Built on Facts

Maxwell’s Equations #2

In our examination of the first of Maxwell’s four equations, we saw that magnetic charge doesn’t exist as far as we can tell. On the other hand, electric charge permeates every aspect of our existence. The motion of charged electrons is one of the central pillars of modern civilization. The way that electric charge creates an electric field is the subject of the second of Maxwell’s equations. In its full mathematical glory:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

The triangle and the dot represent the divergence, exactly as in the first Maxwell equation. The letter E is the electric field. Unlike the magnetic field divergence equation, the divergence of the electric field is not zero everywhere. Instead, it’s equal to the charge density denoted by the Greek letter rho, divided by the electric constant, Greek letter epsilon.

Time for a quick review of divergence. It’s a property of a vector field (such as an electric field) that describes the sources of field lines. Field lines are not physically real but instead serve as a schematic representation, with their direction pointing along the field and their closeness representing the field strength. Draw a little box around a region of space, and if there’s more field lines leaving, you have positive divergence. If there’s more entering than leaving, you have negative divergence. If they’re the same, you have zero divergence.

All three possibilities are pictured in this snazzy Wikipedia image:

i-345d4bfac76c6cf18b72f80c7601408b-charge.png

There’s two charges, one positive and one negative. In the space outside the charges, there’s no charge density. Therefore according to our equation, the divergence has to be zero. Sure enough if you draw a little box there’s just as many field lines entering as there are leaving. But draw a box around one of the charges and the situation is entirely different. There’s an excess of field lines leaving the positive charge (positive divergence) and an excess of field lines entering the negative charge (negative divergence).

This result was known well before Maxwell, and is in fact separately named Gauss’ Law in honor of the great Carl Gauss who did pioneering work in this kind of mathematics. If you know the appropriate calculus, it’s a snappy two-line calculation to derive the more elementary Coulomb’s Law version of the field of a point particle. One minor technicality of Gauss’ Law is that it gives the electric field divergence in terms of a continuous charge density. While this is a near-perfect approximation for macroscopic charge distributions, point particles technically don’t have a finite charge density by virtue of not having volume. This can be mathematically fixed by the method of the delta function.

You might notice that in the sample picture we’ve picked the field lines are closer together in the area between the charges. This is a typical feature of many charge configurations, related to the close mathematical correspondence between the “flow” of field lines down voltage gradients and the flow of water down a hill. If you build up charges on your body by rubbing the floor with your socks, you can feel this process in the form of a zap when the charges in your body come near to the charges in a door knob. The high field from the close proximity breaks down the resistance of the air and makes a spark.

All right, that’s two equations down, two to go. In fact at this point we’re done with electrostatics. However our world is one where charges move and currents flow, so we and Maxwell have two more equations to go before we’ve covered the whole sweeping panorama of electromagnetism.

Comments

  1. #1 CCPhysicist
    August 11, 2009

    First, a comment directed toward Jason@6 of the previous post: The concept here is best visualized by considering fluids, which are also governed by a similar equation as Matt mentioned. Let water run onto a flat surface, and there will be a positive divergence at the point where the stream of water hits and spreads out. If there is a drain nearby, there will be a negative divergence at the point where it enters the drain. The divergence is zero for the points where there is no source or sink (drain) for the vector field.

    This last statement leads to the “minor technicality” Matt alludes to: divE=0 almost everywhere in the picture he shows. It is zero where the background is yellow because there is no charge in those areas. It is only non-zero in the purple areas where the + and – charge is located. That makes it instructive to compare this picture to the one in the previous article. The solenoid produces a magnetic field that does not have any “source” or “sink” points. The magnetic field lines form closed loops … while the electric field lines shown here do not.

  2. #2 Jason A.
    August 13, 2009

    CCPhysicist: Thanks. I know how to calculate divergence but it never really clicked to me what divergence is. Now maybe you can explain curl :P

  3. #3 jdhuey
    August 14, 2009

    Just a historical note: these four ‘Maxwell’s Equations’ are actually due to Oliver Heaviside. He reformulated the cumbersome way Maxwell had presented his equations (20 equations with 20 unknowns) into the differential equation form that is being presented here.

    http://en.wikipedia.org/wiki/Oliver_Heaviside

  4. #4 Torbjörn Larsson, OM
    August 17, 2009

    FWIW on an old thread:

    I don’t think it is entirely productive to think of the two charge equations as electrostatics, or describing such situations. It is, as you will undoubtedly cover later, true by virtue of choosing suitable reference frames. However the magnetic field is, as you also probably will cover, a (low speed) relativity effect, so electrodynamics is already implicit in the equations.

    It is AFAIU the actual choice of reference frames that can produce situations where electrostatics applies. It’s a “simplest analysis” choice, as far as relativity goes.

    “source” or “sink” points

    One reason I like to think of EM in english (besides the text books) is these beautiful pair of words. Not so nice in my native language.

    the cumbersome way Maxwell had presented his equations

    I think there is a history there that, if known, ought to deepen one’s understanding.

    IIRC odds and ends of it, Maxwell, at least at first, managed to visualize his equations by a mechanical model of many a virtual gear and axis in place of the then non-existing field concept. Maybe he wasn’t geared [sic!] to simplify the notation away from an explicit 3D representation.

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