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Archive 1 |
Listed on Wikipedia:Votes for deletion Feb 20 to Feb 26 2004, redirected. Discussion:
-- 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)
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?
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?
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 ;)
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)
...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)
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)
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)
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))
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))
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)
A constant begins to appear half way down page without any definition nor word of explanation. âPreceding unsigned comment added by 138.40.94.135 ( talk) 18:14, 15 January 2008 (UTC)
Take the curl of,
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 Ďv 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)
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)
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)
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)
George, here's the equation, straight out of a textbook:
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)
Well, pick an r, any r. Here's the vector field in question:
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)
Hmm. I was talking about the version of the Biot-Savart law:
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)
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)
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)
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:
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)
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)
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)
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)
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)
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)
How is the sum of many vector fields not a vector field? Headbomb ( talk) 12:17, 17 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 rl (Steve used r'), then take the curl with respect to r, .
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)
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)
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-rl)".
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)
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)
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 , an exemple of a 3 dimensional vector field would be a vector such as . N-dimension vector fields are fields of the form
or more succinctly, .
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
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:
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 rl carrying current I.
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 rl coordinates):
For a solenoid with N turns, this gives the familiar expression
The divergence of a constant is 0 (which is peachy because the divergence of any B field is 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 ).
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 (cylindrical coordinates). The divergence of this is zero, which is peachy. The curl of this is zero everywhere, except at Ď=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 Ď, and it'll be zero there too. Headbomb ( talk ¡ contribs) 20:03, 23 April 2008 (UTC)
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)
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,
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)
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)
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)
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)
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. âPreceding unsigned comment added by 144.173.6.74 ( talk) 08:54, 6 May 2008 (UTC)
Where is defined? Â Thanks, Daniel.Cardenas ( talk) 14:33, 2 March 2009 (UTC)
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)
This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Listed on Wikipedia:Votes for deletion Feb 20 to Feb 26 2004, redirected. Discussion:
-- 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)
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?
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?
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 ;)
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)
...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)
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)
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)
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))
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))
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)
A constant begins to appear half way down page without any definition nor word of explanation. âPreceding unsigned comment added by 138.40.94.135 ( talk) 18:14, 15 January 2008 (UTC)
Take the curl of,
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 Ďv 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)
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)
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)
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)
George, here's the equation, straight out of a textbook:
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)
Well, pick an r, any r. Here's the vector field in question:
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)
Hmm. I was talking about the version of the Biot-Savart law:
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)
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)
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)
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:
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)
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)
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)
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)
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)
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)
How is the sum of many vector fields not a vector field? Headbomb ( talk) 12:17, 17 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 rl (Steve used r'), then take the curl with respect to r, .
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)
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)
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-rl)".
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)
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)
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 , an exemple of a 3 dimensional vector field would be a vector such as . N-dimension vector fields are fields of the form
or more succinctly, .
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
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:
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 rl carrying current I.
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 rl coordinates):
For a solenoid with N turns, this gives the familiar expression
The divergence of a constant is 0 (which is peachy because the divergence of any B field is 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 ).
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 (cylindrical coordinates). The divergence of this is zero, which is peachy. The curl of this is zero everywhere, except at Ď=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 Ď, and it'll be zero there too. Headbomb ( talk ¡ contribs) 20:03, 23 April 2008 (UTC)
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)
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,
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)
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)
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)
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)
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. âPreceding unsigned comment added by 144.173.6.74 ( talk) 08:54, 6 May 2008 (UTC)
Where is defined? Â Thanks, Daniel.Cardenas ( talk) 14:33, 2 March 2009 (UTC)
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)
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