Electric Field Induced by Varying Magnetic Field

[Dale]

it is my understanding that experiment has shown time variable charge (which is also a current) creates fields and that those fields travel outward from that source at the speed of light. In other words the fields are "retarded". That includes E, D, B, H, A etc all of which are retarded at a distance from the current sources. Thus your assertion that fields and charges all exist at the same time is not correct.

And how do you obtain a current? Typically by using a field, usually either an electric field in a battery or a magnetic field in a generator.

Locally, it is clear that the fields and charges do all exist at the same time. And from the structure of the equations themselves there is no reason for you to single out one and say that this causes that but the rest can't cause each other because they are simultaneous. Once you stop looking at the equations and start looking at the retarded solutions then you can still have that a retarded field causes a current or a charge, in fact that is how receiving a radio signal works.

Again, I understand your point, but I think that you are applying it inconsistently when you say categorically that charges and currents are the cause and fields are the effect and that fields can't cause each other because they are simultaneous.

 

[kcdodd]

First, this is wrong:The A potential can't be specified consistently over all spacetime with randomly chosen values, but over a space-like 3D submanifold--for instance, the lab frame at some time, t.I made a notational error. A = (phi,A), where phi is the electric potential and A is the magnetic potential.

All classical electromagnetism can be delt with using exterior derivatives and differential forms in four dimensions. My internet connection is acting up or I would give you a specific link. Wikipedia has some mention of maxwell's equations in differential forms, I believe.
The actual equations are simple and elegant but meaningless if you don't know the notation first.

But notice that A(t,x,y,z) = (phi,A)(t,x,y,z) is a true Lorentz invariant vector field over spacetime.

J(t,x,y,z) = (rho,J)(t,x,y,z) is also a Lorentz invariant vector field over spacetime. J is the current density and rho is the charge density.

The electric and magnetic fields are first derivatives of A. One solution of the derivatives of the electric and magnetic fields results in the wave equations of light. So the wave equations of light are third(edited) derivatives of the 4-vector field, A(t,x,y,z).

(The vacuum wave equation is expressed as 0=d*d*dA (edited). Elegent but meaningless without the decoder ring.)

What can be done with A can be done with J except for the fact that charge is attached to mass. Naively ignoring this for a moment we would conclude that wave equations in J propagate at the same velocity as those of A. But, of course, they don't.

Yes, please post a reference because I have never heard of a matter wave interpretation. And I believe that the wave equations are the second, not third, derivative of A. In fact I don't know of any third derivatives anywhere.

And how do you obtain a current? Typically by using a field, usually either an electric field in a battery or a magnetic field in a generator.

Locally, it is clear that the fields and charges do all exist at the same time. And from the structure of the equations themselves there is no reason for you to single out one and say that this causes that but the rest can't cause each other because they are simultaneous. Once you stop looking at the equations and start looking at the retarded solutions then you can still have that a retarded field causes a current or a charge, in fact that is how receiving a radio signal works.

Again, I understand your point, but I think that you are applying it inconsistently when you say categorically that charges and currents are the cause and fields are the effect and that fields can't cause each other because they are simultaneous.

I would like to point again to a post I made earlier...

\vec{E} = -\nabla\int_v d^3r'\frac{\rho}{4\pi \epsilon_0|r' - r|} - \nabla\times \int_v d^3r'\frac{\partial_t \vec{B}}{4\pi |r' - r|}

\vec{B} = \nabla\times \int_v d^3r'\frac{\mu_0 J}{4\pi |r' - r|} + \nabla\times \int_v d^3r'\frac{\mu_0 \epsilon_0 \partial_t \vec{E}}{4\pi |r' - r|}

I am fairly sure this is accurate, with the caveat that this is to be integrated using a retarded time. In other words, the global field as viewed from a point must be self consistent within the past light cone of that point. helmholtz theorem really would not make sense otherwise (you surly can't integrate outside the past light cone). There are two interpretations to this. Either (a) all field and sources are "simultaneous" within the past light cone (we are simply solving a differential equation on a space-time surface), or (b) all fields and sources on the past light cone "caused" the field value at the point to which the cone is attached (since the entire integral exists in the past of that point). It seems both are valid views, but both exist for fields and sources, not just one.

 

[Phrak]

Yes, please post a reference because I have never heard of a matter wave interpretation. And I believe that the wave equations are the second, not third, derivative of A. In fact I don't know of any third derivatives anywhere.

Please don't misunderstand me. The mater waves--actually, charge waves, are unphysical; they only occur with massless charge. They have come up, or should eventually come up in solid state physics, a branch of physics I know nothing about. Maybe Zapper Z can comment.

Second derivatives with respect to space and time of E and B, d'Alembertian E=0 and d'Alembertian B[/b]=0, are the vacuum equations, of course. E and B are both first derivatives of A and phi. So the wave equation is third order in A.

But nevermind all that. I've become curious about your previous post. I don't know what the physical significance of d'Alembertian A is. Any ideas?

I know that there is one second derivative of A--being first derivative in E and B, that yields the charge continuity equation. I don't know if there are others.

 

[Dale]

Either (a) all field and sources are "simultaneous" within the past light cone (we are simply solving a differential equation on a space-time surface), or (b) all fields and sources on the past light cone "caused" the field value at the point to which the cone is attached (since the entire integral exists in the past of that point). It seems both are valid views, but both exist for fields and sources, not just one.

My interpretation would be closer to b. The definition of "simultaneous" described by a is not symmetric (meaning that if event A is simultaneous with B that does not imply that B is simultaneous with A), which is not the usual way to think of simultaneity.

 

[kcdodd]

Phrak: When someone says the EM wave equation, this is what I think of. Not of E and B, but of A.

\nabla^2\phi - \partial^2_t\phi = -\rho
\nabla^2\vec{A} - \partial^2_t\vec{A} = -\vec{J}

DaleSpam:

I see your point. The value of B depends on A, but not the other way around, since we don't take the advanced solution.

 

[Phrak]

Phrak: When someone says the EM wave equation, this is what I think of. Not of E and B, but of A.

\nabla^2\phi - \partial^2_t\phi = -\rho
\nabla^2\vec{A} - \partial^2_t\vec{A} = -\vec{J}

I could be horribly mistaken, but though these are certainly wave equations, they are not the wave equations of light.

For instance, the homogeneous solution of the wave equation of E is,

\Box \textbf{E} = 0 \ .

\textbf{E} = - \nabla \phi - \partial _t \textbf{A}

\Box (\nabla \phi + \partial _t \textbf{A}) = 0

The last equation is certainly third order in all terms.

It's interesting to note the authors of electromagnetic wave equation on Wikipedia agree with you.

 

[kcdodd]

I see your point. This is straight out of Jackson, and he seems to side-step the peculiarity of the inhomogeneous wave equations for E and B. (I changed to natural units)

\Box \vec{E} = \nabla \rho + \partial_t J
\Box \vec{B} = - \nabla \times \vec{J}

Which he states has the solution of Jefimenko's equations. So I suppose you are correct that A is implicitly third order here. Which I suppose is ok since the sources are now first order. I'm sure there is some deep meaning here, like the derivative of a wave is another wave but it is getting late and I can't recall a specific relation.

 

[Phrak]

kcdodd, you've still brought up an interesting equation, none the less. At one time I was catalouging Lorentz invariant (with other nice properties) equations and gauge invariances of a vector field imposed on spacetime. Somehow I seem to have missed the wave equation for a generalization of the potential, A=(A,phi), or it's not Lorentz invariant (nor connection free). So, thanks for that.

 

[Dale]

To my understanding, the Lorentz gauge condition is Lorentz invariant (but not many other gauge conditions, e.g. the Coulomb gauge condition is not Lorentz invariant). I don't know about the connection. It would be nice to know if the Lorentz gauge is unique in being Lorentz invariant, or if it is one of a whole class of gauge conditions which are Lorentz invariant, but that is beyond my knowledge currently.

 

[Phrak]

Dale, my apologies for being misleading. I know remarkably little about electromagnetism in vector calculus and gauge fixing--or even electromagnetism in mixed tensor notation. As far as I can tell, everything in electromagnetism is expressible as antisymmetric covariant tensors in its native four dimensions. Equations written in these are by design, Lorentz invariant and connection free. So this is what I've studied instead of the tools of vector calculus.

Every tensor field that is over scalar-valued has a regauging field of one less index--and it may be possible to bend the rule on scalars, come to think of it. The charge-current density for instance, can be expressed as either a covariant vector or a tensor of three lower indices.
The last is regaugable by an electromagnetic field tensor.

In generalizing the covariant 4-vector potential to complex entries, an interesting effects on regauging occurs; the imaginary part of the vector potential regauges the electromagnetic field tensor, and the real part regauges it's dual, G_{mu nu} = (1/2)epsilon_{mu nu}^{rho sigma}F_{rho sigma}. Maxwell's equations obtain their maximum symmetry in a single expression, magnetic monopoles appear, and now the problem is getting rid of them.

A scattering of other symmetries and conservation laws come up as well whether using real or complex fields, all beginning with the simple notion of applying a 4-vector field to spacetime.

Sorry for all the greek. The first two chapters of Sean Carroll's Notes on General Relativity provide the background for all this. As much as the 4-velocity provides the unification of mass, energy and momentum, the covariant 4-vector potential does the same for electromagnetism.

 

[Dale]

As much as the 4-velocity provides the unification of mass, energy and momentum, the covariant 4-vector potential does the same for electromagnetism.

Hmm, I should look into that more. It sounds like something that I would enjoy.

 

[Dunnis]

Phrak: When someone says the EM wave equation, this is what I think of. Not of E and B, but of A.

\nabla^2\phi - \partial^2_t\phi = -\rho
\nabla^2\vec{A} - \partial^2_t\vec{A} = -\vec{J}

Maxwell: -"To find the rate of propagation of transverse vibrations through the elastic medium, on the supposition that its elasticity is due entirely to forces acting between pairs of particles

V=\sqrt{m\over p}

where 'm' is the coefficient of transverse elasticity, and 'p' is the density."
The "wave equation" you are talking about is nothing else but the 'wave equation for vibrating string': - "The speed of propagation of a wave in a string (v) is proportional to the square root of the tension of the string (T) and inversely proportional to the square root of the linear mass (μ) of the string:

\frac{\partial^2 y}{\partial x^2}=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2} => v = \sqrt{T \over \mu}

". - http://en.wikipedia.org/wiki/Vibrating_string

 

[Phrak]

Hmm, I should look into that more. It sounds like something that I would enjoy.

I hope you do. I don't know anyone conversant in this.