Spacetime, Phil. Space and Time

Maxwell Equations


All electro-magnetic phenomena are governed by Maxwell Equations; but what are these equations? It is hard to tell in a word, so that we need some preparation.

Maxwell Equations (today's form) consist of four differential equations, (1) the law of electro-magnetic induction (Faraday), (2) Ampere-Maxwell's law, (3) Gauss's law of electric field, and (4) Gauss's law of magnetic field. Maxwell Equations are invariant with respect to Lorenz transformation; this may suggest a close relationship of these equations to Einstein's considerations on relativity.

The first two are expressed in terms of an operation called "rotation", either on electric field E or on magnetic field H, each expressed by a vector (bold letters express a vector in the following).

(1) rot E = -İB/İt

(2) rot H = (İD/İt) + i

where D is electric flux density (“d‘©–§“x), B is magnetic flux density (Ž¥‘©–§“x), and i is electric current density (“d—¬–§“x), which will be explained later.

The latter two equations are expressed in terms of an operation called "divergence" of D and of B.

(3) div D = ƒÏ

(4) div B = 0

where ƒÏ is electric charge density (“d‰×–§“x).


Now, let us turn to a brief explanation of basic concepts of electro-magnetic theory. First, there is a basic electric quantity, called electric charge, having the sign of either plus or minus. Two objects with the same sign of electric charge repel each other, and two objects with the opposite sign of electric charge attract each other, which is known as Coulomb's law. The total quantity of electric charge in a space is zero, thus we have a conservation law for electric charge. Moreover, the quantity of electric charge of any object is the same from any reference frame, not depending on the state of motion of the observer.

Figure 1. Coulomb's Law

Then, loosely speaking, electric charge produces electric field around it; or electric charge exert a force through electric field around it. This is the way of thinking established by Faraday and Maxwell. And since forces and fields have three components in a space, it is convenient to use vector in order to express them.

If you move an object with a quantity of electric charge against the force it exerts, the energy you spent for that movement is preserved in the "space" or field as electric energy. Such preserved energy is called the electric potential of that point of space (field). The difference of the electric potentials of any two points are usually expressed in terms of volt.

Now, strange as it may appear, a point with electric charge either (in case the charge is plus) emanates or (in case the charge is minus) absorbes a number of lines (Faraday called "lines of electric force", but the concept is somehow modified today), called electric flux. This idea also comes from Faraday. A point with charge Q emanates Q lines arount it. The number of such lines contained in a unit area is called electric flux density. Gauss's law of electric field (3) says that, on a closed surface, the number of lines of electric flux going through that surface equals the total quantity of electric charge contained within it; or (3) is nothing but this, expressed in terms of a differential equation.

Figure 2. Electric Flux and its Density

Faraday also considered lines of magnetic force; however, lines of magnetic flux are not idencal with them. Lines of magnetic force, outside of a magnet, are identical with lines of magnetic flux. However, lines of magnetic flux do not have origins, and are either a closed curve or come from infinity and goes to infinity. Gauss's law of magnetic field (4) is nothing but an expression of the fact that magnetic flux does not have origins.

Figure 3. Magnetic Force and Magnetic Flux

Lines of magnetic force (left) and lines of magnetic flux (right)

(figure from w•¨—ŠwŽ«“Tx966C”|•—ŠÙ)

The notion of electric current is already familiar in our ordinary life. But, theoretically, it is defined as a movement of electric charge. As is the case with other kinds of density, electric current density i can be defined appropriately. There are finer distinctions of electric current, but we will disregard them.

What is important is that electric current across a number of lines of magnetic flux suffers a force, and that electric current produces magnetic field, clockwise, around itself. And what is more, electric charge moving perpendicular to orthogonal E and B suffers a force generated by the electric field E' produced in the frame moving with the electric charge.

Figure 4. Current, Magnetic Flux, and Magnetic Field

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With this prepareation, we can have a glimpse of the meaning of Maxwell equations. Intuitively, the four laws express the following properties, respectively. The law of electro-magnetic induction says that if magnetic flux going across a closed circuit changes, an electromotive force is produced, to the direction for preventing that change. Ampere's law gives a quantitative determination of the magnetic field produced by a constant current of electricity; and Maxwell extended this law to the case where the electric current is not constant. Gauss's law of electric field relates the number of lines of electric flux to the electric charge, a basic quantity of electricity, and Gauss's law of magnetic field says that the number of lines of magnetic flux going out of a closed surface is, all in all, zero (lines going out and lines coming in cancel out).

Further, the force in the last figure is called Lorentz Force. However, seen from the frame (red) moving with velocity v, electric charge Q is at rest; and since the produced force F must be equal to the force seen from the frame at rest (black), this relation can be expressed (in vector notation) by

F = QE - QvB = QE', and hence E' = E - vB.

This suggests a sort of relativity of electric field E and magnetic field B; they depends on the state of motion of the electric charge, seen from a frame, and electric field, e.g., is described differently in different frames. If we pursue a geometric version of this Lorentz Force law, in the spirit of Einstein's general relativity, we will come across a unified object, electromagnetic field tensor (called Faraday by Misner, Thorn, and Wheeler), which is frame independent, unlike E or B.

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This article owes most of its materials to the following books; but for all modifications and errors (if any), I am responsible.

•¶Œ£

w•¨—ŠwŽ«“T@‰ü’ù”ŁxA”|•—ŠÙA1992B

‹´–{³OwŠG‚Å‚í‚©‚é“dŽ¥‹CŠwxAƒI[ƒ€ŽĞA1993B

For a readable illustration of Maxwell's derivation of the velocity of electromagnetic wave (and thereby the identification of light as a kind of electromagnetic wave), see the following.

’†’J‰F‹g˜Yw‰ÈŠw‚Ì•û–@xŠâ”gV‘A1958”NA169-175B

For a geometric version of Lorentz Force law, see:

Misner, Thorn, and Wheeler (1973) Gravitation, chapter 3.


Phil. Space and Time


Last modified March 30, 2003. (c) Soshichi Uchii

suchii@bun.kyoto-u.ac.jp

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