Dear Sofia,

    If you make change the electric field in time, you obtain a magnetic field changing in space and an electric current. Therefore the things are more complex that you said. In fact Gauss law is only for electric field no dependent of time.

First of all, your question is excellent !

The general answer to this question is that neither the differential form of the laws, nor the integral, is absolutely better than the other form. The choice of the form depends on the configuration that we have to solve.

In what follows I want to call attention on a problem that we have with the Gauss' law if the electro-magnetic field is time-varying. I stress that the analysis below is restricted to the Gauss' law, and doesn't say that the differential form of the Maxwell laws is generally better than the integral form.

Consider a small volume V delimited by a closed surface S. The integral Gauss law says

(1) ∫S E • n dS =  ∫V ∇E dV,

where n is the unit vector normal to the element of surface dS. We also denote,

(2) ∇E = ρ / ε . 

where ρ is the charge density. The law (1) is derived from the so-called divergence theorem (see Wikipedia), which is due to Lagrange (1762) then rediscovered by Gauss in 1813, in 1826 by Ostrogradsky, and there are others too. So, as you say, it appeared in integral form first. However, nothing in (1) says what happens if E varies in time.

    Then, let's see together what happens if E is time dependent. For this purpose we'll take a small rectangular volume in which the field E is along the direction z. Two vertical surfaces S1 and S2 of this volume are perpendicular to the field, are separated by a distance ∆z, and have area ∆x∆y. The other surfaces are parallel to E. In this case the equality (1) becomes, after dividing both sides by ∆x∆y∆z, 

(3) [E(x + ∆x, t) - E(x, t)] / ∆z =  ρ(t) / ε .

It's almost the differential form of Gauss' law.

Though, there is a problem hidden here: if at the time t the field changes on the surface S1 (i.e. at x), what is the field on S2 (i.e. at x+∆x)? And what is the value of the charge density ρ? Well, it takes a time ∆t = ∆x/v for the change in the field to reach the surface at x+∆x, where v is the velocity of the electrical signal. At the end of this time-interval we will have the charge density ρ(t+∆t). Though, during this time-interval, on the surface S1 the field is likely to change again. Therefore, ρ(t+∆t) is in fact given by the relation

(4) [E(x + ∆x, t+∆t) - E(x, t)] / ∆z =  ρ(t+∆t) / ε .

It appears that the differential Gauss' law is better in cases where the field varies rapidly in time, despite the historical order of the development, because in the times when the integral form was introduced the finite velocity of the light was ignored.

Please note again: this answer doesn't treat the other Maxwell laws, the aim here is only to show a particular problem with the integral form. There is no aim at cutting that the differential form is generally better, in all the possible cases, than the integral form.

Dear Sofia,

thanks for your answer.. (1) ∫S E • n dS =  ∫v ρdV

"it takes a time ∆t = ∆x/v for the change in the field to reach the surface at x+∆x"

yes the reconfiguration should take time. There is a dynamic, a delay according to the speed of light. Usually infact during the transitory a EM wave is generated.

"in the times when the integral form was introduced the finite velocity of the light was ignored."

this is for sure...

So according to your opinion at least for the static electric field law, the differential form should be preferred because it is not influenced by the reconfiguration speed. While the Gauss' law cannot have a physical counterpart because it is something defined in a limited volume (charge) which affects something at any indefinite distance (field over a surface), like the Newton's law..

    Dear Sofia,

    If you make change the electric field in time, you obtain a magnetic field changing in space and an electric current. Therefore the things are more complex that you said. In fact Gauss law is only for electric field no dependent of time.

Dear Stefano,

As you saw, even if we consider a small volume we have problems with the integral form of Gauss' law. Now, imagine yourself what would happen if we would have a big volume surrounded by a surface and the field throughout the volume and surface would vary in time. Definitely the differential form is better for such cases. I don't know in detail the history of the integral form, but I guess that it was developed for fields which are static in time.

Now, you mention Newton's law. I don't know any form of the divergence theorem for gravitational field, but it seems to me logical that it should exist, and it would be very interesting to find real examples of that. So, if you'd meet such examples, I'd be glad to hear from you.

Dear peoples,

An integral law like that of Gauss can only say what was the situation inside the Gaussian surface in a time-retarded sort of way. The same with the integral form of Faraday's law where the magnetic field could change inside a Faraday loop due to changes in current density but the emf around the loop wouldn't immediately respond. The integrals laws could the thought of as "local" in the sense that they say what is going on at the surface or loop in their reference frame. I realize this observation is not really an addition to the discussion so much as just an "in other words".

Dear Sofia and Stefano,

Let me try to answer your question in more detail, trying to avoid ambiguities:

1. The equation div D=0 only contains the information of the Coulomb law where the velocity of propagation of the electromagnetic field is assumed to be infinite as in electrostatics (Poisson equation). All the textbooks of electromagnetism starts with the definition of the electric field and its static equations. The same considerations could be made for div B=0.

2. Maxwell notice that if he introduced the displacement currents in Ampere's equation, then he could use the Gauss law for obtaining the local conservation of the charge. And what is more important, he could couple the Faraday and Ampere modified, for giving a wave equation, where the electromagnetic field would have a finite velocity well known in optics at those times.

3. Another point that the question touches is that in fact Maxwell equations are integral equations instead of differential equations. This is true because there no simple differential manifolds where the differential equations do not give all the information. For instance, we could have locally that div B=0 if we used non simple connected spacetime although globally we had magnetic monopoles. Notice that in this case the Gauss integral theorem couldn't be applied as Stefano wrote where the div operator is substituted by the boundary operator of the manifold of integration.

4. But the magnetic monopole is not the only case, there are everyday cases. For instance, let put wires carrying electric current which cross making knots, try to calculate the magnetic field using only the differential Maxwell equations and you would see inmetidatally that you need to put them in integral form.

I'm a bit confused by this thread.  Div E=rho is true whether time dependent or not.  The integral equations are consistent with this and causality.  If you go to nontrivial topology then you can end up with restrictions on the kinds of bounding surface used in the integral form and the particular charge distributions that are consistent with the equations e.g. you need equal positive and negative charge on an S^3 space.  Modulo these concerns I think these forms are still equivalent.  

There are dynamical and constraint elements of Maxwell's equations by the dynamical ones propagate the constraints assuming you start with them holding.  If you vary your surfaces as above what it is telling you is that you have a current carrying charge into your volume.  \partial_\mu j^\mu=0 is one of the consistency constraints.  

When I first saw this thread I though it would be a discussion on dipole changing effects like the Shockley-James paradox or the angular momentum paradox.  How about the Feynman disk?  

Dear Clifford,

My opinion of the Shockley-James paradox  is that it is rather trivial if you consider how the charge-magnetic dipole system is assembled. You can start with the charged disks not rotating and the charges at "infinity". There is no momentum in these components of the system at this point. You employ an external agent to start the disks rotating and bring in the charges to their assumed locations. If you calculate the momentum the agent has to acquire to perform this assembly by imparting momentum to the system, it is exactly the same as the so-called (and non-existent in my view) hidden momentum in the system. I don't understand (1) why no one has seen this before and (2) why it is so opposed by many physicists. They claim a system that is at rest and contains electromagnetic momentum has to have hidden momentum. To me, this is saying that the system has been created by magic, perhaps by Merlin or Tinker Bell? I have used the same type of calculations to show that there is no hidden momentum in many of the models, for example, those by Giffiths, Babson, and others, for which hidden momentum is claimed. Getting this published is another matter.

The key to most of these is in keeping track of the electromagnetic momentum and fringe fields.  It is not so trivial though.  There is a delocalized field that contributes to the p of the charge itself so disentangling this can be tricky.  I believe Babson et al only claimed the momentum of a parallel plate capacitor in a magnetic field was "hidden" in the fringe fields.  Not really hidden as much a boundary condition error for naive calculations.  I know there was some discussion of lag times in when the forces were applied by Boyer to get the correct forces.  

Actually, the angular momentum paradox continues to perplex me.  I've seen some attempts at it but haven't come to a final resolution for myself.  

Clifford,

I don't recall anything in Babson et al about fringe fields. I do remember that they claim the hidden momentum is mechanical. But, wherever it is, i claim that you can't ignore how a system comes into being. If an electromagnetic system "at rest" and containing EM momentum has to also contain hidden momentum such that when the EM momentum is released the system remains stationary, you ignore the fact that some agent had to have assembled it in the first place and, if the agent kept the system stationary, there was an exchange of momentum between the agent and the system during assembly. The system doesn't just pop out of a clear blue sky with its energy and momentum acquired by fiat.

Take for example the system of Shockley and James. Start with a positive and negative charge very far from the rotating, charged disks, but connected to the "pill box" in such a way that the system can hold its shape. At this point there is neither linear nor angular momentum in these components. Move the charges to the positions described by the model. When this is done there are Lorentz forces acting on the charges due to motion through the magnetic field and hence impulses applied to the charges. There is also an increasing displacement current at the site of the disks due to the motion of the charges which produces a Lorentz impulse on the disks equal to that on the charges and in the same direction. So, the system acquires mechanical momentum upon assembly (unless some agent holds it stationary). However, the system also contains EM momentum equal and opposite to the mechanical momentum, hence total momentum is zero. When you look at the system in its rest frame, you are not looking at it in the same reference frame in which it was assembled. My calculations show that when the disks are braked to a stop, the system returns to rest in its original rest frame due to the forces acting on the charges from the collapse of the magnetic field (and loss of EM momentum). No force acts on the disks (unless, as shown by Coleman and Van Vleck, the current consists, not of charges on rigid rotating disks, but charges moving in a circle on frictionless paths; when that occurs relativistic kinetic energy effects come into play, but this does not save momentum conservation).

I should look at his article again someday.  I remember his talk when he first found his result (we were in grad school together) which came down to spotting a mistake in Griffiths.  I think Griffiths then cranked out some rotating sphere model.  The fringe fields are definitely key though for the problem with an error.  I'm very familiar with all these examples.  Should try to write them up someday but not sure there is anything really new to say.  

Dear Sofia,

I agree with Daniel,

"And what is more important, he could couple the Faraday and Ampere modified, for giving a wave equation, where the electromagnetic field would have a finite velocity well known in optics at those times."

yes the first two equations are just static and this avoids to consider any speed of perturbations...

THere are similarities between E-M laws with the Stokes-Navier equations of fluid dynamics.

In fluid dynamics the analitical functions (functions of complex variable) are used in order to model the field. In E-M this opportunity is not exploited at all, complex variables are used only to model oscillations.

The similarity of a circuitation of B around a wire  determinig the J, with value of the residues of an analitical functions found around its singularities is quite evident.

Clifford,

I think i now recall the fringe fields as being mentioned in the capacitor inside a solenoid study, but my impression is that those who argue for hidden momentum claim it is mechanical as did Coleman and Van Vleck. (However i think i've got off topic here in this string, so i'm shutting up. :-)

I think we are missing a lot about Quantum mechanics.....and maybe with the original Hamilton's Quaternion formalism we can  renew the junction between Maxwell equations and quantum mechanics which is already in place in some other formalisms.

Dear All,

a very interesting paper from W. Engelhardt...

Article Relativity of time and instantaneous interaction of charged particles