Maxwell's Equations

So far we have deduced the
existence and description of the electric and magnetic fields and of the forces
associated with them. Now we want to sum up that knowledge. It turns out, as
James Clerk Maxwell showed in the Nineteenth Century, that we can condense that
knowledge into four simple-looking equations.

I: Gauss's Law of Electricity

If we have an electric charge Q concentrated at a point, then Coulomb's law tells us that it generates an electric field extending radially away from it with strength

(Eq'n 1)

at every point a distance R from the charge. Imagine that we have a perfectly spherical shell of radius R with the charge occupying its center. With that image in mind, we need to take a slight detour.

We define the flux of an
electric field in the following manner: a thin film suspended in region of space
pervaded by an electric field defines an area. A grid of fine lines subdivides
the film into minuscule elements of area, each so small that we have a
vanishingly small difference between its curvature and that of a perfectly flat
plane (zero curvature). From the center of each such element we draw an arrow
perpendicular to the perfectly flat plane tangent to the element at that point
and we make its magnitude, d**a**, represent the area of the
element. To calculate the flux (or "flow") of the electric field across that
element we multiply the electric field **E** at that point by the
elemental area vector via the vector inner (or dot) product, so we have

(Eq'n 2)

We then calculate the total electric flux, M, by integrating those contributions over the whole film.

In the case of the spherical shell with an electric charge at its center, we have for the total electric flux

(Eq'n 3)

in which S represents the surface over which we carry out the integration.

Next imagine a set of
straight lines jutting from the center of the spherical shell and passing
through the shell in such a way that the points where they pass through the
shell define a closed line on the shell. Let the part of the shell enclose
within that boundary line comprise a very small portion of the whole shell and
imagine moving that little bit of area radially inward or outward to a new
radial distance R' from the center of the shell, so expanding it or contracting
it as it moves that its edge remains in contact with the straight lines that
define its boundary. In that situation the area of that little patch changes in
the proportion R'^{2}/R^{2} and the strength of the electric
field changes in the proportion R^{2}/R'^{2}, so the amount of
electric flux passing through the patch, which the patch contributes to the
whole shell, does not change.

If we tilt the patch through some angle 0(eta), again so stretching it that its boundary remains in contact with the straight lines that define it, then we increase the area of the patch by the factor 1/Cos0. But when we calculate the vector inner product of the electric field with patch's area vector, we must multiply the magnitudes of the electric field and of the area of the patch together and then multiply their product by the cosine of the angle between them, in this case Cos0. Thus we infer that tilting the patch as described above does not change the contribution that the patch makes to the electric flux over the spherical shell.

We can subdivide our spherical shell into minuscule patches and then move those patches radially and tilt them to transform the sphere into a closed surface of arbitrary size and shape with the charge Q at any point inside. And yet, we now know, the total electric flux through that surface remains unchanged from the simple number that we obtain by dividing the electric charge inside the surface by the electric permittivity of space. Further, the superposition principle tells us that if we have more than one electrically charged particle within the closed surface, each particle contributes its own flux to the total, unaffected by the other particles, so the Q in Equation 3 represents the net electric charge enclosed within the surface S.

Now consider an electrically charged particle at the common geometric center of two thin hemispherical shells of differing radii. Imagine further that we have so oriented the hemispheres that one lies inside the other and that we can connect them to each other by attaching a flat annular skin to their edges, thereby creating a closed surface that encloses a thick hemispherical shell of space. We now note explicitly that where the electric field points into the enclosed space it makes a negative contribution to the flux and where it points out of the enclosed space it makes a positive contribution to the flux. In this case the flux through one hemisphere equals the negative of the flux through the other hemisphere and the flux through the annulus equals zero (because the electric field has no components perpendicular to the surface of the annulus), so the total flux through the whole surface equals zero.

As we did above, we can subdivide the hemispheres into patches and then expand or contract their radii and tilt them, all without changing the flux through the surface. We can do that in such a way that the annulus shrinks down and vanishes, leaving us with a closed surface of arbitrary size and shape, through which the net electric flux equals zero. So now we know that electric charges outside a closed surface contribute nothing to the electric flux through that surface and, therefore, that the Q in Equation 3 refers to the net charge enclosed within the surface S and only the net charge enclosed within the surface S. With that fact in mind, we can say that Equation 3 expresses Gauss's Law of Electricity. More generally we have

(Eq'n 4)

in which D represents the density of electric charge inside the volume V enclosed by the surface S.

II: Gauss's Law of Magnetism

We know that magnetic monopoles, the magnetic analogue of electric charges, do not exist in Reality, so we infer of the magnetic flux through a surface S that

(Eq'n 5)

But we also have magnetic fields generated by electric currents and we don't yet know whether those fields must conform to Equation 5.

Imagine that an "infinitely"
long straight wire carries an electric current and thereby generates a magnetic
field. If we trace the direction of the magnetic field so generated we get an
array of concentric circles with the wire passing through their centers. We
define surfaces S_{1} and S_{2} as two flat planes parallel to
each other and perpendicular to the wire, which penetrates them. Define surfaces
S_{3} and S_{4} as two concentric cylindrical sheets of
differing radii with the wire's center coinciding with their common axis. And
define surfaces S_{5} and S_{6} as two flat planes that cross
each other at some angle on a straight line that coincides with the center of
the wire. Those surfaces enclose an annular segment V, so we define only those
parts of the surfaces that bound that volume as S_{1}', S_{2}',
S_{3}', S_{4}', S_{5}', and S_{6}'. We then have
the boundary of V as S_{V}= S_{1}'+S_{2}'+S_{3}'+S_{4}'+S_{5}'+S_{6}'.

By symmetry we know that the
magnetic field of the wire has no components perpendicular to S_{1}', S_{2}',
S_{3}', and S_{4}', so those surfaces contribute zero to the
magnetic flux passing through S_{V}. Likewise we know that the magnetic
flux through S_{5}' equals the negative of the flux through S_{6}'
and thus cancels it out. So the net magnetic flux through S_{V} equals
zero.

Now let a set of circles, all
centered on the wire, define a torus of arbitrary cross section. Where it passes
through S_{5}' and S_{6}' that torus cuts patches S_{5i}'
and S_{6i}' out of those surfaces. A sufficiently dense set of such tori,
all contiguous, completely subdivides S_{5}' and S_{6}' into
appropriately minuscule patches. Making sure that the edge of the patch remains
in contact with the circles that comprise the torus that defines it, stretching
the patch if necessary, we can move each patch and/or tilt it without changing
the contribution that it makes to the magnetic flux through S_{V}, just
as we did with the electric flux. If we begin with S_{1}' and S_{2}'
far enough apart and with S_{3}' and S_{4}' having appropriate
radii, then we can so manipulate the patches of S_{5}' and S_{6}'
in such a way that S_{1}', S_{2}', S_{3}', and S_{4}'
shrink down to zero area and the altered S_{5}' and S_{6}'
become a surface of arbitrary size and shape. We do that without changing the
net magnetic flux through the surface, so Equation 5 holds true in this case.

Next consider the case in
which the wire passes through the surface S_{V}'. On the assumption that
the radius of S_{4} is greater than the radius of S_{3}, we
define that surface as S_{V}'=S_{4}+S_{1}+S_{2}-S_{1}'-S_{2}'-S_{4}';
that is, S_{V}' encloses the volume of the cylinder, centered on the
wire, from which we cut V. As before, we can subdivide S_{5}' and S_{6}'
into minuscule patches and then move and tilt those patches until S_{4}-S_{4}'
and S_{3}' vanish, thereby leaving a body of arbitrary shape except for
the flat, circular end faces, S1'' and S'', that remain. And we know that we
still have zero net magnetic flux through that body. If we then slice the body
into vanishingly narrow sections parallel to the end faces, we can shrink the
radii of those sections without changing the contribution that they make to the
net flux: even though the strength of the magnetic field encountering the rim of
the section changes, it does so in a uniform way that does not change the net
flux through the section. Thus we can start with the sections midway between the
end faces and progressively shrink the sections as we move closer to those ends
until the sections at the faces vanish altogether. Now we have a body of
arbitrary size and shape with a current-carrying wire running through it and we
still have zero net magnetic flux through the surface of that body, in
accordance with Equation 5.

That fact also remains true to Reality if we make the wire differentially short, shrinking it to length dl. We can now invoke the principle of superposition to state that the magnetic fields of a large number of such elements each yield zero flux through a given arbitrary closed surface, so the net field also yields zero magnetic flux. Thus we infer that Equation 5 holds true in general, even with curved, even kinky, wires. It tells us then that in the presence of any array of electric currents Reality's rules must so shape the magnetic field that its flux through any closed surface always equals zero. It also expresses the fact, already deduced, that magnetic monopoles do not exist.

III: Faraday's Law of Electromagnetic Induction

Now that we have added a new term, the negative time derivative of the magnetic vector potential, to our basic description of the electric field, we should check that new description and make sure that it conforms to the law of conservation of energy. We accomplish that task by calculating the vector dot product of the electric field at a point with a differential element of distance going through that point () and then integrating that product over a set of distances that comprise a closed loop. We have, then

(Eq'n 6)

We already know that

(Eq'n 7)

so Equation 6 becomes

(Eq'n 8)

in which equation I have exploited the fact that the differentiation with respect to time commutes with the integration over distance.

To calculate the line
integral of the A-field around a closed loop let's confine our attention to the
x-y plane and note that the elements of distance along a rectangular loop point
in the counterclockwise sense. Let the loop enclose an area that extends from
x=0 to x=x_{0} and from y=0 to y=y_{0}. We thus get four
integrals

(Eq'n 9)

But we can express the difference between A_{y}(x=x_{0})
and A_{y}(x=0) as

(Eq'n 10)

Likewise we have

(Eq'n 11)

So Equation 9 becomes

(Eq'n 12)

But that equation gives us only the x-y surface integral of the z-component of the curl of the A-field,

(Eq'n 13)

If we extend that analysis to the y-z and z-x planes, then we discover finally that

(Eq'n 14)

which means that the line integral of the A-field around a loop C equals the integral of the curl of the A-field over the surface S bounded by C. In this equation we have one manifestation of Stokes' Theorem, one of the aspects of the Fundamental Theorem of the Calculus in three-dimensional space.

If we substitute that result into Equation 8 and recognize that , then we have

(Eq'n 15)

That equation expresses Michael Faraday's law of electromagnetic induction, which tells us that a magnetic flux changing through some area S generates a net electromotive force on the boundary C defining that area. On first impression that statement looks like it offers us the possibility of violating the law of conservation of energy, but we really should test that impression.

We start by imagining that we have laid a wire N times around C. If we recognize the integral on the left side of Equation 15 as representing the voltage V induced in the wire and recognize the integral on the right side of the equation as representing the magnetic flux enclosed within the area bounded by the wire, then we can rewrite Equation 15 as

(Eq'n 16)

That equation presents Faraday's law in the form known to engineers as Lenz's Law (named for Russian physicist Heinrich Lenz). It tells us that if we do anything that changes the magnetic flux enclosed by the wire, that change will induce in the wire a voltage that will, in accordance with Ohm's law (V=IR), produce an electric current in the wire.

Now let's set up a different experiment to get a proper feel for Lenz's law. Imagine that we have a very large, very solid magnet in which we have cut a narrow gap so oriented that its broad faces run parallel to the x-y plane. In that situation we effectively have a magnetic field, of uniform strength B pointing in the positive z-direction, only in the gap. We also have a rectangular copper loop, of length x and height y, that we can move in the x-direction, into and out of the gap.

`With the loop outside the gap and with one of its vertical arms just inside the gap, we pass an electric current I through the loop in such a way that it appears to use to move around the loop in the clockwise sense when we look in the positive z-direction. Where it interacts with the B-field that current exerts a force upon the loop: the components of the force exerted upon the horizontal arms of the loop are equal and oppositely directed, so they contribute no net force to the loop, and the force exerted upon the vertical arm in the gap acts to pull that arm deeper into the gap.

That force has magnitude F=IyB. If the loop moves in the x-direction at the speed dx/dt=v, then the force does work upon the loop at the rate dW/dt=Fv=vIyB. We can transfer that work via some mechanical linkage to another system and thus remove it from our simple magnetic motor. But the loop must move in order to harness that work and that motion, interacting with the magnetic field, induces a voltage in the loop in a way that opposes the current flowing in the loop. We calculate that voltage as V=vyB. Moving against that voltage, the current must absorb energy from the circuit at the rate dE/dt=-VI=-vIyB, which exactly equals the work that we extract from the system.

If we modify that experiment, hold the loop stationary and let the magnet move, then we get the same result, although we must connect the magnet rather than the loop to our linkage. We must use the same numbers and calculate the same rates of work being done. In this reversal we see a reflection of the imaginary experiment with which Einstein opened his 1905 paper "On the Electrodynamics of Moving Bodies". In both aspects of this experiment, I need to point out, we have created a changing magnetic flux within the loop by increasing the area occupied by the magnetic field.

What happens if, instead of changing the area occupied by a magnetic field, we change a magnetic flux by changing the strength of a magnetic field occupying a fixed area? Does that situation still conform to the law of conservation of energy?

To answer that question let's set up a wire loop
of radius R and assume that we have an electric current I flowing in the wire.
Let's also imagine that we have put a thin disc of radius R inside the loop so
that the edge of the disc rubs up against the wire at every point in its
circumference. And then let's assume that we have spread a magnetic charge P
uniformly over the disc so that it has a surface density of
F=P/BR^{2}.

From the disc a magnetic field B emanates and crosses the wire in the loop and disc's common plane. We have B = Pf(R), in which f(R) represents a geometric factor that depends only on the radius R of the disc. I have attempted to calculate the form of that geometric factor from the magnetic analogue of Coulomb's law and the representation of the disc as a set of concentric rings and have thus far not succeeded. From my experience I can infer that the solution of the integrals describing the field of a uniformly charged ring, never mind the full disc, lies somewhere between extremely difficult and intractable. Fortunately, we do not need to know the actual algebraic representation of f(R) in order to continue this discussion.

The disc's magnetic field interacts with the current in the wire to generate a force pushing the wire in the direction parallel to its axis. We describe that force as

(Eq'n 17)

At the same time the magnetic field generated by the current in the wire pushes on the magnetic charge on the disc. We have that force as

(Eq'n 18)

Newton's third law of motion necessitates that those two forces have equal magnitudes and opposite orientations, so we can equate the magnitudes and then solve for the magnetic flux in the interior of the loop. We obtain

(Eq'n 19)

In accordance with Equation 16, we know that a change in the magnetic flux enclosed by the loop will produce a voltage in the loop, one that will push the current in the loop in a way that opposes the change in the flux. Thus we have in this case

(Eq'n 20)

Moving against that voltage, the electric current must absorb power at the rate

(Eq'n 21)

If we substitute from Equation 20 and carry out the implicit integration, we find that the total energy absorbed by or given up by a change in the flux equals

(Eq'n 22)

We must put that much energy into increasing the current that generates the magnetic flux and we will get that much energy back when we diminish the current in the same amount; that is, the induced voltage will oppose the current when we increase it and will drive the current when we try to diminish it. We have no way to get around that fact, so we may conclude that Lenz's law encodes the conservation of energy.

IV: The Ampere-Maxwell Law of the Magnetic Field

We already know the rule describing how an electric current I generates a magnetic field;

(Eq'n 23)

So we know that the magnetic induction field
integrated around a closed loop C equals the product of the magnetic
permeability of vacuum and the integrated flux of the electric current density
**i** over the surface S bounded by C.

As we have seen, the structure of Reality gives us a remarkable freedom to deform surfaces and volumes of integration without altering the description of physical law that those integrations encode. Now I want to exploit that fact to re-examine the breathtaking deduction that James Clerk Maxwell made in the 1860's.

Suppose that we have a long,
straight wire with an electric current I flowing in it and that we construct a
circle of radius R centered on the wire in such a way that the wire passes
through the plane in which the circle lies in the direction perpendicular to
that plane. Equation 23 then tells us that 2RB = µ_{0}I. Following
Maxwell's lead, we imagine cutting a segment of length dl out of the wire and
welding flat metal plates to the wire's bare ends, orienting the plates
perpendicular to the wire. In the present case, then, electric charge will
accumulate on or drain out of each of the plates at the rate dQ/dt = I.

In that setup the wire still
penetrates the flat surface S_{1} bounded by the circle C on which we
integrated the wire's magnetic field. I now define a second surface S_{2}
bounded by C, drawing it out from C like a soap film parallel to the wire at
first and then taking it into the gap between the plates to close it. Now we
have a closed surface S_{1}+S_{2} that contains an electric
charge Q(t), so by Gauss's law we have

(Eq'n 24)

We can differentiate that equation with respect to time, with the proviso that the bounding surface does not change in any way, and we get

(Eq'n 25)

Because we can bring the
plates arbitrarily close together, we can state that the electric field between
the charges on the plates remains almost completely confined to the gap between
the plates with a vanishingly small amount of the flux through S_{2}
extending beyond the edges of the plates. On that basis we can state that the
electric flux through S_{1} equals zero and we can therefore drop that
surface from our calculation. Thus Equation 25 becomes

(Eq'n 26)

Because the circle C bounds S_{2} we can
substitute that result into Equation 23 and get

(Eq'n 27)

the magnetoelectric analogue of Faraday's law of electromagnetic induction.

We can generalize that result
by noting that if a second electric current entered our arrangement and passed
through S_{2}, it would add its own separate contribution to the
equation. Expressing the current in terms of the current density passing through
S_{2}, we have (after re-labeling S_{2} as S for generality)

(Eq'n 28)

which expresses Ampere and Maxwell's law of the magnetic field.

V: Maxwell's Equations

So now we have Maxwell's Equations of Electromagnetism:

(Eq'n 29)

(Eq'n 30)

(Eq'n 31)

(Eq'n 32)

But those are integral equations, favored by engineers. Physicists prefer differential equations, so now I want to transform those equations into their differential equivalents.

The Fundamental Theorem of
the Calculus, as it plays out in three dimensions, tells us that for any
vectorfield **M** the geometric structure of Reality requires

(Eq'n 33)

the Ostrogradsky-Gauss theorem, and

(Eq'n 34)

the Kelvin-Stokes theorem. These comprise examples in Euclidean 3-space of the more general Stokes' theorem of differential geometry.

Let's take Gauss's law of electricity and replace the first integral by its equivalent from Equation 33. We get

(Eq'n 35)

If two integrals calculated over the same range equal each other, then their integrands equal each other, so we have

(Eq'n 36)

By the same reasoning Gauss's law of magnetism becomes

(Eq'n 37)

Substituting from Equation 34 transforms Faraday's law into

(Eq'n 38)

In Euclidean space, we know, space itself does not change, so the area S, once we define it does not change. That fact and the fact that we can commute the operations of differentiation with respect to time and of integration over the area S lead us to infer that

(Eq'n 39)

And again the same reasoning gives us the differential form of Ampere and Maxwell's law as

(Eq'n 40)

Thus we have the four
fundamental equations of electromagnetic theory.

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