Thread: Gauss' Law for spherical shell vs Coulomb's law, regarding reativity

Shalom

We are used to hearing that Coulomb's law doesn't settle with the relativity principle that nothing moves faster than the speed of light, in the sence that it embeds 'Action in a Distance'. Meaning that if somthing changes in r1 at time t1, and we write the law for any t before t1+(|r1-r2|/c), (r2 is the where the test charge is) then the law doesn't represent reality, because the 'knowledge' about the change hasn't reached r2 yet.

And it is often said that Gauss' law fixes that because of its local nature. But what I can't figure out is:

(1) how does one settle that with one of the famous implementations of Gauss' law, the one for a spherical shell with a charge in its center. When we use the Divergance law to find the same form of Coulomb's law, resulting from Gauss' Law. How does this implementation not violate the relativity principle, violated by Coulomb's law (nothing travels faster than the speed of light)?

(2) another related question is how does Gauss' law, or Maxwell's laws express that the information about the change in r1 travels at speed c? (other than the wave equation please, and other than being local). I just can't see how Gauss' law shows that principle, which Coulomb's law couldn't.

And I would really love to hear from anyone who might have a better vision and be able to see what I seem to be missing here.

Thank you in advance

(2) another related question is how does Gauss' law, or Maxwell's laws express that the information about the change in r1 travels at speed c?

The Maxwell equations already imply the electromagnetic wave equations, which in turn imply that the propagation speed is always exactly c. This can easily be shown mathematically.

(1) how does one settle that with one of the famous implementations of Gauss' law, the one for a spherical shell with a charge in its center. When we use the Divergance law to find the same form of Coulomb's law, resulting from Gauss' Law. How does this implementation not violate the relativity principle, violated by Coulomb's law (nothing travels faster than the speed of light)?

Again, for the same reason. Static fields do not require the exchange of information, whereas changes can only propagate at c. Remember that Gauss's law and Coulomb's law are not enough to describe the behaviour of the fields; one needs to consider the entire set of Maxwell equations. It can be shown that these equations do not allow changes in the fields to propagate at anything other than c.

An alternative approach to the whole thing would be to use QED instead of Maxwell's equations. The physical implications are the same.

Originally Posted by Markus Hanke

The Maxwell equations already imply the electromagnetic wave equations, which in turn imply that the propagation speed is always exactly c. This can easily be shown mathematically.



Again, for the same reason. Static fields do not require the exchange of information, whereas changes can only propagate at c. Remember that Gauss's law and Coulomb's law are not enough to describe the behaviour of the fields; one needs to consider the entire set of Maxwell equations. It can be shown that these equations do not allow changes in the fields to propagate at anything other than c.

An alternative approach to the whole thing would be to use QED instead of Maxwell's equations. The physical implications are the same.

Thank you, I feel I should clarify the question a bit:

1. I know how to derive the wave equation with speed c from Maxwell's equations. But I'd like to know if there's another approach to see that Maxwell's laws DO express the limitaion on the ability of information to propagate no faster than light, as opposed to coulomb's law which doesn't.

2. My focus on Gauss' law is in its implementation on a sphrical shell, using the divergence theorem, to derive a formula which is identical to Coulomb's law. i would very much like to know if when doing that onelimits the case to static states, thereby waiving all the fundamental difference between Gauss' law and Coulomb's law, and emptying Gauss' law from its advantage.

thank you in advance

Originally Posted by Ammah

Thank you, I feel I should clarify the question a bit:

1. I know how to derive the wave equation with speed c from Maxwell's equations. But I'd like to know if there's another approach to see that Maxwell's laws DO express the limitaion on the ability of information to propagate no faster than light, as opposed to coulomb's law which doesn't.

2. My focus on Gauss' law is in its implementation on a sphrical shell, using the divergence theorem, to derive a formula which is identical to Coulomb's law. i would very much like to know if when doing that onelimits the case to static states, thereby waiving all the fundamental difference between Gauss' law and Coulomb's law, and emptying Gauss' law from its advantage.

thank you in advance

I'm sorry, but I really don't know what you are getting at. The changes in non-static fields are electromagnetic waves, so the very fact that the equations for them follow from Maxwell's laws already proves the point. The only other way I can think of is via the differential forms formalism of the Maxwell equations :





Since the Hodge star explicitely depends on the metric of the space-time in question, this pretty much rules out any instantaneous information transfer. I admit though that I'd be hard pressed to prove that formally

Originally Posted by Ammah

2. My focus on Gauss' law is in its implementation on a sphrical shell, using the divergence theorem, to derive a formula which is identical to Coulomb's law. i would very much like to know if when doing that onelimits the case to static states, thereby waiving all the fundamental difference between Gauss' law and Coulomb's law, and emptying Gauss' law from its advantage.

thank you in advance

Since Gauss's law applies in any frame you may set up, it is valid in complicated situations. However, in other than static frames getting that information out in usable form may be impracticable. In particular, the derivation of Coulomb's Law from Gauss's Law uses a static symmetry argument. You choose the Gaussian surface to be a sphere whose center is at the point charge. Then the electric field must be the same at all points on the sphere because of the rotational symmetry of the setup. Given a constant electric field you can remove its value from the integral over the surface, and Coulomb's Law pops out. If the charge is moving in the frame you have chosen, the symmetry doesn't hold since it is broken by the arbitrary direction of the charge's movement and, except for one instant, by the charge not being at the center of the sphere. Moreover, the value of the charge of the point charge cannot change because of charge conservation. So even though Gauss's Law isn't static, its use to derive Coulomb's Law is static.

Gauss's Law in a nonstatic system isn't all that simple anyway. The integral over the surface is to be taken over the entire surface at a constant time in the reference frame. For an observer in that frame but not at the center of a spherical surface, the information about these values arrives at different times since it is carried by light traveling different distances at a constant speed. Relativity is almost never simple.