Can Electric Fields be Discontinuous? - Conservation of Charge

[mfduqued]

I show the conservation of charge from the equations of MAxwell, but I suppose that an electric field is continuous at space and time, therefore I can exchange the derivates from Gauss' law, at time's derivate of this law..

My question is, At nature Can I have an electric field discontinued, therefore I can't exchange the derivates?

 

[mfduqued]

Hi, my question What is more fundamental, the maxwall'equations or charge conservation?, because I obtain from gass' law and Ampre-Maxwell's law of charge conservation, but this law is appart from other laws.

Then I can say, but I can prove this law (charge conservation) in continuous situations, this law is apart from the other equations of Maxwell?

 

[jambaugh]

The "what is more fundamental" question an be tricky, "fundamental in what sense?" is the begged question.

From an empirical operational point of view we measure E-M fields and thence confirm Maxwell's equations using test charges and so we must first observe charge as a conserved quantity.

From a theory development perspective you have a set of mutually necessary conditions and you can pick where you start and so pick what is fundamental vs what is derived however the current modern approach is to start with a gauge symmetry which in a very immediate way implies a conservation principle via Noether's Theorem. One then derives Maxwell's equations. But there is no reason to believe the gauge approach is starting at a more fundamental level in some other sense. It is simply a means of encoding a class of theories which contain certain common and commonly observed features such as conservation of charge.

At a metaphysical level the universe simply is as it is an doesn't order the features of nature as more an less fundamental, that is really an ordering we impose based on the way we conceptually decompose natural phenomena to better understand it.

We can view charge as geometric artifacts of the fields, or the fields as propagation of charge displacements and it becomes a chicken and egg loop.

That having been said, the answer to your questions is... most definitely charge conservation is more fundamental!

 

[mfduqued]

Thank jambaugh for your response,

But I have a question,
I always can to obtain the equation of charge conservation, from Maxwell's equations, although the field is not continuous at space-time?

This subject is because I think at some situations to obtain the charge conservation is not easy, as,
$\frac{∂∇\bullet D}{∂t}=\frac{∂ρ}{ε_0 ∂t}=\frac{∇\bullet ∂D}{∂t}$
and I use the Ampere-Maxwell 's equation

 

[physwizard]

I show the conservation of charge from the equations of MAxwell, but I suppose that an electric field is continuous at space and time, therefore I can exchange the derivates from Gauss' law, at time's derivate of this law..

My question is, At nature Can I have an electric field discontinued, therefore I can't exchange the derivates?

If the field is discontinuous you would not even be able to differentiate it, leave alone exchange derivatives. So you would not be able to apply the Maxwell equations in differential form, leave alone charge conservation. In practice most of the situations would be where the field is continuous everywhere except at a few isolated points where it may not be due to the point charge concept. If you are worried about charge conservation in such cases, you don't have to be, because you can integrate the current density over an arbitrarity small closed surface surrounding the point and use the integral form of Maxwell's equations to prove charge conservation around that point.
Frankly, I would suggest focus more on the Physics, and don't get boggled too much by mathematical technicalities.

 

[mfduqued]

Thank physwizard for your response.

Yes, I had this doubt when I couldn't use the Maxwell's Equations Differential.

 

[physwizard]

Frankly, I am not an expert on this subject. In the case of the field blowing up at a few isolated points, I feel that you can still continue to use Maxwell's equations in differential form and the concept of delta functions will facilitate this.
You could try reading up a bit on delta functions and the theory of distributions.
You have raised a valid point and I encourage you to pursue it further. If you gain a greater understanding of this, do post here on the forum!