Gauss's law for electrodynamics

[hercules68]

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 !

 

[Khashishi]

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.

 

[the_emi_guy]

Gauss's law is a specific case of Stoke's theorem.
http://en.wikipedia.org/wiki/Stoke'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.

 

[yungman]

I am not good in definitions but I did look into Gauss Law. I really don't see the relation of Stokes and Gauss. Even in Gauss law for magnetism:

http://en.wikipedia.org/wiki/Gauss%27s_law_for_magnetism

It only said $\nabla \cdot \vec B = 0\;$ where it states there is no mono magnetic pole.

Gauss law is mainly used in Divergence theorem where $\nabla \cdot \vec E=\frac {\rho_v}{\epsilon}$ Where:

$$\int_v \nabla\cdot \vec E dv'=\int_s \vec E\cdot d\vec s'=\frac Q {\epsilon}$$

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:

$$\int_s \nabla X\vec B\cdot d\vec s'=\int_c \vec B \cdot d \vec l'= \mu I$$

that relate current loop with field through the loop.

 

[Meir Achuz]

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 !

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.

 

[andrien]

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)

 

[yungman]

I can't think of any direct prove on Gauss 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 $\int_s \vec E\cdot d\vec s'$.

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:

$$\nabla X \vec E=-\frac{\partial \vec B}{\partial t}$$

 

[andrien]

I just want to say that gauss law follow immediately from maxwell's fourth eqn when combined with continuity eqn for charge density.

Let us see,
c²(∇×B)=j/ε₀+∂E/∂t
now,
c²{∇.(∇×B)}=∇.j/ε₀+∂(∇.E)/∂t
USING ∇.j=-∂ρ/∂t and the fact that gradient of curl vanishes.
one gets,
∇.E=ρ/ε₀