How to deduce Gauss' law from Gauss Divergence Law

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Homework Statement


Gauss Divergence Law:
df53cbd418aaded3f2ad2d2fe1d60f2d.png

Gauss' law
914f57946de3c30d15e1a9778b276842.png


Can we obtain the Gauss' Law from Gauss Divergence Law?

Homework Equations



In Spherical coordinates,

electric field strength

(Q/4[itex]\pi[/itex]εr^2,0,0)
Then ∇[itex]\cdot[/itex]E=0+0+0=0

The Attempt at a Solution



We can not obtain the Gauss' Law from the general mathematical law?
 

Answers and Replies

  • #2
UltrafastPED
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So the volume integral of div E [application of Gauss' theorem to Gauss' law) is proportional to Q, the total charge included.

But Q is the volume integral of the charge density ... so we have a volume integral on both sides.

Then the integrands must be equal, which gives div E = densit/eps_0.
 
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  • #3
vela
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Homework Statement


Gauss Divergence Law:
df53cbd418aaded3f2ad2d2fe1d60f2d.png

Gauss' law
914f57946de3c30d15e1a9778b276842.png


Can we obtain the Gauss' Law from Gauss Divergence Law?

Homework Equations



In Spherical coordinates,

electric field strength

(Q/4[itex]\pi[/itex]εr^2,0,0)
Then ∇[itex]\cdot[/itex]E=0+0+0=0

The Attempt at a Solution



We can not obtain the Gauss' Law from the general mathematical law?
There's a singularity in ##\vec{E}## at r=0 you need to account for.
 

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