {x}}$

The divergence of a constant is 0 (which is peachy because the divergence of any B field is 0 ($\nabla \cdot {\vec {B}}=0$), and so is the curl (which is also peachy, because inside the solenoid, there is no current, and the E field is constant so the curl should be zero $\nabla \times {\vec {B}}=\mu {\vec {J}}+\mu \epsilon {\frac {\partial {d}E}{\partial {d}t}}$).

That fields are produced by "sources" is completely irrelevant. That there's a singularity is completely irrelevant because we are not interested in the dB field, its curl, or its divergence. We are interested in the B field. The B field of wire of infinitesimal ridius is given by ${\vec {B}}({\vec {r}})={\frac {\mu I}{2\pi \rho }}{\hat {\phi }}$ (cylindrical coordinates). The divergence of this is zero, which is peachy. The curl of this is zero everywhere, except at ρ=0, where it is undefined (but that's because current density is infinite). You can define it at zero if you treat the current density as a delta function in ρ, and it'll be zero there too.Headbomb (talk · contribs) 20:03, 23 April 2008 (UTC)

Headbomb, I apologize if I was a bit sloppy with my maths terminologies but I don't see this issue here as being resolved by going deep into maths and delta functions.

By your own admission there is something peachy surrounding the interpretation of div B = 0 in relation to the Biot-Savart law. That's all I have been saying to Steve. The situation is not as clear cut as some people think.

Hence the need for good quality naming terminology in relation to div B = 0. Stick to the good old fashioned 'No Magnetic Monopoles'.

While there may be a theoretical case to back up the increasing modern usage of the term 'Gauss's law for magnetism', I believe that somebody who has looked closer at the Biot-Savart law would not wish to use that name. Biot-Savart is too complex for the purposes of using the name 'Gauss's law'.

As I said to Steve, I am not interested in amending the main article here. The main article is quite good and it presents the facts as they are presently understood, in quite a clear manner.

Steve has agreed in principle that the curl of Biot-Savart leads to Ampère's circuital law.

There is no point in carrying this discussion any further because it will end up in a battle of opinions, and since the Biot-Savart law is official orthodox physics, then this is not the place for me to be arguing about the extent of its application.

If you still think that the definite integral can patch up the sources, then we'll just have to agree to differ. George Smyth XI (talk) 06:40, 24 April 2008 (UTC)

Peachy... as in everything's fine.

Yea, seems like you misunderstand the term "peachy". You seem to have interpreted it as "fishy", maybe.

The "equivalence" of the Lorentz Force and Faraday's law is your original research, as is your use of "curl E=-dB/dt" with a total derivative instead of partial derivative as a meaningful and distinct equation, as is your assertion that the two words "Faraday's law" can have one and only one correct God-given meaning, despite widely-used modern textbooks explicitly disagreeing. The page WP:RS makes it clear that Griffiths and Feynman are extremely reliable sources, while Maxwell is not at all. So by deleting information from extremely-reliable sources, and replacing it with your own opinion, you're continually engaging in original research, which is fine to do, but not on Wikipedia. To justifiably delete information which is explicitly stated in reliable sources, your task would be to find even more reliable, and even more explicit, sources to back yourself up, something that you've been showing little inclination to do.

I hope you make the effort, if possible, to talk in person to a practicing physicist about the Biot-Savart law and any of your other unique opinions about electromagnetism. These types of conversations are a lot easier and more productive at a blackboard than they are over the internet. :-) --Steve (talk) 01:02, 28 April 2008 (UTC)

The summation B

Headbomb, maybe I ought to further clarify that the summation B of the definite integral is not the same as the B that is associated with dB in the Biot-Savart law.

The dB implies a small change in B at that point due to a single element of current. Both this B and dB will be inverse square law dependent under the terms of Biot-Savart.

The summation B that you seem to be wanting to focus on is a summation of all such B's in the above paragraph, around a closed loop of current. The summation, or definite integral will give a total value for B at that point. But we cannot take the curl of that value.George Smyth XI (talk) 04:55, 15 April 2008 (UTC)

Magnetic Monopoles

Steve, I saw your speculative generalized Lorentz force on the Magnetic monopoles page. I like that kind of speculation. But I can assure you that the analogy with the actual Lorentz force doesn't exist just as nicely as you would like. In fact, the correct equation for magnetic force is,

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

and there are no extra terms. B is actually magnetic force within the context of magnetic charge. I think that once again, your mistake in assuming the extra terms, lay in the fact that you were ignoring the contents of the E term of the real Lorentz force. That is the Coulomb force and the -(partial)dA/dt bit.

There is actually no equivalent to the -(partial)dA/dt term in the magnetic analogy. You could perhaps put down a version of Coulomb's law for magnetic charge beside the above expression. George Smyth XI (talk) 08:15, 14 April 2008 (UTC)

Uh, shouldn't this be on the magnetic monopole talk page? Headbomb (talk) 17:35, 14 April 2008 (UTC)

Maybe. But it is relevant to this article too. Maxwell didn't include either Faraday's law or the Biot-Savart law in his original eight equations. I want to show Steve that Ampère's circuital law is to Biot-Savart as Faraday's law is to the Lorentz force.

Interestingly, Maxwell criss-crosses this relationship by opting for the Lorentz force and Ampère's circuital law. One might have thought that the Biot-Savart law was the most closely parallel to the Lorentz force. But then I suppose that since displacement current was his main theme, then Ampère's circuital law highlights that better than Biot-Savart.

I also wanted Steve to see that magnetic charge is an ideal, and that the actual symmetry in Maxwell's equations (Heaviside versions) is only as good as that ideal can ever become a reality. In reality, there is no magnetic charge and no magnetic monopoles and so Maxwell's equations are not perfectly symmetrical. Likewise, Biot-Savart is not perfectly symmetrical to the Lorentz force and that fact should be taken note of as regards Steve's projected magnetic equivalent of the Lorentz force. George Smyth XI (talk) 17:10, 14 April 2008 (UTC)

George, if you have a problem with the equation for the Lorentz force including monopoles, write a letter to J. D. Jackson. I took the equation straight out of his textbook. It's also, in cgs, here, so you'd better write a letter to that author too, and to the journal that accepted his nonsense publication. When your version is subsequently published in a papers and widely-used textbooks, then of course we can change the magnetic monopole page. In the meantime, I'm going to continue to assume that Jackson is right, and therefore, that you're wrong. When you start getting your papers published about the Biot-Savart law, Coulomb gauge, monopole force law, and any other topics where your understanding is flatly contradicted by the entire physics community, then perhaps I'll read what you have to say more carefully. In the meantime, you're at the start of either the most revolutionary physics career since Einstein, or sore disappointment when you eventually come to understand that maybe you're not always wiser than the accumulated knowledge of thousands of physicists over many decades. --Steve (talk) 17:31, 14 April 2008 (UTC)

Steve, I wasn't planning on changing anything on the monopoles page. Those are highly speculative topics. I was merely pointing out to you for your own interest, where the breakdown in the symmetry lies. You seem to be interested in that topic.

What's ultimately important is that you realize that Biot-Savart is to Ampère's circuital law what the Lorentz force is to Faraday's law.George Smyth XI (talk) 04:47, 15 April 2008 (UTC)

George, you have some understanding of the Lorentz force that leads you to say something about its relation to Faraday's law. This is the same understanding of the Lorentz force that makes you "assured" that the (repeatedly published and widely accepted) expression for the force on monopoles is a "mistake". Therefore, it seems to me that your understanding of the Lorentz force is demonstrably at odds with the understanding of the physics community. Until you correct this discrepancy, either by convincing the physics community that it is wrong about monopole forces, or by doing some more research yourself to understand the physicists' point-of-view better, you'll find that your opinions about the Lorentz force carry little weight with me, and probably with other professional physicists that you may have occasion to talk to in your life. Same goes for your understanding of the Biot-Savart law. --Steve (talk) 16:41, 15 April 2008 (UTC)

Steve, if you take the curl of the Lorentz force, you get Faraday's law. It's as simple as that. George Smyth XI (talk) 07:10, 16 April 2008 (UTC)

Vector sign?

Mat, physics student, I could be wrong, but if there is an r hat in the equation, then doesnt the dB require a vector sign? I would change it myself, but I dunno how. —Preceding unsigned comment added by 144.173.6.74 (talk) 08:54, 6 May 2008 (UTC)

Boldface is an alternate way to denote that something is a vector. It means the same thing as a vector sign. I added a note to the equation explaining that. Does that clear things up? --Steve (talk) 20:17, 6 May 2008 (UTC)

Where is $K_{m}$ defined?

$\mathbf {B} =K_{m}{\frac {q\mathbf {v} \times \mathbf {\hat {r}} }{r^{2}}}$

Where is $K_{m}$ defined?   Thanks, Daniel.Cardenas (talk) 14:33, 2 March 2009 (UTC)

I looked at an old version of the page and found the answer. Daniel.Cardenas (talk) 14:37, 2 March 2009 (UTC)

AGL v. BSL

If Ampere's and Gauss' laws for magnetism (AGL) may be derived from the Biot-Savart law (BSL), but not the other way around, shouldn't we think of BSL as more fundamental than AGL? Why say that BSL "will always satisfy" (or not contradict) the AGL when, according to your reference text, the AGL is derivable from the BSL, and instead say that the AGL is ancillary to BSL?Toolnut (talk) 00:59, 15 June 2011 (UTC)

BSL certainly can be derived from AGL. Why do you say it can't? It's the same proof as in the article, plus maybe one or two extra sentences about the uniqueness of solutions to differential equations under certain conditions. :-)

BSL is less fundamental than AGL, because BSL is only true in magnetostatics, whereas AGL are true always. In some textbooks like Griffiths they start with magnetostatics, give BSL, and "derive" AGL from BSL. But then when extrapolating to the general case (changing currents, moving charges, light propagation, etc.) they have to come clean and say that AGL are the fundamental laws that are true in every situation, whereas BSL was just a special case, the time-independent solution to AGL for static charges/currents. :-) --Steve (talk) 01:48, 15 June 2011 (UTC)

I am not yet convinced of the derivation of AGL from BSL, let alone the other way, having only seen the unconvincing proof on Wikipedia: I will be looking at your source, soon, to see what I'm missing in the math. The part that I question is the derivation of Ampere's law; the Gauss law satisfaction is settled.

Also, nowhere in the derivation have I seen any step that was advanced only through assumption of constant current. So if Ampere's law is derivable from BSL with no such assumption of magnetostatics being made, and if in your mind Ampere's law holds true in the time-varying case, so also must BSL, when so derived. In both cases, the magnetic field would have to be derived completely at each time step, as well as its effect on a time- and spatially-varying current source. At wavelengths comparable to the length of the conductor, Ampere's law, just as BSL, would have to be modified to accomodate the time of flight (at the speed of light) of action at a distance by a source of current, as well as the coupling of both laws to the other half of Maxwell's equations that gives rise to wave propagation.

BSL is more elementary, but more precise, if one needs to be, though I only recall using Coulomb's superposition integral (similar to the BSL integral, but with action at a distance by charges), with instantaneous point charges, in my antenna design class. If it's possible to use only AGL and Farady's Law to precisely solve any electromagnetics problem with the finite-difference-time-domain method, including steady-state statics problems, then there should be a way to derive BSL from AGL that has not yet been discussed: i.e., AGL is true if, and only if, BSL is true; one implies the other.

In my mind, there are a lot of unanswered questions in electromagnetics that I'd like to try to get a handle on, as this stuff gets crammed into us as students without the time to convince ourselves of their truths.Toolnut (talk) 07:53, 15 June 2011 (UTC)

"Also, nowhere in the derivation have I seen any step that was advanced only through assumption of constant current." The key step is: "...using the fact that the divergence of J is zero (due to the assumption of magnetostatics)..." When the divergence of J is nonzero, e.g. a linear antenna, Biot-Savart law is false.

"Ampere's law, just as BSL, would have to be modified to accomodate the time of flight (at the speed of light)..." I don't think this is true. Ampere's law does not have to be modified to accomodate time of flight. Can you show me what you think the modified form is, and somewhere that you saw it? Ampere's law is one of Maxwell's equations, and Maxwell's equations predict finite speed of light and do not contain any action at a distance! This is a famous and important fact, well-known in light of the historical role of Maxwell's equations in the development of special relativity. (Biot-Savart law does, however, predict action at a distance. The way to "correct" the Biot-Savart law to the general case is to turn it into Jefimenko's equations, which hold in every situation because they are essentially equivalent to Maxwell's equations.) :-)

(By "Ampere's law" I always mean "Ampere's law with Maxwell's correction".) (I'm glad you're taking time to convince yourself of the truth! Very worthwhile! Sorry you haven't found all your answers on wikipedia. Maybe all your answers is too much to hope for--wikipedia can't do everything a textbook can do--but wikipedia surely always has room for improvement!) :-) --Steve (talk) 13:38, 15 June 2011 (UTC)

Thank you for pointing out the assumption of statics being made by setting $\nabla \cdot \mathbf {J} =0$. I wonder how much harder it would be to establish the equivalence b/w Jeffimenko's and Maxwell's equations.

I guess by "time of flight" I was referring to "retarded time," as found in Jefimenko's equations. The original Ampere's equation is modified by inclusion of displacement and induced currents, and a similar modification was applied to the BSL, the product having been renamed "Jefimenko's equation for the magnetic field." It would therefore be more beneficial to be shown the proof of the equivalence of the latter form, of which the BSL is a special case.

Finite-difference numerical methods allow us to apply Maxwell's equations in sufficiently small cubes, one cube and one sufficiently small timestep at a time, with the conditions within each cube being looked at in isolation of all except those of its common boundaries with neighbors, starting from given initial conditions. Using Jefimenko's equations, instead, allows us to avoid computing the field at unnecessary points in space. I guess the latter equations are what I was referring to as being used in antenna design, though in a simpler form for the steady-state frequency domain. Toolnut (talk) 18:38, 20 June 2011 (UTC)

I'm not familiar with the derivation of Jefimenko's eqn. Even without the derivation, I'm definitely in favor of putting Jefimenko's eqn in this article and explaining the differences between that and BSL.

What you just said about finite differences sounds correct to me. :-) --Steve (talk) 13:02, 26 June 2011 (UTC)

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