Maxwellian Deduction of Coulomb=s Law
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In his discussion of a magnetostatic field emanating from poles, in AOn Physical Lines of Force@ (1855 - 56), James Clerk Maxwell laid out a derivation of the inverse-square law of static forces vastly more elegant than the derivation that I laid out in ADeducing the Newton-Coulomb Law@. More mathematician than physicist (and certainly a better mathematician than I am), Maxwell led his readers on a stroll through the realm of mathematics, only lightly touching the realm of physics, but matching it perfectly where he did. If you want to know what it feels like to comprehend directly a Platonic Form, this will take you remarkably close.
Imagine that two point-like particles exert a force each upon the other. If we hold one particle stationary and move the other particle around it, we can measure at each point the strength and the direction of the force that the stationary particle exerts upon the moving particle. At each point, then, we can draw a minuscule arrow to represent the force exerted at that point. If we draw a line, going from point to point by following the direction that the arrow at each point gives us, we will thus trace what Maxwell called a line of force. If we draw enough lines of force, we will depict the forcefield emanating from the stationary particle and thereby see an example of what mathematicians call a vector field.
Whatever form the forcefield takes, it must conform to the requirement that the forces exerted between the two particles obey the laws pertaining to conservation of linear momentum and conservation of angular momentum. Conservation of linear momentum (Newton=s third law of motion) necessitates that the particles exert equal and oppositely directed forces upon each other. Conforming to that requirement ensures that the two particles, taken together as a single system, do not generate a net force that would make the system=s linear momentum increase/decrease continuously. Conservation of angular momentum necessitates that the forces exerted between the particles not push the particles into revolving about any point or make a wheel uniformly covered with such particles spin faster or slower. We call a field that conforms to that requirement irrotational.
Here=s how Maxwell took my breath away:
If a point-like particle extends a forcefield
then the requirement that the field be irrotational corresponds to
Expanding that equation to display the components of the curl shows us that
Maxwell recognized those equations as the criteria for the dot product EA dx to be an exact differential of a scalar functionφ; that is,
(See the appendix for a proof). Comparing the expanded expressions in that equation term by term gives us
which makesφ the potential function associated with the vector field E. To the extent that E represents a force, to that same extent φ represents a potential energy associated with that force.
Next we must refer to Helmholtz=s vectorfield theorem, which mathematicians call the fundamental theorem of vector calculus. If we have a vector field V such that
in which s represents the source density of the field and c represents the circulation density of the field and if the source and circulation densities go to zero at all points infinitely far from our field point, then the vector field has a unique representation as the linear sum of an irrotational (curl-free) component field and a solenoidal (divergence-free) component field. In the case of an electrostatic field the circulation density equals zero everywhere, so we must have at least one point where the source density does not equal zero, which tells us that
Maxwell recognized that equation as an example of Poisson=s equation, whose solution he knew. As he put it, ANow it may be shewn that [Poisson=s equation], if true within a given space, implies that the forces acting within that space are such as would result from a distribution of centres of force beyond that space, attracting or repelling inversely as the square of the distance.@ In that statement Maxwell reveals his inference that the mathematical description of the strength of an electrostatic field emanating from a single Acentre of force@ has the form
in which r represents the distance between the Acentre of force@ and the point at which we measure the strength of the field and G represents some proportionality factor pertaining to the ability of the Acentre of force@ to exert a force.
Now we can say that Equation 9 represents the force that two point-like particles exert each upon the other if each particle carries one unit of what Maxwell called free electricity (and we call electric charge), however big that unit may be. For the sake of consistency with the modern representation of the law we redefine the proportionality constant as
in whichε0 represents the electric permittivity of vacuum, whose value we can only determine through measurements made on suitable experiments. If we put more than one unit of charge on particle A (the number of such charges represented by qA), each of those charges responds separately to the field emanating from the single unit of charge residing on particle B, so we calculate the force acting on particle A by multiplying Equation 9 by qA. In accordance with Newton=s third law of motion, the same force acts on particle B, though in the opposite direction. If we now put a number qB of unit charges on particle B, each one will respond to the multiplied field emanating from particle A, so we calculate the force that particle A exerts on particle B (and vice versa, of course) as
along the straight line passing from one particle to the other. That equation expresses Coulomb=s law of the electrostatic force. Using that equation to describe the electric field emanating from a single charged particle and applying Gauss=s law lets us rewrite Equation 8 as
the first of the four Maxwell equations of electromagnetism, in whichρ represents the density of electric charge at the point where we calculate the divergence of the electric field.
We must feel sheer astonishment when we first learn that Maxwell worked out that derivation (though not quite in that condensed form) by describing the velocities and pressures in an incompressible fluid moving through a resistant medium, the fluid coming out of point-like sources and going into point-like sinks (Maxwell=s Acentres of force@). That a fluidic model should give us the laws of electromagnetism properly astounds us. That concepts derived from direct perception (of running water) should match the laws that govern things several steps removed from direct perception (electric and magnetic forcefields) hints at a great profundity in the relationship between mathematics and Reality yet leaves to us the task of discovering its nature.
In his paper Maxwell had to assume one proposition, one that he could not deduce from more fundamental principles. He had to assume that electric charge obeys a conservation law and he expressed that law in the form of a continuity equation, which we write in modern vector notation as
That equation tells us that at any point in space where we find a non-zero divergence in the electric current density we must also find an equal accumulation or dissipation of the electric charge density. We can convert that statement into an equivalent statement, one that resembles Newton=s third law of motion, by stating that for every increase or decrease that we have in the amount of positive electric charge at some point we must have an equal increase or decrease in the amount of negative electric charge at that point. Let=s now deduce that statement.
Assume that nought but a single unit of electric charge exists in the entire Universe. That charge=s electric field extends throughout all of space and, we assume, ends on the boundary of space. That abrupt ending corresponds to a non-zero divergence of the field, which necessitates, in accordance with Equation 12, that the boundary of space possess or manifest an electric charge of a polarity opposite that of the assumed charge. But because it touches Absolute Nonexistence, the boundary of space cannot possess or manifest any properties whatsoever, so we must infer that the assumed charge=s field does not reach the boundary. Because the field spreads throughout all space, we must assert the existence of a charge of polarity opposite that of our assumed charge. An observer with either charge will see it lying the same distance from the boundary of space in all directions, so the charge=s field has the same strength wherever it approaches the boundary. The two fields will then cancel perfectly at the boundary, but if and only if the asserted charge has the same magnitude as has the assumed charge. That remains true to Reality regardless of how many charges we assume, so we infer that the electric charges in the Universe must add up to a net zero at all times. That inference means that any phenomenon that creates or destroys a certain amount of positive electric charge must necessarily create or destroy, at the same time and place, an equal amount of negative electric charge. Q.E.D.
That deduction mimics the deduction of the conservation law pertaining to linear momentum. But this deduction has a special consequence that we didn=t get with linear momentum. Substitute the charge density from Equation 12 into Equation 13; we get
We know that the divergence of a curl always equals zero, so we can assert the existence of some vectorfield H such that
Of course, in this derivation we have no hint that H represents a forcefield, but we know, nonetheless, that if we multiply that equation by the magnetic permeability of vacuum, we will obtain Maxwell=s version of Ampere=s law of magnetic induction. Thus, our derivation of the conservation law pertaining to electric charge shows us, in part, how electric and magnetic fields must relate to each other.
Appendix: The Exact Differential
If we have a function of more than one variable, we confront a problem that we don=t have with functions of a single variable: does that function have an exact differential or an inexact differential? The problem originates in the fact that the variables represent a multidimensional space that allows us to integrate a function over an infinite variety of paths. Let F represent a function of more than one variable; then dF is an exact differential if and only if the integral
does not depend upon the path that we follow in the integration. We must carry out the integration over all of the variables in the function, so we must rewrite the integral as a sum of integrals,
in three dimensions for example. If the integral of Equation A-1 truly does not depend upon the path of integration, then it must give us F as a unique function of the variables. That function must have a unique differential with respect to those variables;
If we are given the components P, Q, and R, we know that Pdx+Qdy+Rdz is an exact differential if we also know that
But if we are given only P, Q, and R, how can we know that?
To answer that question Leonhard Euler devised a simple test based on the fact that in calculating a second derivative the differentiation with respect to one variable commutes with the differentiation with respect to the other variable; that is, for illustrative example, the fact that
Thus, we know that Pdx+Qdy+Rdz gives us an exact differential if
If we have a vectorfield E that has components Ex=P, Ey=Q, and Ez=R, then Equations A-6 represent respectively the z-, x-, and y-components of the field= s curl. So now we know that if L H E=0, then the dot product of E with the minuscule element of displacement, EA dx=Exdx+Eydy+Ezdz=dφ, gives us an exact differential, as Maxwell stated and exploited.
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