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What are the FULL classical electrodynamic equations?

  1. Jul 18, 2006 #1
    Most (or all) solvable EM problems are either (1) given a fixed static or periodically-varying EM field, find the motions of charged particles, OR (2) given a fixed static or periodically-varying charge distribution, find the resulting EM field.

    That is, either (1) solve the Lorentz force equation or (2) Maxwell's equations. But what does the complete problem (1) + (2) look like? Oh, I know, I know, its not solvable. I'd just like to see the differential equations once.

    The problem is that all my references give Maxwell's equations in terms of a continuous charge distribution ([tex]\rho[/tex] and [tex]J[/tex]) while the Lorentz force is given for point particles. If we stick to a continuous charge distribution, what I need then, I guess, is the Maxwell equations (got 'em) + the continuity equation (got it) + plus the continuous version of the Lorentz force in terms of [tex]\rho[/tex] and [tex]J[/tex] and depending on E and B (need it).

    If someone could provide that last piece, I'd appreciate it.

    Interestingly, in quantum theory this is no big deal. It falls right out of SU(1) gauge invariance, replacing the derivatives of the fields with the "covariant derivative", i.e. the derivative plus a term proportional to the EM vector potential (which really just amounts to the classical p --> p - eA). I am wondering how the classical version compares with the quantum case.

  2. jcsd
  3. Jul 19, 2006 #2

    Consider that a continuous charge distribution is made of individual charges.
    A macroscopic averaging of the "microscopic" equations is all that you need, theoretically.
    Unfortunately the result may not be so simple than it sounds.
    For more details you should read some book on plasma physics.
    Plasma physics is precisely the study of fluids of charged particles.

    This field is far from being closed. There is a lot of physics behind.

    Take for example the -well known- theory of collisions in plasma. The collisions of charged particles is a long-range interaction, which is very different from collisions of "neutral" particles. The theory involves a shielding effect by opposite charges (Debye shielding). Some hypothesis or some "philosophy" are needed to drift from microscopically reversible equations to the irreversible macroscopic equations. The end result is a collision term in the averaged equations (Fokker-Plank for example) responsible for some dissipative behaviour like resistivity, viscosity, ... . This is related to the roots of the 2nd law of thermodynamics, see the difficulty ?

    There are also many different possible approximations depending on the situation considered. For example MHD is a kind of equivalent to the Navier-Stokes equation when viscosity and dissipation do not play a significative role, quite an interresting subject (magnetic flux tube keep their topology frozen). Resistive MHD is the same when the effect of resistivity is taken into account (flux tubes can disappear or coalesce). The Vlasov approximation also neglects dissipation but accounts for details of the particle phase-space distribution, it is essential to study some field-particles resonnances, like the famous "Landau damping". See the books for more.

    Nothing similar could pop up in quantum mechanics unless -again- a macroscopic system is studied (quantum plasma physics, semiconductors, ...). At the microscopic level, the charge distribution in QM is the density probability, something totally different (an electron cannot collide with itself!).

    In summary, the classical charge density is something totally different from the QM probability density.
    It is a macroscopic view on a large quantity of individual charged particles.



    I translated your question by: "I know how it is in QM thanks to groups theory, but how is it in classical mechanics". But I did not understand your QM approach, could you explain it further?

    Reading here on PF, I am daily surprised by how abstract QM (groups theory, QFT, ...) seems to become always more popular than classical physics, not to mention thermodynamics or dynamo and thermic motors !!!

    The advantage may be that old difficulties with quantum physics may simply be forgotten, maybe forever. Because these difficulties were essentially due to teaching classical before quantum.
    Last edited: Jul 19, 2006
  4. Jul 19, 2006 #3
    You can see Maxwell's equations at


    Notice how these equations are given in terms of densities, i.e. current density and charge density.

    In tensor form Maxwell's equations are expressed as

    [tex]\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}[/tex]

    The most general form of Maxwell's equations are the Proca equations. In tensor form they are

    [tex]\partial^{\alpha}F_{\alpha\beta} + {\mu}^2A_{\alpha} = \frac{4\pi}{c}J_{\alpha}[/tex]

    where [itex]\mu[/itex] is proportional to the photon's proper mass. Setting this to zero gives the equations in that link above.
    The Lorentz force density, in tensor form, relates the 4-force to E, B, [itex]\rho[/itex] and J. They are

    [tex]f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}[/tex] = [itex](\frac{1}{c}[/itex]J*E, [itex]\rho[/itex]E + [itex]\frac{1}{c}[/itex]JxB)

    Last edited: Jul 19, 2006
  5. Jul 19, 2006 #4

    The equation that you state are well-known, except for the exotic photon proper mass. However, you still need to indicate how the Lorentz force acts on the charge densities and currents.

    This link, I think, cannot be established without explicit reference to the charged particles and their law of motion.

    For example, a current in an wire cannot be seen as a motion of "the" charge density, but instead it is due to the motion of the electronic density, while the total charge density remains zero. You need to consider (at least) ion and electron charge densities separately. In general you can have many different charged species.

    This is to stress that in classical physics (like in QM) there is no evolution equation for total charge density and current.

    Beside that, the picture is not very different in QM.
    In QM a charge looses its localisation and gets a smooth distribution.
    In CM the distribution is a Dirac, that's all. No big deal either.
    As an exercice, you might try to replace the equations of motion (on the particles coordinates) by an equation on the density (that will keep it a Dirac, of course).
    It may look very familiar. (would it be Vlasov ?)

    Last edited: Jul 19, 2006
  6. Jul 19, 2006 #5
    The EM equations which include a non-zero proper mass can be found in one of the most popular texts on EM. That text being Classical Electrodynamics, J. D. Jackson. Often here I've seen people post the lower bound on the photon's mass when the topic of "photon mass" arises in this forum. Whether it is well-known will depend on a person's experties as well of the field of physics they are versed in.
    The OP wished to know the differential equations for the force/field relations which are the density equivalent of the Lorentz force equation. This is the equation I posted.
    The explicit reference is in the charge density [itex]\rho[/itex] and the current density J. These quantities appear explicitly in the equation I posted.
    The current in a wire is a supperposition of the current due to the positive charges as well as the negative charges. If the charge density is zero in one frame then it need not be zero in another frame (and usually isn't). To obtain these densitities one needs more than the laws of electrodynamics but the law that the equations that govern nature (e.g. laws of electrodyanamics) must be expressible in invariant form, i.e. as tensor equations. To obtain the charge and current densities compared to that in another one simply applies the relavent transformation. The tensors are frame dependant whereas the charge and current densities are frame dependant. The 4-current [itex]J^{\alpha}[/i] = (c\rho, \boldface J)[/itex] is the geometric object which appears in these equations. Only after one chooses a particular observer can one define either charge density or current density. Thus once one has the 4-current expressed in one frame it can be found in all other frames of reference by a mere change in spacetime coordinates.

    For an example of where the charge density is zero in one frame but non-zero in another please see the web page I made for this at


    You will notice that while the total charge is zero the charge densities need not be zero.
    I posted the answer to this above, i.e.

    [tex]\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}[/tex]

    This may not look like an equation which gives the time evolution of charge and current density, but it is. The equation above is a set of 4 equations. One of them is found by letting [itex]\alpha[/itex] = 0. The partial derivative is then becomes [itex]\partial^{\alpha}[/i] = [itex]\partial^{\0}[/i] which is a time derivative.
    Actually it is the opposite. In classical EM equations one can deal with either particles or distributions whereas in QM one deals only with particles. In any case I am not versed in quantum electrodynamics and am thus not able to post on that theory so I'm posting only the classical equations.
    I've already done that.

    Note: There is another equation but I don't know how to post that using the symbols I know of in Latex.

    Last edited: Jul 19, 2006
  7. Jul 19, 2006 #6

    I agree with probably everything you said. But I did not understand your comment on the Maxwell's equation :

    [tex]\mathbf{d^*F} = 4\pi\mathbf{J}[/tex]​

    What's the magic I missed?
    Do you mean that the Lorentz force on particles can be derived from this equation?
    I do not reject the idea, but I would greatly appreciate further explanation or a proof if you confirm, or a reference. For sure, I always tend to identify fields and particles (I mean in electrodynamics!): fields reveal the particles and vice-versa. I find it fascinating that particles could somehow vanish from the fields equations (any comment on that?).


    Last edited: Jul 19, 2006
  8. Jul 19, 2006 #7
  9. Jul 19, 2006 #8

    Because of the way I edit here on the PF, you reacted to my post before I really finished it (maybe it is not yet finished!). My point was not about the algebra or the notations, but about the physics: do the Maxwell's equations imply the Lorentz force?


  10. Jul 19, 2006 #9


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    One generally describes matter by an electromagnetic (4-) current density [itex]J^{a}[/itex] and a stress-energy tensor [itex]T^{ab}[/itex].

    The sum of the stress-energy tensors of the matter and electromagnetic field must vanish. One can then show that
    [tex]\partial_{b} T^{ab} = - F^{ab} J_{b}[/itex]
    This describes how matter couples to an electromagnetic field. The right-hand side could be interpreted as a force density if you like.

    The evolution equation for the current is obtained by taking the divergence of the Maxwell equation [itex]\partial_{b} F^{ab}=J^{a}[/itex]. The result is just the continuity equation
    [tex]\partial_{a} J^{a} =0[/tex]

    This set of equations (plus Maxwell's) isn't fully determinate, but that's just a reflection of the fact that different types of matter move differently. You also need constitutive equations describing the type of matter under consideration to complete the problem (i.e. give an "equation of state").
  11. Jul 19, 2006 #10

    Obviously the "motion" of the charge density cannot be derived from the Maxwell's equations. This was quite obvious since the mass of the charged particles (or the mass density) don't appear anywhere.

    However there was at some old time this dream to get the mass of the electron from its field energy density. Since then things have been elaborated quite a lot ... and I don't know if the dream has been updated to cover all particles ...

  12. Jul 19, 2006 #11
    There are three equations which cover EM. Maxwell's equations, the Lorentz force

    Equation #1 - the two equations, one for the divergence of E and one for the curl of B are combined into the single covariant equation

    [tex]\partial^{\alpha}F_{\alpha\beta}= \frac{4\pi}{c}J_{\alpha}[/tex]

    Equation #2 - the two equations, one for the divergence of B and one for the curl of E are combined into the single covariant equation

    [tex]\partial_{\alpha}G^{\alpha\beta}[/tex] = 0

    where [itex]G^{\alpha\beta}[/itex] is the dual field-strength tensor defined as

    [tex]G^{\alpha\beta} = \frac{1}{2}\epsilon^{\alpha\beta\mu\nu}F_{\mu\nu}[/tex]

    Equation #3 - The Lorentz force density, the EM field tensor and the 4-current

    [tex]f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}[/tex]

    To trace the path of a single particle in the resultant EM field one employs the Lorentz force equation. These three equations give you everything that you need to know to solve any problem in EM. That includes the time evolution of source charges and source currents. As far as
    the answer is no. The Lorentz force equation must be stated as a seperate equation rom Maxwell's equation to find the force on a test particle moving in an EM field. The three equations above give you all you need to know to find the EM field and the given charge and current densities as a function of time and space. The Lorentz force will give you the force on test particle moving in this field. A test charge is defined as that charge which does not significantly alter the field in which the charge is located.

    Note: From the questions posted above it appears that one needs to solidify in one's mind what it is we are seeking. As such one needs to distinguish source charges and source currents and the test particles which move in the fields created by these sources.
    This is still the case in classical EM. One can calculate the mass of a particle by considering both the bare mass (mass that exists when the charge is zero) and the electromagnetic mass (mass due to charges field due to EM energy). This has been done succesfully in Rorlich's text on EM. This is still the case in Classical EM. Such "particles" have a finite diameter and are refered to as "particles" since one assumes a particle radius which can be neglected.

  13. Jul 19, 2006 #12
    Thanks, guys. There is a lot for me to digest above.

    Pete, I think I almost have everything I need. In your equation [tex]f^{\alpha} = \frac{1}{c}F^{\beta\lambda}J_{\lambda}[/tex], what is the left-hand-side? A force density? Relate that to time-derivatives of charge density and current density and I think that's it.

    The question of whether we use continuous charge distribution or point charges is almost a matter of taste, since on the scale at which we notice that charges are discrete classical EM fails anyway. I would choose continuous distributions since then we don't run into the infinite self-energies for the field of a point particle.

    Hmm. Maybe the Lorentz force density equation by itself is not adequate as I hoped. Since what we are talking about here are the movement of charge distributions which DO significantly alter the field.

    lalbatros, I will try put together some LaTex for the quantum version of this for the Klein-Gordon field (the easiest case) soon.

    Later edit: btw, I would be happy to have simply a reference that covers this adequately. I don't find it in Jackson's Classical Electrodynamics.
    Last edited: Jul 19, 2006
  14. Jul 20, 2006 #13

    That is not possible. This Lorentz force applies to charged particles. Generally, you need to calculate the motion of these charges first, and get currents and densities by summing on the particles. This may eventually lead to an evolution for the current density, but this constitutes then a model for the material considered. Note that you may write the requested equation for each charged species separately. But remember my previous remarks concerning dissipative effect that you need to take also into account and that result from microscopic details. I also like to mention resonant absorption of field energy by particles, the Landau damping, because I find it a wonderful subject and also because it shows you that the full distribution function of the charged particles can also play a role in the evolution of the current densities ... and the fields. This shows clearly that a complete physical picture cannot avoid the details of the density and current "composition", even not only the species but the full distribution function.

    As an example, the motion of a thin wire in a magnetic field can indeed be given by an evolution equation for current density. In this case the material model is simply that the magnetic term of the Lorentz force applies to the bulk mass of the wire. The electrons carrying the current are bond to the wire.

    An other example I know very well is the magnetized plasma. In this case the individuality of charged species, all sort of ions and electrons cannot be skipped off. The theory of electromatic waves in a magnetized plasma could illustrate that in much detail. Obviously you could get for example some Larmor resonnance on some sort of ions, this would act on the evolution of charge density and currents, in the corresponding frequency domain. There are numerous examples in this field of physics.

    You might consider a lot of other examples from physics or chemistry. Consider the hall effect for example or take electrochemistry.

    Personally, I have two references on this subject, that are very complementary:

    Landau & Lifchitz (Fields Theory) is more concise and straight to the point and faster to read, but can be sometimes harder. The "fields" cover electromagnetism as well as gravitation.

    Wheeler & al (Gravitation) is long but extremely funny and fascinating to read, specially for the first time. If you are not familiar with modern differential geometry notations, you need to spend some time to learn it, can be funny. It is more difficult to use as a reference book because it is more pedagogy-oriented and for a second reading usually you don't want pedagogy any more. Despite the title "gravitation", it covers electromagnetism too and many other topics, a must have!

    The choice depends on the time you make available for reading.

    Finally, to repeat myself, plasma physics is essentially dealing with this topic since is deals with free charged particles. My prefered reference is "Principles of Plasma Electrodynamics" that deals mainly with waves. Another very nice book is "Electrodynamics of Particles and Plasmas". As you can already see from those titles, it is related to your question!


    Last edited: Jul 20, 2006
  15. Jul 20, 2006 #14
    Yes. It is force density.
    The EM fields are time dependant and this time dependance comes from the first two EM equations I posted. The force density is a function of mass density and 4-acceleration. The 4-acceleration in term tells the charged particles (with finite mass described by the mass density and charge density) how to move. There is no requirement that I know of which demands time derivatives of sources to be given in a differential equation which describes EM. Alot of it is implicit in the equations as I've addressed above and below.
    I posted that to describe a test particle moving in a field created by the sources. It was never intended to address charge/current density sources.
    See the section of Jackson which describes Maxwell's equations in covariant form. In Jackson's third edition this section is 11.9 Invariance of Electric Charge; Covariance of Electrodymamics. You'll see how Maxwell's equations are defined. The equations of EM are defined as Maxwell's equations and the Lorentz force equation. Given this set of equations all the information to describe the EM field and the time developement of the sources becomes well defined and, along with initial conditions, has a unique solution.

    Last edited: Jul 20, 2006
  16. Jul 20, 2006 #15
    The sources must evolve according to Maxwell's equations and the Lorentz force equation. This can only be omitted when one gives sources which exist independantly of the fields in which they are in.

  17. Jul 20, 2006 #16

    I was thinking to a metal (electronic conductor) or a semiconductor or a resistive plasma far from any resonnance. Take the metal, clearly the current density (and charge) do react directly to the fields. For example by an increase of the current by an emf or even by a displacement of the conductor. Note that these two responses already correspond to different "equation of evolution" for the currents. This was the whole point of my post: current densities are not "autonomous variables", there are usually many more degrees of freedom in materials interacting with em fields and therefore much more physics. I hope I could stimulate some interrest in some of this physics and drift this discussion a little away from the initial question whose answer is simply 'no it works in another way'.

    Of course I totally agree that this all reduces ultimately to the Lorentz force and Maxwell's equation.

    Last edited: Jul 20, 2006
  18. Jul 20, 2006 #17
    I took another look at Jackson and see that section 6.1 Maxwell's Displacement Current; Maxwell Equations. You will note that with regard to Maxwell's equations Jackson states
    This appears to be the textbook statement that you're looking for.

    It is useful to know that there must be a mechanism which, in the static case, holds a charge distribution together. E.g. consider a sphere which is uniformly charged. Without something holding the charges together to retain the charge density the charge distribution would not be static. This force is known as the "Poincare stress." An interesting paper on this is

    Mass renormalization in classical electrodynamics, Griffiths and Owen, AJP, 51(12), December 1983, pp 1120-1126
    Last edited: Jul 20, 2006
  19. Jul 20, 2006 #18

    A quantum field theory example looks like the following. Herein h-bar=1 and c=1 and I am not being very careful with the other coeffecients and signs. This is just conceptual.

    The free Klein-Gordon field has a Lagrangian (density) like


    Actually, this is a Lagrangian for two fields: [tex]\phi_{1}=\phi+\phi^{\dag}[/tex] and [tex]\phi_{2}=\phi-\phi^{\dag}[/tex]. Together they have a conserved "charge" current as if the fields have opposite charge. But without the EM field "turned on", the two fields don't interact.

    The free Maxwell Lagrangian is [tex]\mathcal{L}_{EM}=-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}[/tex] where [tex]F^{\alpha\beta}=\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}[/tex].

    To get the interaction we replace [tex]i\partial_{\mu}\phi[/tex] with [tex]\left(i\partial_{\mu}-eA_{\mu}\right)\phi[/tex] (which is like the familiar p --> p - eA in classical Lagrangians) to get a full Lagrangian

    [tex]\mathcal{L}=\left(\partial_{\mu}-ieA_{\mu}\right)\phi^{\dag} \left(\partial^{\mu}+ieA^{\mu}\right)\phi-m\phi^{\dag}\phi-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}[/tex]

    The Euler-Lagrange equations in terms of two fields phi and A give the differential equations describing the evolution of the system.

    (The p --> p - eA procedure follows from a deeper consideration of requiring local gauge invariance under SU(1) transformations, btw.)

    So now let me ask a more straightforward question, or at least, a question which is more clearly defined.

    What is the Lagrangian of a classical system consisting of EM fields and charges? Not a Lagrangian just for the particles in which the potentials are a given but a Lagrangian in which the fields are allowed to vary as well as the charges.

    Pete, the trouble I have with the quote from Jackson is that the Lorentz force is for point particles while Maxwell's equations are given in terms of continuous charge distributions. There is a continuous version of the Lorentz force given in a post above but I don't know the continuum analog to Newton's second law.

    The Lagrangian method is preferable since it gives the differential equations directly without the intermediary concepts of "force" or "force density".

    The Lagrangian was my ultimate goal all along, anyway. I just wanted differential equations so that I could construct a Lagrangian that would yield them. I was curious how it would compare to the quantum Lagrangians.
    Last edited: Jul 20, 2006
  20. Jul 20, 2006 #19

    As simple as it looks in Landau-Lifchitz, it is awfull for me to write it in full with this tex stuff.
    So I translate the action from L-L to a non-usual style, and I also drop all factors:

    S = -path(mc ds) - 4space(A.j dW) - 4space(F.F dW)​
    path() is an integral over the path of the particle(s)
    4space() is a volume integral over 4-dimentional space
    ds is a particle path element
    dW is a 4-space volume element​

    I think you can introduce the current density(ies) and convert the first term to a 4-space integral.
    I don't have the result right now.
    Note that in Landau-Lifchitz the second integral was already obtained by converting a path integral to a volume integral by introducing the current(s). The initial form of the second term was:


    (where dx is the path vector in the 4-space)​

    Finally, I guess a 1-1 correspondance with the Lagragian you gave.
    Note the second term in your Lagrangian and how it corresponds to the first term here, given the quantum charge density.

    Please give us your analysis.

    Last edited: Jul 20, 2006
  21. Jul 20, 2006 #20
    By Lorentz force Jackson is refering to the force exerted on a charged particle by the EM field. Thiss can be adapted to the case of a continuous charge distribution by a limiting process, the result of which would yield the Lorentz force density.
    One can express the Lorentz equation in terms of derivitives as well. One simply replaces the force density by the mass density times the acceleration of the matter which constitutes the charge distribution. By Lagrangian method I assume you're refering to the Lagrangian density function, is that correct? In any case the equations turn out to be identical to each other in the long run. In any case the force is in Lagranges equation anyway. I don't see how you conclude that the Lagrangian method is preferable because the force is not explicitly in Lagrange's equations. They are implicitly in those equations.

    Last edited: Jul 20, 2006
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