Why I doubt the generality of Gauss' law: A Gaussian sphere 1 light year across

[Matterwave]

But the source free Maxwell equations can't have a charge inside your surface...? So how does that apply to Gauss's law in what we are talking about here? I guess I don't quite get your argument then.

 

[Ben Niehoff]

The source-free Maxwell equations apply to everywhere where the charge isn't. That is, ALL of space except for the one point where the charge is.

Or in other words, it so happens that J=0 except at one point.

 

[Matterwave]

Certainly one would like Gauss's law to hold even if you are inside a continuous charge distribution though. How does that work then in that case?

 

[Ben Niehoff]

Right. So, with sources, we have

d * F = * J
So now, merely integrate this over some 3-volume V:

\begin{align*} \int_V d * F &= \int_V * J \\ \int_{\partial V} * F &= \int_V * J \end{align*}
which is Gauss' Law.

The reason I focused on the source-free equations, is because it is only when J = 0 that the result of integration doesn't care about the choice of surface. Obviously, if J \neq 0, then different surfaces might contain different amounts of charge, hence giving different results. Gauss' Law still holds, though.

 

[Matterwave]

I guess my question, what does F being harmonic have to do with field lines? It seems to me that Stokes theorem is sufficient in both (empty space and continuous charge) cases to justify the use of field lines.

 

[ARAVIND113122]

i completely agree with kmarinas86. the field SHOULD continue to change for 1 year[approx].so far,none of the members have posted a convincing reason why it shouldn't.after all,information does take time to travel,and nothing happens instantaneously.once the charge is taken out of the sphere,the 'data' from the other side of the sphere would take time[almost 1 year] to reach the side that was initially closer to the charge

 

[Ben Niehoff]

I guess my question, what does F being harmonic have to do with field lines? It seems to me that Stokes theorem is sufficient in both (empty space and continuous charge) cases to justify the use of field lines.

You're right. Harmonic forms are special for other reasons that don't necessarily apply here.

 

[Dale]

i completely agree with kmarinas86. the field SHOULD continue to change for 1 year[approx].so far,none of the members have posted a convincing reason why it shouldn't.after all,information does take time to travel,and nothing happens instantaneously.once the charge is taken out of the sphere,the 'data' from the other side of the sphere would take time[almost 1 year] to reach the side that was initially closer to the charge

The Lienard Wiechert potential is the convincing reason. The LW potential depends only on the motion of the charge at the retarded time. So the field at the near side updates with ~1 ns delay and the field at the far side updates with ~1 year delay. The field does not change outside of the times as indicated above.

If you are not willing to accept standard solutions like the LW potential, then there is little that can or should be done to convince you.