Gauss's law can be proved qualitatively by proving that the field inside a charged closed surface is zero. However Maxwells' equations says that gauss's law holds true even for electrodynamics. how can this be verified experimentally? Thanks in advance !
Gauss's law, is a specific case of Stokes's theorem. http://en.wikipedia.org/wiki/Stoke's_theorem edit: I interpreted Gauss's law to mean the divergence theorem, which is a mathematical statement. My mistake; that would probably be called Gauss's theorem.
Gauss' law is a law of physics that relates electric charges to electric fields. Stoke's theorem is a purely mathematical statement, like the commutative property of addition.
I am not good in definitions but I did look into Gauss Law. I really don't see the relation of Stokes and Guass. Even in Guass law for magnetism: http://en.wikipedia.org/wiki/Gauss%27s_law_for_magnetism It only said [itex]\nabla \cdot \vec B = 0\; [/itex] where it states there is no mono magnetic pole. Guass law is mainly used in Divergence theorem where [itex]\nabla \cdot \vec E=\frac {\rho_v}{\epsilon}[/itex] Where: [tex]\int_v \nabla\cdot \vec E dv'=\int_s \vec E\cdot d\vec s'=\frac Q {\epsilon}[/tex] http://phy214uhart.wikispaces.com/Gauss%27+Law http://phy214uhart.wikispaces.com/Gauss%27+Law The only one that remotely relate magnetic field through a surface is: [tex] \int_s \nabla X\vec B\cdot d\vec s'=\int_c \vec B \cdot d \vec l'= \mu I [/tex] that relate current loop with field through the loop.
1. The charged closed surface must be a conductor. 2. I don't know of any direct experimental test for a time varying E field. The fact that its inclusion in Maxwell's equations leads to many verifiable results is an indirect proof of its general validity.
I just want to say that gauss law follow immediately from maxwell's fourth eqn when combined with continuity eqn for charge density.(just take the divergence)
I can't think of any direct prove on Guass surface with varying charge inside. But I cannot see anything wrong that the total electric field radiate out of a closed surface varying due to vary charge enclosed by the closed surface still obey [itex]\int_s \vec E\cdot d\vec s'[/itex]. The difference is with varying charges generating the varying electric field, a magnetic field MUST be generated to accompany the varying electric field according to: [tex]\nabla X \vec E=-\frac{\partial \vec B}{\partial t}[/tex]
Let us see, c^{2}(∇×B)=j/ε_{0}+∂E/∂t now, c^{2}{∇.(∇×B)}=∇.j/ε_{0}+∂(∇.E)/∂t USING ∇.j=-∂ρ/∂t and the fact that gradient of curl vanishes. one gets, ∇.E=ρ/ε_{0}