On this webpage, we wish to demonstrate some of the problems with Maxwell's electrodynamics. We are concerned here with the mathematical consistency of the system of equations known as Maxwell's electrodynamics (to be defined below). We are not concerned here with the associated physics.
Again we state that we are working under the restrictions:
UPDATE (2018-07-06): Thanks to the input of several reviewers of this website, and other people, we have come to realize that the way we present Maxwell's electrodynamics below is not the traditional point of view of the physics community.
However, we feel that the traditional point of view is, respectfully, a "dodge" that allows physicists to ignore the important issues outlined here. For that reason, we keep the presentation as is but provide detials to (what I am told is) the traditional point of view at the end of this webpage under the heading "Traditional Point Of View".
Your input continues to be important to us. We have made several changes throughout this web site based on such input. Thank you all.
We will now state what we mean by Maxwell's system of equations for electrodynamics. This consists of at least the following set of equations. (SI units will be used.)
Maxwell's Differential Equations:
| 1) | Gauss's Law, Electric: | \( \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho_f \) |
| 2) | Gauss's Law, Magnetic: | \( \nabla \bullet \vec{B} = 0 \) |
| 3) | Faraday's Law: | \( \nabla \times \vec{E} = - \dfrac{\partial}{\partial t} \vec{B} \) |
| 4) | Ampere's Law: | \( \nabla \times \vec{B} = \mu_0 \vec{J}_f + \dfrac{1}{c^2} \dfrac{\partial}{\partial t} \vec{E} \) |
Potential Function Definition Of The Fields:
| 5) | Electrodynamic Field: | \( \vec{E} \equiv - \nabla \Phi - \dfrac{\partial}{\partial t} \vec{A} \) |
| 6) | Magnetic Induction Field: | \( \vec{B} \equiv \nabla \times \vec{A} \) |
Potential Function Definitions:
| 7) | Scalar: | \( \Phi \equiv \dfrac{1}{4 \pi \epsilon_0} \iiint \rho \dfrac{1}{r} d\tau \) |
| 8) | Vector: | \( \vec{A} \equiv \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \dfrac{1}{r} d\tau \) |
Lorentz Force Equation:
| 9) | \( \vec{F} = q \vec{E} + q \vec{v} \times \vec{B} \) |
Continuity Equation:
| 10) | \( \nabla \bullet \vec{J} = - \dfrac{\partial}{\partial t} \rho \) |
where \( r \) is the distance from the differential volume elment \( d\tau \) to the observation position, \( \rho_f \) is the free charge density function, \( \vec{J}_f \) is the free current density function, \( \mu_0 \) is the permeability of free space, and \( 1/c^2 = \epsilon_0 \mu_0 \) where \( \epsilon_0 \) is the permittivity of free space.
Again, we are considering this mathematical system of equations as mathematics only. We are not considering the physics or physical interpretation.
By performing the calculation indicated on the left-hand side of Gauss's Law (Eq. (1) above), we find that the result does not match the right-hand side of Eq. (1). The resulting equation is $$ \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0}\rho_f - \nabla \bullet \dfrac{\partial}{\partial t}\vec{A} $$ and this is obviously not Eq. (1) above.
That is, Maxwell's system of equtions is easily shown to be mathematically inconsistent, as written, without any further clarifications.
The problem appears to be that Gauss's Law is derived for electrostatics and not for the more general electrodynamics. Yet it is listed, unchanged from the derivation in electrostatics, in the set of Maxwell's equations for electrodynamics. See, for example, excerpts from Jackson's book Classical Electrodynamics.
A simple resolution is to define the Coulomb field (also called Coulomb's Law in Jackson's book) $$ \vec{E}^{(C)} \equiv - \nabla \Phi $$ and to write the electrodynamic field as $$ \vec{E} \equiv \vec{E}^{(C)} - \dfrac{\partial}{\partial t} \vec{A} $$
It is then possible to show that $$ \nabla \bullet \vec{E}^{(C)} = \dfrac{1}{\epsilon_0}\rho_f .$$ This is Gauss's Law, not Eq. (1) above.
In some textbooks the authors state that the restriction \( \partial \vec{A} / \partial t = \vec{0} \) is to be applied to Gauss's Law. This is a perfectly good restriction. But then Gauss's Law, Eq. (1), should be written as $$ \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho_f \;\; restriction: \partial \vec{A} / \partial t = \vec{0}, $$ and then this restriction needs to be applied to all other equations.
Another approach in the textbooks is to assert that they are working in the Coulomb gauge such that \( \nabla \bullet \vec{A} = 0 \). Again, this is a perfectly good restriction. But then Gauss's Law needs to be written as $$ \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho_f \;\; restriction: \nabla \bullet \vec{A} = 0, $$ and this restriction needs to be applied to all equations. This is effectively stating that Maxwell's system of equations, Eq. (1) through (10), is only valid in the Coulomb gauge. We don't think that was the intention of imposing the restriction, but it is a consequence.
We have provided additional details for the correction of Gauss's Law showing that a notational change is necassary for mathematical consistency and clarity.
Another way to approach this problem is to first consider performing a mathematical expansion on \( \vec{E} \). We could then write $$ \vec{E} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + ... + \vec{E}_N = \sum_{i=1}^N{}\vec{E}_i .$$
We also call this expansion a "decomposition". For example, we could use a Helmholtz decomposition of a vector field. But here, for this problem, we use the decomposition $$ \vec{E}_1 = -\nabla \Phi \\ \vec{E}_2 = -\dfrac{\partial}{\partial t}\vec{A} $$ so that $$ \vec{E} = \vec{E}_1 + \vec{E}_2 = -\nabla \Phi -\dfrac{\partial}{\partial t}\vec{A} $$
We now simply relabel \( \vec{E}_1 \rightarrow \vec{E}^{(C)} \) for convenience.
Is it not obvious, from the rules of formal logic and mathematics, that we can not relabel \( \vec{E}_1 \) to be \( \vec{E} \) because \( \vec{E} \) has already been utilized, assigned to be the electrodynamics electric field? Seriously, would you accept the relabeling \( \vec{E}_1 \) to be \( \vec{E} \) and write $$ \vec{E} = \vec{E} + \vec{E}_2 = -\nabla \Phi -\dfrac{\partial}{\partial t}\vec{A}\;? $$ This appears to be exactly what has happened to the electric field in classical electrodynamics, as shown in the excerpts from Jackson's texts. The symbol \( \vec{E} \) has inappropriately been used for both \( -\nabla \Phi \) and for \( -\nabla \Phi -\partial\vec{A}/\partial t. \)
NOTE: This "second inconsistency" does not depend on the first inconsistency above being correct. Although I understand that the way I have written about it below makes it seem like it does, it really doesn't. It doesn't because, again, I am simply performing the calculation indicated in Maxwell's equation, using the Maxwell definitions, to determine what restrictions and definitions are needed to make the equation mathematically correct.
The obvious question is now: "Which of \( \vec{E} \) or \( \vec{E}^{(C)} \) is to be used in Eq. (1), (3), and (4)?"
By performing the calculations indicated in Eq. (1) through Eq. (4), we have checked the consistency of these equations using both \( \vec{E}^{(C)} \) and \( \vec{E} \). The closest match we have found to Eq. (1) through Eq. (4) with the fewest restrictions is
Corrected Maxwell Differential Equations:
| 11) | Gauss's Law, Electric: | \( \nabla \bullet \vec{E}^{(C)} = \dfrac{1}{\epsilon_0} \rho_f \) |
| 12) | Gauss's Law, Magnetic: | \( \nabla \bullet \vec{B} = 0 \) |
| 13) | Farady's Law: | \( \nabla \times \vec{E} = - \dfrac{\partial}{\partial t} \vec{B} \) |
| 14) | Ampere's Law: | \( \nabla \times \vec{B} = \mu_0 \vec{J}_f + \dfrac{1}{c^2} \dfrac{\partial}{\partial t} \vec{E}^{(C)} \)
\( restrictions: \partial \rho / \partial t = 0, \;\; \vec{v} = \vec{0} \) |
where \( \rho \) is the charge density function and \( \vec{v} \) is the velocity of the test charge particle, or detector, or observation position.
Note the "restrictions" in the, now corrected, Ampere law. We have found that these two restrictions, \( \partial \rho / \partial t = 0 \) and \( \vec{v} = \vec{0} \) must be applied for Ampere's law to be mathematically correct. Without these restrictions, and the use of \( \vec{E}^{(C)} \), Ampere's law is mathematically inconsistent.
Here are the details for the derivation.
One of the consequences of correcting Maxwell's system of equations is that the \( q \vec{v} \times \vec{B} \) term of the Lorentz force equation, Eq. (10), must always be zero.
Don't panic!
Although we have shown that Maxwell's electrodynamics is inconsistent with the Lorentz force equation, when the \( q \vec{v} \times \vec{B} \) term is not equal to zero, it turns out that Weber's electrodynamics, with Weber's force equation, is consistent with the Lorentz force equation (meaning that the Lorentz force equation is two terms of the more complete (having more terms) Weber force equation).
Whenever someone argues that "Experimental results prove that the Lorentz force equation is correct, and therefore, is a validation of Maxwell's electrodynamics." we can now mathematically prove that 1) Maxwell's system of equations (including the scalar and vector potential function definitions) are not mathematically consistent, and 2) any experiment that validates the Lorentz force equation is a validation of Weber's electrodynamics and not a validation of Maxwell's electrodynamics because Maxwell's electrodynamics is inconsistent with a non-zero magnetic induction term in the Lorentz force equation.
As we describe on other webpages, Weber's electrodynamics starts with the discrete sources case and builds up to the continuous sources case. It is the discrete sources case that is the more fundamental since all charged bodies are discrete. However, Maxwell's system of electrodynamics is not fully written out for discrete sources, and a smooth transition from the discrete to the continuous case is lacking in some topics. (Note that Maxwell's four differential equations are not presented for the discrete sources case. Why?)
Based on feedback of the above presentation of Maxwell's electrodynamics, we have come to understand that the way we have presented Maxwell's electrodynamics is not the way electrodynamics is meant to be presented and understood. The clarification, from these other researchers, is summarized (I hope accurately) as follows:
1) Maxwell's equations (SI units),
| Gauss's Law, Electric: | \( \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho_f \) |
| Gauss's Law, Magnetic: | \( \nabla \bullet \vec{B} = 0 \) |
| Faraday's Law: | \( \nabla \times \vec{E} = - \dfrac{\partial}{\partial t} \vec{B} \) |
| Ampere's Law: | \( \nabla \times \vec{B} = \mu_0 \vec{J}_f + \dfrac{1}{c^2} \dfrac{\partial}{\partial t} \vec{E} \) |
are postulated to be correct. Therefore, they are not subject to mathematical verification.
2) It is further postulated that the fields can be written in terms of scalar and vector potential functions when these potential functions are not specified explicitly in terms of charge density and current density functions. That is,
| Electrodynamic Field: | \( \vec{E} \equiv - \nabla \Phi - \dfrac{\partial}{\partial t} \vec{A} \) |
| Magnetic Induction Field: | \( \vec{B} \equiv \nabla \times \vec{A} \) |
are postulated to be correct for some unspecified \( \Phi \) and \( \vec{A} \).
3) The following equations can then be shown to hold for the scalar and vector potential functions $$ \nabla^2 \Phi + \dfrac{\partial}{\partial t}(\nabla \bullet \vec{A}) = -\dfrac{1}{\epsilon_0}\rho_f , $$ $$ \dfrac{1}{c^2}\dfrac{\partial^2}{\partial t^2}\vec{A} - \nabla^2 \vec{A} +\nabla (\nabla \bullet \vec{A}) + \dfrac{1}{c^2}\dfrac{\partial}{\partial t}(\nabla \Phi) = \mu_0 \vec{J}_f , $$
where the scalar and vector functions are left unspecified. The issue of how to express the scalar and vector potential functions in terms of charge density and current density functions are not to be specified at this level.
4)When the scalar and vector potential functions are specified in terms of the charge density and current density functions, then Maxwell's equations are not mathematically correct for the general case (this is known). They no longer form a system of differential equations. For example, Gauss's Law is not valid, in general, in terms of the scalar and vector potential functions written as
| Scalar: | \( \Phi \equiv \dfrac{1}{4 \pi \epsilon_0} \iiint \rho \dfrac{1}{r} d\tau \) |
| Vector: | \( \vec{A} \equiv \dfrac{1}{4 \pi \epsilon_0 c^2} \iiint \vec{J} \dfrac{1}{r} d\tau \) |
However, Gauss's Law is valid for these potential functions for the special case of the Coulomb gauge condition given as $$ \nabla \bullet \vec{A} = 0 .$$
But, there is no guarantee that other Maxwell's equations are valid for this gauge selection. Therefore, Maxwell's equations are no longer necessarily a consistent system of equations.
5) Maxwell's electrodynamics is not written for discrete point charges because the field energy of a point charge is infinite. However, in the approximation of continuous charge density and current density functions, the field energy is finite and Maxwell's electrodynamics (continuous source case only) is then valid.
Again, these are other researchers' points of view.
With these points of view, one might be forgiven for thinking that physicists, at some point, simply gave up the idea that electrodynamics should (or could) form a coherent, mathematically verifiable, system of equations. Fortunately, this issue is resolved by Weber's electrodynamics that is a verifiably consistent mathematical system of equations. Sadly, the idea that electrodynamics should be a mathematically verifiable, consistent, system of equations at all levels (fields and potential function definitions) appears not to be of interest to most physicists anymore.
We take the point of view, as expressed previously on this web page, that Maxwell's electrodynamics should be treated as a system of equations, verifiable when given the definitions of the scalar and vector potential functions in terms of the charge and current density functions. And, since the discrete sources case is more fundamental than the continuous sources approximation, it is even more important to have a mathematical theory of electrodynamics consistent for discrete sources. These are all satisfied with Weber's electrodynamics.
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