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
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, da, 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
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
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/R2 and the strength of the electric field changes in the proportion R2/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
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
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 S1 and S2 as two flat planes parallel to each other and perpendicular to the wire, which penetrates them. Define surfaces S3 and S4 as two concentric cylindrical sheets of differing radii with the wire's center coinciding with their common axis. And define surfaces S5 and S6 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 S1', S2', S3', S4', S5', and S6'. We then have the boundary of V as SV= S1'+S2'+S3'+S4'+S5'+S6'.
By symmetry we know that the magnetic field of the wire has no components perpendicular to S1', S2', S3', and S4', so those surfaces contribute zero to the magnetic flux passing through SV. Likewise we know that the magnetic flux through S5' equals the negative of the flux through S6' and thus cancels it out. So the net magnetic flux through SV equals zero.
Now let a set of circles, all centered on the wire, define a torus of arbitrary cross section. Where it passes through S5' and S6' that torus cuts patches S5i' and S6i' out of those surfaces. A sufficiently dense set of such tori, all contiguous, completely subdivides S5' and S6' 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 SV, just as we did with the electric flux. If we begin with S1' and S2' far enough apart and with S3' and S4' having appropriate radii, then we can so manipulate the patches of S5' and S6' in such a way that S1', S2', S3', and S4' shrink down to zero area and the altered S5' and S6' 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 SV'. On the assumption that the radius of S4 is greater than the radius of S3, we define that surface as SV'=S4+S1+S2-S1'-S2'-S4'; that is, SV' encloses the volume of the cylinder, centered on the wire, from which we cut V. As before, we can subdivide S5' and S6' into minuscule patches and then move and tilt those patches until S4-S4' and S3' 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
We already know that
so Equation 6 becomes
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=x0 and from y=0 to y=y0. We thus get four integrals
But we can express the difference between Ay(x=x0) and Ay(x=0) as
Likewise we have
So Equation 9 becomes
But that equation gives us only the x-y surface integral of the z-component of the curl of the A-field,
If we extend that analysis to the y-z and z-x planes, then we discover finally that
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
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
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/BR2.
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
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
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
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
Moving against that voltage, the electric current must absorb power at the rate
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
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;
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 = µ0I. 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 S1 bounded by the circle C on which we integrated the wire's magnetic field. I now define a second surface S2 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 S1+S2 that contains an electric charge Q(t), so by Gauss's law we have
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
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 S2 extending beyond the edges of the plates. On that basis we can state that the electric flux through S1 equals zero and we can therefore drop that surface from our calculation. Thus Equation 25 becomes
Because the circle C bounds S2 we can substitute that result into Equation 23 and get
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 S2, it would add its own separate contribution to the equation. Expressing the current in terms of the current density passing through S2, we have (after re-labeling S2 as S for generality)
which expresses Ampere and Maxwell's law of the magnetic field.
V: Maxwell's Equations
So now we have Maxwell's Equations of Electromagnetism:
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
the Ostrogradsky-Gauss theorem, and
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
If two integrals calculated over the same range equal each other, then their integrands equal each other, so we have
By the same reasoning Gauss's law of magnetism becomes
Substituting from Equation 34 transforms Faraday's law into
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
And again the same reasoning gives us the differential form of Ampere and Maxwell's law as
Thus we have the four
fundamental equations of electromagnetic theory.
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