Built on Facts

Maxwell’s Equations #4

Fundamentally Maxwell’s equations describe the origins of electric and magnetic fields. Given a set of conditions on the right hand side of the equations, you’ll have fields described by the left hand side. Between the four equations the fields are uniquely specified, and there is nothing more to include. Thus far we’ve seen electric fields generated by electric charge, electric fields generated by changing magnetic fields, and magnetic fields not generated by magnetic charge because there isn’t any. Conspicuously absent is any way to generate magnetic fields. It is this fourth equation that closes the circle and describes the origin of the magnetic field.

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

The left hand side is the curl of the magnetic field, precisely analogous to yesterday’s curl of the electric field. The right hand side contains two terms, for there are two ways to generate magnetic field circulation. The first is via an electric current, with current density given by J. The electromagnets that heft huge piles of metal scrap at junkyards work on this principle. Run current through a wire, and the wire will develop a magnetic field surrounding it. Via Wikipedia:

i-049468aca4e09027b976e8a26897c2ca-wire.png

The second term on the right is of the same type as in the equation for the curl of the electric field. In this equation, a changing electric field produces a magnetic field of specified curl. This is entirely without reference to any charge. While the J term creates a magnetic field in the presence of flowing charge, it’s not required. So long as there’s a changing electric field, the derivative on the right will be nonzero and you have yourself a magnetic field.

Historically the equation with only the first term on the right is called Ampere’s Law. This limited equation turned out to fail in explaining a large number of physical phenomena, such as the magnetic field between plates of a charging capacitor. Between the plates there is no current, but there is a field. It was Maxwell who first correctly accounted for this, wrote the complete equation, and worked out the consequences of the four combined equations that now bear his name.

The four equations are almost symmetric in their form, with the symmetry broken only by the absense of magnetic charge. If it existed a magnetic charge density in the magnetic divergence equation and a magnetic current density in the electric curl equation would complete the symmetry. This sat as a somewhat aesthetically unsatisfactory condition for many years, despite the fantastic explanatory power of the theory of electromagnetism. It took another brilliant physicist to (mostly) put this enduring question to bed. Though the classical mechanics of Newton required relativistic correction, Einstein showed that Maxwell’s theory was already consistent with the theory of special relativity. Electricity and magnetism turn out to be literally the same phenomenon in different reference frames. Rewritten in a relativistically natural way (the term of art is covariant), the broken symmetry vanishes and the slight lopsidedness of the equations as we’ve seen them vanish into the ether. Well, as it were.

All the countless thousands of pages of classical electromagnetism textbooks are lagniappe following from these four equations. The consequences of those equations are numerous and profound, and of course those textbooks are indispensable as no one is clever enough to find all of their subtleties for themselves. In fact, we have four equations and five days in the working week. It would be a shame to waste the last. Tomorrow we’ll look at the most profound of the implications of these equations; the one Maxwell is most famous for discovering, the one we see every time we see.

Comments

  1. #1 Uncle Al
    August 13, 2009

    Between the four equations… “Among the four equations…” “Between” is a binary comparison.

    Pyotr Ufimtsev, Metod Kraevykh Voln v Fizicheskol Teorii Difraktsii, (Moscow: Sovetskoe Radio), 1962. Ufimtsev eventually emigrated to the US where he became faculty at Caltech. On 17 January 1991 he was astounded to discover – on TV! – what Denys Overholser had done with his 242 page essentially unreadable treatise on an abstract application of Maxwell’s equations.

  2. #2 Jim_K
    August 13, 2009

    I am just now getting to understand Maxwell’ Equations. Aren’t epsilon0 and mu0 constants whose product is 1/c*c?
    (I hope you understand what I meant, I don’t have much symbology on my iPod.)

  3. #3 Nathan Myers
    August 13, 2009

    The most profound that we know of. There are bound to be more interesting implications hiding in there than nature has got around to exploiting!

    Uncle Al: No, “between” is right, here. The rule you are using is insufficiently subtle.

  4. #4 Paul Murray
    August 14, 2009

    I only vaguely get this. Sure, a magnetic field is generated by current, but there’s no current outside the wire. Why does the magnetic field extend beyond it?

    I expect it’s because the only way to solve the four equations simultaneously requires it. Is that right?

  5. #5 rob
    August 14, 2009

    Paul Murray: the magnetic field extending out from the wire where there is no current density isn’t really strange. other fields behave the same way. earth’s gravitational field exists outside of it’s radius. a charge distribution has an electric field that exists outside the volume of the distribution. etc.

  6. #6 CCPhysicist
    August 14, 2009

    The four equations are almost symmetric in their form, with the symmetry broken only by the absense of magnetic charge.

    Or, you could say the symmetry is broken by the PRESENCE of electric charge, which has the virtue of enabling an easy transition to the empty space solution that comes next. Oh, and you might not misspell presence….

    The funny thing about reading a blog is that you notice the possible flaw in things you have been saying a particular way for years, mainly because that is the way it was first said to you. I’ll have to try saying it a different way this semester.

    BTW, the symmetry is also broken by the minus sign, but that turns out to be a really important detail that leads to a traveling wave solution.

    @2: The annoying asymmetry of those constants will go away if you choose the “correct” units. Those factors simply fix a mismatch that results from a particular choice for things like volts and tesla. Ah, if they only made a statvolt meter. Or not.

    @4: The same reason an electric field extends beyond a charge? Both are action at a distance. Also, once you learn how relativity makes this happen (the easiest place to see this is in the Berkeley physics volume 2 text), the two are not all that different.

  7. #7 hosting
    August 16, 2009

    other fields behave the same way. earth’s gravitational field exists outside of it’s radius. a charge distribution has an electric field that exists outside the volume of the distribution. etc.
    ow yes

  8. #8 Paul Murray
    August 17, 2009

    Yes, I understand that the field does extend, but I don’t see howw you get that from the equations.

    The current outside the cylindrical volume of the wire is zero. Is the electric field also zero? I suppose so – a charged particle would “see” the wire as electrically neutral.

    I suppose the situation, then, is that the curl of the magnetic field must be zero, with the boundary condition that it has a particular value at the surface of the wire, and those two together entail the magnetic field looking as it does.

  9. #9 Torbjörn Larsson, OM
    August 17, 2009

    First, thanks Matt for this very good series of posts! I hope you add this to the SB series of “basics”. [In which case you probably would want to elaborate on what an EMF is in #3.]

    Second, as there is no measure on “profoundness”, my own list would be slightly different:

    #1: If you integrate the electric charge equation to find the force from a charge (Coulomb’s law) you will find that it vanishes as r^2.

    That can only happen if the force is long-range, which IMO is a profound result. EM is one of the few (two) interactions we know of with unbounded range.

    #2: Relativity shows that magnetism is a relativistic pseudo-force, as much as gravity is.

    I.e. it is convenient to think of gravity as a newtonian field with a force, but it is really curved space that causes it (which massless particles follows without feeling pseudo-forces).

    It is also convenient to think of magnetism as a field with a force, but it is really moving charges that causes it (which magnetic less charges follows; dunno if the analogy holds further).

    [The later would of course be void if we observe these mysterious magnetic charges, monopoles, that you mentioned in a previous post's thread. I'm not conversant with quantum field theory, but my naive instinct would be to bet that they don't exist, at least as local charges.

    Choosing between a rationale for the quantization of the EM field that you mentioned, and a simpler relativistic theory (one less field, one broken symmetry) I would naively and a priori choose the later. As relativity is a constraint on QM that it follows, not the other way around.]

    #3: Magnetism is a low speed relativistic effect! This is #1 on my “awesomeness” list, btw.

    We are used to think of magnetism as a field and relativity as high speed not-everyday effects. Not so!

  10. #10 Torbjörn Larsson, OM
    August 17, 2009

    @ CCPhysicist, FWIW this late:

    Do you mean physical presence?

    Because AFAIU symmetry equals having a charge by way of Noether’s theorem. Possibly a universal boundary constraint “charge”, but more often local ones it seems.

    In which case IMHO it makes more sense, at least classically and naively (not more “basically” probably, but it is tempting :-o), to speak of an absent or broken symmetry as an absent charge. Unless you are speaking of actually realized symmetries and charges or, as you imply, technicalities of QFT.

    @ Paul:

    - The wire doesn’t *have* to be neutral, you can think of shooting electrons in a beam through space. Not too many, the like charges will tend to break “the current wire”, disperse the beam. So in practice, yes.

    (And conveniently conducting solids, liquids, gases and most plasmas are basically electrically neutral, so no problem.)

    - What you say is IIRC true (yesh, it has been too long!), provided you use the div B = 0 result on the magnetized volume. Ie you need the whole package (of equations) for EM dynamics.

  11. #11 Brian
    August 19, 2009

    @Paul
    The short reason why the magnetic field has nonzero components outside of the wire:
    The curl of the magnetic field is not zero. In integral form this means if I were to draw a loop around the wire (an amperian loop). The line integral of the magnetic field at this loop must be equal to the current enclosed within the loop. Therefore the magnetic field CANNOT be zero in the presence of a constant electric field and some current.

    The long reason:
    It is a result simply because of this fourth law.
    We can try to solve for B using this equation. When we assume we have an infinitely long wire, and assume a constant current we can deduce two other pieces of information.

    1. The current is constant and the wire is electrically neutral, therefore the electric field outside the wire is zero (from gauss’s law) and is constant for all time.

    2. Choosing a cylindrical coordinate system (s,phi,z) where the wire is at the z axis means that the current density J points along the z axis. This (along with gauss’s law for the magnetic field) means the magnetic field can only point in the phi direction (the direction pointing around the wire).

    Next we can solve for the magnetic field. From maxwell’s law, we see that the current density isn’t zero, while the derivative of the electric field is zero, so the curl of B isn’t zero. Taking a surface integral of the curl of B over a surface dA and likewise for the current density leaves us with an integral perfectly set up for stokes’ theorem. A surface integral over J leaves us with the total current I.

    This leaves us with: (line integral) B.dl = mu * I

    Knowing that B has only a phi direction, and is also independent of phi allows us to solve for B.
    B = munot I /( 2 * pi * s)

    It should be noted that the reason this field extends beyond the charge IS NOT the same reason why gravity and the electric field extend beyond their respective charges. Gravity and the E field are a consequence of Gauss’s laws (The divergence of the electric field is not zero because some charge is enclosed within some volume). This magnetic field will always have the divergence be equal to zero, HOWEVER the curl will not be zero in the presence of a constant electric field and a nonzero current density.

The site is currently under maintenance and will be back shortly. New comments have been disabled during this time, please check back soon.