Talk:Maxwell's equations/Archive 1

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Personally, I like Gauss' law better if the ε₀ is moved over to the other side, under the integral. That way, it still works if the permitivity isn't constant; you just replace ε₀ with a function ε --AxelBoldt

What do you mean if ε_o isn't constant? Do you mean for cases where there is matter between the surface and the charge, and thus you need to account for a drop in the E field due to the permittivity of that matter? --BlackGriffen

Yes, that's what I meant. If there's matter with varying permittivity ε around, you need to integrate ε multiplied with E in Gauss's law. --AxelBoldt

Ok, I'm adding a note about it now.

Upon further reflection, that is wrong. The dielectric constant of matter doesn't just magically reduce the electric field. The dielectric constant (I had the wrong name previously) is a measure of how easy it is to separate the molecules of matter in to a dipole. To show why is relatively easy (now that I think about it properly). Consider a small positively charged sphere. The electric field outside this sphere is is the same is if it was a point charge: kq/r². Stick a neutrally charged spherical shell around it. The electric field of the sphere creates dipoles within the shell that surrounds it. The net effect is like two thin shells of charge have formed; a negative one on the inner surface of the shell and a positive one on the outer surface. The charges of these shells have to be precisely equal due to conservation of charge. The net effect? Everywhere but on the inside of the walls of the neutral shell, the electric field still looks like kq/r². Within the walls of the shell the electric field is weaker, but as long as the surface entering that region removes no net charge, the decrease in the electric field is compensated for by two factors: a change in the area of integration, the fact that the charge shells are approximations of microscopic dipoles means that there is still a net surface charge that compensates for the inner charge. Even in the limiting case, metals, where the surface charges are great enough to reduce the electric field in the body of the metal to zero, Gauss's law holds. --BlackGriffen

Is it worth mentioning that the elegant formulations of Maxwell's equations were not developed by Maxwell, but by another man ? Maxwell had the right idea, but he was definitely not elegant in his math.... I've done a bit of web-searching to validate this idea and currently cannot vouchsafe it. I recall a history of science teacher describing it in great detail when I was younger, but have no way to verify/validate it.

Okay, I found a page that claims: 1884: Oliver Heaviside expresses Maxwell's Equations as we know them today ie: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heaviside.html therefore this validates my recollection. (it said Maxwell's equations were originally 20 equations in 20 variables instead of two equations in two variables) Now I can go to sleep...

I deliberately left the history empty because I did not know it. By all means, add a history section. I do know that the wave equations for light that can be derived from them led to relativity. (the term describing the velocity of the wave was 1/(μ_oε_o)^.5 which is equal to c, and the fact that it didn't contain a term for the velocity of the observer is what sparked Einstein's imagination/lead to his postulate that c is a constant to any observer.

Also, 4 Maxwell's equations with 4 variables (time, charge density, the electric field, and the magnetic field). Where do you get two? --BlackGriffen

I think it would be nice to mention in the main article how ε_o, μ_o and c are related, so that people realize that the speed of light occurs in Maxwell's equations and that therefore the conjecture that electromagnetic radiation is light is not too abstruse. --unknown

Nice idea, but this article needs to remain focused on these equations because that is all it is for. A better place for that connection would be in an electromagnetic radiation/waves/light article. --BlackGriffen

Oh, there's also a minor oversight: ε is used as the permittivity and also as the electromotive force around a loop. --unknown

EMF is supposed to be a scripty E. Anyone know how to do one of those? --unknown

I understand that, but there are only so many symbols in the english language. I used ε instead of ΔV or Δφ_E for three reasons: first, φ and/or V are used in electrostatics to represent the electric potential as a scalar function in space, and any closed loop integral over a continuous scalar function in space has to be zero; second, ε is the closest thing (almost exactly the same, in fact, to the scripty thing described above); and third, the limited number of symbols means that what the symbol represents has to be labeled each time anyway. To give you another couple of overloaded characters in physics: p represents both momentum and pressure (in mathematics p also represents the period of the wave); v is used for velocity, volume, and voltage, velocity is generally lower case, volume is upper, and voltage is usually upper if it's constant and lower if it's time varying. I've really beaten that horse to death, but I wanted to make it crystal clear that I had considered the conflict when I wrote the article. --BlackGriffen

And one last thing: I don't quite understand why the last paragraph mentions cgs versus mks units? How could the units possible change the equations? --AxelBoldt

If you use kg for mass, m/s² for acceleration, and lbs for force, Newton's second law takes on the form F=kma, k a constant. Choosing a better system makes k go away, simplifying the equation. It's the same deal with CGS and MKS, a lot of the constants go away in the former system. --Unknown

Precisely, I'll add more to the main page presently, but it's all about clairity. --BlackGriffen

I would love it if someone more knowledgeable than I would add concrete examples to each equation description, to make the descriptions more accessible to non-physicists. Such an example might perhaps be: as you move away from a sphere charged with static electricity, the charge density in space drops by four for every doubling in the distance from the center of the sphere (just as gravity drops when you move away from the Earth, due to the equidistant spreading of lines of force from a sphere). [Please, please forgive me for the mistakes in this--I just wanted to illustrate what I meant by a 'concrete example'.] David 16:31 Sep 17, 2002 (UTC)

On my computer (Windows 98/IE/Arial), the symbolic manipulations in the subject page all show a character that looks like a vertically-oriented rectangle. Here is an example: ∇×E = - 1/c ∂B/∂t. The rectangles are before the E, B, and t. The browser appears to have translated them to Unicode, which is not supported by Windows 95 and 98.

I have created and uploaded an image for the Del symbol and have edited this page to reflect the change. Here is an example: $\nabla$ E = 0. David 16:27 Oct 7, 2002 (UTC)

    I use lynx to browse, so I would prefer the word epsilon to a picture of a squiggly e.  As long as there is text saying what each variable stands for, using the plain letter e for epsilon is clear as well.
         --- Urushiol

The math formulae had had great big \bullets added to them: I have removed them, and cleaned up the layout. The Anome 17:59 10 Jun 2003 (UTC)

Maxwell's Ether exists!

The Ether's impedance z and Planck's Constant h are related, making them both Quantum Constants. z= m/q and h=mq where m is the ether magnetic charge in webers(volt seconds) and q is the ether electrical charge in coulombs. Knowing the value of h and z , m=500 atto webers and q = 1.326 atto coulombs or 8.28 electrons.

The three constants, c, z and h unify Quantum, Relativity and Electric Theory. Wardell Linday

Maxwell's Equations Derived and Revised!

This is an excellent article on Maxwell's Equations. However the entire discussion of Maxwell's Equations and electricity and Magnetism would be much simpler and more correct using quaternions.

The complete and correct Equations of Electricity and Magnetism is given by the Homeostasis Condition: 0=XE

where E = Es + IEx + JEy + KEz = Es + Ev is a quaternion electric field and

where X = d/cdt + Id/dx + Jd/dy + Kd/dz = d/cdt + DEL

is my Quaternion Change operator, a quaternion extension of Hamilton's DEL.

"c' is the speed of light and E is related to c and "z" the free space impedance by E = cB = zH = zcD.

X and E are quaternions and follow quaternion multiplication. "Maxwell's " Equations completely are given by:

0 = XE = (dEs/cdt - DEL.Ev) + (dEv/cdt + DEL Es + DELxEv)

substituing E/c=B gives the traditional terms.

0 = XE = (dBs/dt - DEL.Ev) + (dBv/dt + DEL Es + DELxEv)

The observation here is that the first term is scalar of the quaternion and the second term is the vector of the quaternion.

0= XE requires both the scalar term and the vector term to be zero, thus

0 = (dBs/dt - DEL.Ev) and 0 =(dBv/dt + DEL Es + DELxEv)

If I had started with 0=XB = (dEs/cdt - DEL.Ev) + (dEv/cdt + DEL Es + DELxEv) then

0 = (dBs/cdt - DEL.Bv) and 0= (dBv/dt + DEL Es + DELxEv)

Thus one quaternion equation gives Maxwell's four and corrects them.

Notice that dBs/cdt = DEL.Bv, or the divergence or growth of the magnetic field, is not zero, but zdDs/cdt = z rho or z times the charge density rho!

I recommend revising this article to reflect this view.

Wardell Lindsay

Hello Wardell: have you considered using four-vectors? They neatly wrap up Maxwell's equations in a way that is very similar to what you propose. -- The Anome 17:57 2 Jul 2003 (UTC)

The Anome,

Yes I do use four vectors see my webpage:

http://www.geocities.com/wardelllindsay/unification.html

My Interval is a natural fallout of quaternions without introducing "imaginary time". The difference is in the mathematics. Only quaternions provide an associative (a(bc) = (ab)c) division algebra ( ax=b is solvable).

I have not seen a derivation of Maxwell's Equations similar to mine, which poses the stationary condition of the electric field and "one" equation.

A similar equation also describes Quantum Theory, using the "Life" variable L=hc.

Thanks for your comment and interest.

Lindsy

Please read: http://www.innerx.net/personal/tsmith/QOphys.html Maxwell's Quaternions were thrown away from Electromagnetism by Josiah Willard Gibbs at Yale and Oliver Heaviside in England.

and this page is informative http://www.ott.doe.gov/electromagnetic/history.shtml [notice it's a .gov site]

[That depends on one's definition of "informative:" This "history" page is written by the crackpot disciples of the infamous and equally crackpot Lt. Col. Thomas Bearden (USAF, Ret.), who believes in everything from "overunity" free-energy machines, to "scalar-wave" cancer cures, to "Tesla Death Rays," to UFOs. Hence, the fact that this is a ".gov" site simply proves that some UNBELIEVABLY moronic and insane people can manage to get into the civil service, from whence they become nearly impossible to fire...]

Electromagnetic History

more later ... reddi 03:47 7 Jul 2003 (UTC)

First of all, I have no dispute with the historical fact that Gibbs and Heaviside, along with the rest of practicing physicists and engineers, abandoned the quaternion notation. However, you seem to be under the misapprehension that by doing so, they "threw away" something profound, or that Gibbs didn't "understand" it. Quaternions, as proposed by Maxwell (some years after his initial work), were only notation, they contained no special physical content and are mathematically equivalent to the modern formulation. As notation, even Maxwell himself found them inconvenient: in his two-volume treatise on electromagnetism, he devotes all of five pages or so to quaternion notation; he lauds it as a promising notation, but admits that he finds it inconvenient for practical calculations and doesn't use it in the rest of the books. (I just checked this evening.) As a completely separate issue (independent of quaternions), Maxwell used the vector potential explicitly and picked a particular gauge choice (the Coulomb gauge, I think it was); this sort of thing can be (and is) done all of the time in the ordinary vector notation as well, and in "ordinary" physics this vector-potential gauge is not observable. As to the papers on the web site that you mention, you'll notice the telling fact that they're not published in respected peer-reviewed journals (e.g. Phys. Rev.). -- [User:Stevenj|Steven G. Johnson]

Howdy .... if you read in this link .... http://216.239.53.104/search?q=cache:Yc5OJrDDTX8J:www.nku.edu/~curtin/crowe_oresme.do ... or ....http://www.nku.edu/~curtin/crowe_oresme.doc ... it's titled 'A History of Vector Analysis' ... I kinda AM under the "misapprehension" that by doing so, they did "throw away" something profound. As fars as i can tell Gibbs didn't "understand" it. I saw that there were two important functions (or products) called the vector part & the scalar part of the product, but that the union of the two to form what was called the (whole) product did not advance the theory as an instrument of geom. investigation. (gibb's words) .... Heaviside didn't either .... I had the same difficulties as the deceased youth, but by *skipping* them, was able to see that quaternions could be explored consistently in vectorial form. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. ... So I dropped out the quaternions altogether, and kept to pure scalars and vectors.... [heaviside's words] .... Quaternions is one of the simpliest ways to describe space-time. x * y * z * t = 4D ... simple .... maxwell did want to unify EM with it eventually (as you said) ... and i like Einstien when he say, "I like to make things simple, but not one bit simpler" ... (it's also describe this reality ... not a 3D or 2D imaginary world) ... moving on ..... Quaternions are just notation .... no special physical content .... I do doubt that though they are equilivant (but i am not a math guy so until such time i can find any substantial proof ... i'll take your word for it) to modern notation ... though i thoughT, IIRC, that the modern notation cannot handle the _non-linear_ electromagnetic phenonomen that quaternions are naturally suited for (such as scalar waves) ... reguarding "quaternions are burdensome" ...no pain no gain 'eh? =-] ... Finally ... Do all links and book citations on wiki need to be published in respected peer-reviewed journals? From the limited knowledge i have about math, it looks correct to me, no worse than Sweetser quaternions .... mabey we can let the reader decide? more later (bedtime for bonzo here soon) be safe .... reddi 05:42 7 Jul 2003 (UTC)

- ("Sweetser quaternions" probably refers to this which is/was linked at quaternions -- fmr Kwantus)

Your quote from Gibbs just shows that he didn't like the notation, and that he felt it didn't add anything: writing vector fields as quaternions with no scalar part and using the quaternion product with del to get curl and divergence (as Maxwell did) neither simplifies the algebra (arguably) nor exposes any new symmetries, mainly because Maxwell was always forced to treat the scalar and vector parts separately (in which case, why combine them at all?). (The real use of four-component objects comes from 4-vectors and the relativistic inner products of the Lorentz group, which came later and is quite distinct from quaternion multiplication, as well as being different from the component grouping used by Maxwell) Similarly, Heaviside found them inconvenient; there's no reason to think that they failed to "understand" them. Vector notation can handle nonlinear media just fine, by the way (see e.g. Agarwal & Cooley's nonlinear optics book). Regarding citations, I would go further: science articles should stick to generally accepted results (i.e. peer-reviewed stuff that has stood the test of time, with special skepticism when it comes to speculative work on new physical law that is not yet tested). The problem with "letting the reader decide" is that the reader does not have the evidence to do so in a short summary, nor can the casual reader distinguish between the consistent/well-supported/widely-accepted and the crackpot. (And if you don't have sufficient maths to understand something, you should be wary about writing on it yourself.) Steven G. Johnson

[quote from Gibbs] didn't like the notation? Why didn't he like it? hmmm ... probably because he didn't understand them ... you disliek what you don't understand ... basic human nature ... too bad he doesn't see how it add things ... writing vector fields as quaternions as Maxwell did, was NOT the ultimate intention for these equation .... it does simplify the algebra [it mirrors the 4D world we live in, as Hamilton realized) .... Maxwell saw the promise of treating the scalar and vector parts together to refelct nature more elegantly (that's why combine them) ... [snip vectors diversion] .... Similarly, Heaviside didn't understand them either [didn't see the elegance]; if they did, what other reason can there be then? mabey it's a reverse hanlon's razor ... now as to the ability of vectors .... Vector notation can handle nonlinear media ? wha'? yea .. that's why they figured out the geomagnetic nonlinear phenomenon [among others] and [begin] they have that GUF already [/end sarcasm] .... Now over citations, science articles should stick to generally accepted results? Wha'? ... why is progress made in science? ... becasue information conventional AND unconventional is given to ppl [thankfully ppl were able to hear about a college graduate with this totally unconventional idea that the earth surface was on plates and they shifted around ... then it caught the attention of other scientists that realized that yea ... the earth had techtonic plates!] ... (now ... peer-reviewed stuff is good ... but the free flow of ideas is VITALLY important .... isn't that the part of this encyclopedia concept? GPL et al.?] ... skepticism comes with BOTH speculative works and conventional works .... physical laws are only a law till they are broken (or does Newton still trump einstien?)) ... I think i see what the real problem is ... "letting the reader decide" .... we wikipedians are giveing them the evidence, are you proposing that only one side of the evidence is presented? [doesn't seem NPOV to me] ... and it can be done in a short summary [that's a strawman arguement that it can't] ...as to can the casual reader distinguish between "consistent" "well-supported" "widely-accepted" or the "anomolous" "narrowly-accepted" "fringe"? May NEVER know if they dont's see BOTH sides .... (AND i do comprehend some math, I'm just not a mathematician and understand everything. "Why should I refuse a good dinner simply because I don't understand the digestive processes involved" - Heaviside =-) more later ... reddi 00:17 8 Jul 2003 (UTC)

this is a page with an interesting spin to say the least: http://www.cheniere.org/books/analysis/history.htm My degree is in maths with a good dose of physics, and I know enough to attest quaternions are essentially despised in the west; they were used only to provide illustrations for algebra theory, and their use by Maxwell was never mentioned. (Until I happened on that page, and then checked here, my hedgykashun left me the impression quaternions had never been applied to anything.) I know enough of quaternions to know they have subtle differences from vectors, and I know enogh of about how physics is done to know it's quite possible something was simplified away Heaviside's conversion to vectors. Remember that we habitually bash Newton's definition into F=ma even though he originally said F=dp/dt - and we got away with it until relativity gave us time-dependent mass. Remember that physicists habitually throw away the interestig parts of their equations until they cane be solved and then pretend that result answers their original question - that's exactly how what's studied under the chaos rubrik got ignored for so long. I'd have to see a step-by-step review of Heaviside's logic before I'll decide whether his version is exact or simplified. -- user formerly known as Kwantus (PS It also seems odd to read things like "superstring theory is free from quantum anomalies if the spacetime dimension is 10 and the quantum gauge symmetry is SO(32) or E8×E8" or "string theory in a background of 5-dimensional anti-de Sitter space times a 5-sphere obeys a duality relationship with superconformal field theory in 4 spacetime dimensions"[1] but encounter claims there's no physical significance to algebraic structures (somewhere up above)...)

This PDF appears to list the original 20 equations in reals, and reductions to 6 vector+2 real. (1.6) is Ohm's law, (1.4) the Faraday force, and (1.8) the continuity equation, leaving five. Comparing with the General Case (GC) on the main page, (1.1, 1.3) are combined into GC4, and (1.7) matches GC1 except for sign (!). That leaves (1.2) and (1.5) which don't obviously match GC2 and GC3, esp since a variable appears in (1.2) which does not appear in GC. (There may be some way of stirring up Maxwell's eight to get Heaviside's four without loss; I'd just like to see what it is.)

Reddi, Thanks for the references. I had previously read smith.

Here is my take on quaternions the math and quaternions the physics. Math: I think Hamilton was a better mathematician than his contemporaries. But that doen's make his math the end all. I think in the late 1800s, the idea of a fourth dimension was novel and considered useless in a three dimensional world. Thus the useless scalar dimension was jetisoned and the three vectors found work. Maxwell and others were upset over Hamilton's Rules (II= minus 1). Gibbs and others "fixed" this and made II= +1, and voila we have vector Algebra. In a way, the physicists created a "mathematics" that has serious defects. Associativity (AB.C =A.BC) and Closure ( II is not a member of the set of vectors)is lacking. In a physics sense Maxwell complained that when he thought he was computing a maximum, he got a minimum. For example when a rock is displaced in the direction of gravity the sign is negative in quaternions. This sign told Maxwell that this was exergy (outenergy), when he was expecting enegy. Energy is when you displace the rock against gravity!

My point is that mathematics is a very useful tool and physicists need to understand the mathematics they use and should not select defective mathematics. In a sense quaternions represent the only Associative Division Algebra. This means that if physicists want to solve AX=B, the only algebra competent to do this is quaternions or systems isomorphic to quaternions! (Real algebra and complex algebra being sub-algebras of quaternions.)

Tony Smith and others (John Conway and Derek Smith: On Quaternions and Octonions) have shown quaternions to be isomorphic (do the same thing) to the Group Theory view of physics.

PHYSICS: Planck's and Einstein's Quantum Equations and "Theory" are also seen to be derivable from the same quaternion equation as Maxwell's Equation. The variable I call Life L = Ls + Lv, where Ls is the scalar and Lv the vector of Life. I believe Life is the most important variable in the universe and came into being when God said "Let there be Light", and there was Life. Life is related to action by the speed of light, L=ch:

Work = XL = (dLs/cdt - DEL.Lv) + (dLv/cdt + DEL Ls + DELxLv)

work = XL = (dhs/dt - DEL.Lv) + (dLv/cdt + DEL Ls + DELxLv)

Planck and Einstein only considered the scalar equation "(dhs/dt - DEL.Lv)" and physicists have not "discovered" the vector equation.

Planck's Law is the Boundary/conservation condition 0 =XL. Einstein, deals with the internal/non-boundary condition, kinetic Energy = (dhs/dt - DEL.Lv).

The Boundary Condition vector equation is the widely discussed but seldom shown "action reaction equation "0=(dLv/cdt + DEL Ls + DELxLv)".

These I believe are physical facts that have not been "discovered". For example I have not seen the "work function, phi" in Quantum theory described as the Divergence of a vector function with units energy-distance, Lv. I have not seen discovered the vector equation of dependency of the vector radiation the gradient of a scalar and the curl of the Lv. Or to put it in conventional action h terms,

0 = dhv/dt + (DEL Ls) + c(DELxLv) = (dBv/dt + DEL Es + DELxEv)

It may be that this is not true for actio h, but the same equation is true in electromagnetism for the the E field! Maybe the experimentalist should look at this relationship.

Reddi and Steven thanks for your points. I appreciate the critical thinking. Physics is alive abd Life is beautiful.

--- reddi I have these ideas laid out fuller in : http://www.geocities.com/wardelllindsay/unification.html

There is not a lot of text but the math and ideas are there.

It is my understanding that Theodore Kaluza unified electromagnetism with Albert Einstein's theory of general relativity. Apparently, Einstein did not like Kaluza's assumption that the universe is invariant in the 5th dimension he invoked. Therefore, Einstein tried to redo the theory with curled up dimensions. But, he never succeeded. I refer to Chapter 18 of the _Introduction to the Theory of Relativity_ by Peter Gabriel Bergmann.

Joseph D. Rudmin

I saw a an old version of this article at http://www.rare-earth-magnets.com/magnet_university/maxwells_equations.htm I'm just curious, did he copy Wikipedia, or did we copy them? There's is copyright GPL, so I assume they got it from here. Are they obliged to provide a link to Wikipedia or to mention that they got it from Wikipedia? Actually, I just read Wikipedia:Copyrights and it says that Wikipedia must be referenced. What can we do about this minor breach? dave 19:43 16 Jul 2003 (UTC)

One of us should write a curteous request to the owner of the webpage that credit be given, and perhaps a link to this page be provided, since it is occasionally edited. To avoid multiple requests and for consistency, one of the managers of this site should have a designated task of reconciling copyright violations. I personally would look very unfavorably on any punitive action on the part of wikipedia. The wikipedia page http://www.wikipedia.org/wiki/Wikipedia:Copyrights discusses what to do in case of copyright infringement. Your notice is sufficient, I think.Rudminjd 15:15 20 Jul 2003 (UTC) Joseph D. Rudmin

It has been taken care of, and there is such a page: Wikipedia:Sites that use Wikipedia for content. I have sent out a standard letter, which asks for them to provide a link to the original article, and a link to GFDL. I'm still waiting for a response. I'll wait a few weeks and then try contacting them again, perhaps by phone. dave 15:52 20 Jul 2003 (UTC)

I permitted myself to edit "The Source of the Magnetic Field" section. I replaced the vector for the electric displacement field (D) with that of the electric field (E), for 2nd and 3rd equation of this section. Anyone feeling confident enough please counter check. Jerome Peeters (08.08.2003)

Both the previous revision and your correction were incorrect. D should appear in the law, or \epsilon E, but not \epsilon_0 E (except in vacuum) and not not \epsilon_0 D. —Steven G. Johnson

This article needs to clarify the relationship between the microscopic Maxwell's equations (in terms of E and B) and the macroscopic Maxwell's equations (in terms of D and H, which involve macroscopically averaged quantities like the dielectric constant of a material). I find the whole discussion to be currently slightly confused (e.g. I just noticed that the discussion of Gauss's law contained several errors before I fixed it just now) and in need of a much more careful rewriting. Sigh, not that I have the time to do it myself right now. —Steven G. Johnson

Do we need to jump all the way to differential geometry and differential forms to do relativistic Maxwell's eqns? Surely a gentle introduction with 4-vectors first would help. See this treatment -- The Anome 19:20, 10 Aug 2003 (UTC)

I think there was an error in saying that div(mu*B)=0 since the mu is in the wrong place. I allowed myself to remove the mu from the "linear media" equations.

The first tensor equation covers only Conservation of Charge, Coulomb's Law and Ampere's Law. My reference for the 2nd tensor equation (which expresses Faraday's Law and No Magnetic Monopoles) in the tensor version of Maxwell's equations is: Charles F. Stevens 1995, Six Core Theories of Modern Physics p.199, MIT Press ISBN 0-262-69188-4. 169.207.115.28 03:28, 31 May 2004 (UTC)

It's all well-covered in Jackson, which is already referenced. Strictly speaking, the first equation does does not directly express conservation of charge, which is the equation

\partial _{\beta }J^{\beta }=0

, although this can be derived from the first equation by taking the 4-gradient

\partial _{\beta }

of both sides. (I'm not sure if all of the sign conventions are consistent with those in four-vector, by the way, since Jackson uses the opposite sign for g.)