|Aug29-11, 11:18 PM||#18|
independent equations of maxwell equations
Going back to the OP:
Below, I will refer to both the microscopic and the macroscopic Maxwell equations; if anybody is unfamiliar with these terms, please see the first two tables here: http://en.wikipedia.org/wiki/Maxwell%27s_equations.
The OP implicitly talks about the microscopic Maxwell equations. At first glance, indeed there seem to be eight scalar equations for only six scalar unknowns; however, this is actually not the case. The reason is that the two divergence equations are not independent of the two curl equations, i.e. the former can be derived from the latter. Simply take the divergence of Ampere's law and you get Gauss' law. Similarly, take the divergence of Faraday's law and you get the law of no magnetic monopoles. Since the two divergence equations are simply consequences of the two curl equations, we find that there are actually only six independent scalar equations for the six unknown scalar field quantities.
For the macroscopic Maxwell equations, by similar arguments, you wind up with six independent scalar equations for twelve (!) scalar unknowns. The remedy is that the constitutive relations give you six more independent scalar equations.
Hope that clears everything up!
|Aug30-11, 04:34 AM||#19|
OK. better said, there are 8 variables and 8 scalar equations.
however, you’re taking the question to another level. For this it’s a lot simpler after having converted maxwell’s equations to differential forms. It’s far more obvious and natural.
charge density, current density and the fields are completely specified by the 4 vector potential for all spacetime points.
the electric and magnetic fields are specified from F=dA. F is the exterior derivative of A. Using J=d*F, the electric charge and current density are specified.** So there are really only 4 independent variables. J=d*F obtains the 4 maxwell equations and d2J=0 4 more. However, the last is a mathematical identity saying there are no magnetic charges or currents.
going backwards, beginning with the charge and current density, is harder. Integration introduces constants of integration. Physically, these are boundary conditions or gauges. I’m not as comfortable with the integral forms, though the resultant gauging is fairly obvious once you have it.
to get F when given J, schematically, *F = ∫J + θ. The gauge θ is a 4x4 tensor with 6 independent terms.
(*F)μν dxμdxν = (∫Jρμν dxρ) dxμ dxν + θμν dxμdxνwith the gauge included, the differential equations are J = d(*F + θ), where dθ = 0.
so, in the typical problems usually encountered where charge and current densities are the "knowns", to get the fields knowing the charge and currents, 6 additional variables need to be specified by setting a gauge or giving boundary condtions.
also, A = ∫F + φ, where φ is a dual vector, so 4 additional independent terms arise.
Aν dxν = (∫Fμν dxμ) dxν + φν dxνF = d(A+φ), where dφ = 0.
This second gauge is not encountered in the usual problems. It's just known that A is regaugable and probably cannot be directly measured to an exact value, but known only up to a gauge offset.
** reference: exterior derivative, Hodge dual, and “all exact forms are closed”.
For inclusion of magnetic monopoles, 'A' can have complex valued entries, but things are a little more convoluted. F is not only complex but, as I recall, regauges itself, i.e.: *iF regauges F.
|Sep7-11, 08:10 AM||#20|
|Oct5-11, 06:46 AM||#21|
Hi, this old threads interets me.
Maxwell equation for electromagnetic potential are four formula for four unknowns. we can solve this set of equations allowing gauge freedom.
Is gauge freedom is originated from the fact the four equations are redundant thus only three or less number of equations are independent? Or gauge freedom has nothing to do with independence of the four formula? Advise, please
|Sep18-12, 11:09 AM||#22|
I know this thread is old but I have done some analysis and I need to correct my earlier comments. Maxwell's equations contain six unknowns: Ex, Ey, Ez, Bx, By, Bz and seem to contain a linear set of differential equations consisting of 8 scalar equations. The problem therefore seems overspecified. But in fact, the curl equations of Maxwell's equations (Faraday's law and the Maxwell-Ampere law) contain the other equations (Coulomb's law and the no-magnetic-charge law) up to an arbitary unknown function that is constant in time. The six scalar equations contained in the Maxwell curl equations are therefore the complete set of dynamical linear differential equations. We therefore actually have six equations in six unknowns. The two divergence equations in Maxwell's equations are not useless though - they are needed to uniquely determine the extra term that the curl equations cannot determine. From a mathematical standpoint, the two divergence equations of Maxwell's equations are mere initial conditions, not part of the linear system of dynamical equations. Mathematically speaking then, Maxwell's equations contain six scalar equations in six unknowns (Faraday's law and the Maxwell-Ampere law) as well as initial conditions (Coulomb's law and the no-magnetic-charge law), and all is well. There is no over-specification. Perhaps the confusion arises because the divergence equations are pseudo-static (or instantaneous) so that the initial condition constraints end up being valid for all time, so they don't look like initial conditions. I go through all the math here:
|Sep18-12, 11:32 AM||#23|
sweet_springs: Gauge freedom is not a result of some kind of redundancy in Maxwell's equations for the fields, because there is no redundancy in Maxwell's equations for the fields. Gauge freedom is a result of the fact that we are defining the fields in terms of something else non-physical (the potentials). Because the entities are non-physical, we are free to define them however we want (within the usual mathematical constraints of well-behaved functions, etc.) That is the origin of gauge freedom. As an analogy, consider the system of equations:
x2+y2 = 10
xy = 5
I could solve this system outright. But I might find it easier to define the unknowns in terms of new variables. For instance, if I define u = x + y, v = x - y, my system of equations reduces down to:
u2 = 20
v2 = 0
But there is nothing special about my transformation aside from the fact that it makes the math easier. I could have just as easily made the transformation u = x2, v = x2 and arrived at the system:
u + v = 10
uv = 25
I am free to define in terms of new variables how ever I want as long as the new variables do not correspond to physical properties, because they may then be constrained by other laws. So at its heart, gauge freedom is simply a result of the electromagnetic potentials having no physical meaning. Certain choices for the potentials (certain gauges) make the math easier, so they get special attention. Note these concepts are within a classical framework. When you add on quantum considerations (such as the Aharonov-Bohm effect), you can make the argument that the potentials are physical and things get more complicated.
|Sep18-12, 11:19 PM||#24|
Your post (#22) is very similar to my post (#18); however, I like how you refer to the two divergence equations as "initial conditions" (perhaps "boundary conditions" would be a better term, but that's just semantics!). In this light, there seems to be a very important point, which is implicit in our discussions, that we both glossed over. That point is that to derive Gauss' Law from Ampere's Law, one must invoke the Continuity Equation.
But where does the Continuity Equation come from? If we take the two curl equations of Maxwell to be axioms and the two divergence equations of Maxwell to be derived, then the Continuity Equation is an additional axiom that we must add to make the theory physical. But we should note that the Continuity Equation is a relation among the sources, i.e. among our "independent variables" -- it does not add to the "list of equations & unknowns." It seems then that we may consider the Continuity Equation to be a boundary condition that must always prevail.
From the standpoint of applying the Helmholtz Theorem, having equations that explicitly contain the field vectors, or perhaps just for some sort or aesthetic beauty: we consider the fundamental equations of electromagnetism to be the Laws of Gauss, Faraday, Ampere, and the Absence of Magnetic Charge. But perhaps for mathematical simplicity, we should consider the fundamental equations to be the Laws of Faraday and Ampere plus the Continuity Equation.
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