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Explain me the derivation of differential form of gauss law...

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@Demo User

Gauss' law in integral form:

\oint_{S}E\:dS=\frac{Q}{\epsilon _0}

Rewrite the right side in terms of a volume integral-

\oint_{S}E\:dS=\oint_{V}\frac{\rho }{\epsilon _0}\:dV

The divergence theorem says that the flux penetrating a closed surface S that bounds a volume  V is equal to the divergence of the field F inside the volume.

\oint_{S}F\:dS=\oint_{V}(\nabla F)\:dV

Use the divergence theorem to rewrite the left side as a volume integral

\oint_{V}(\nabla E)\:dV=\oint_{V}\frac{\rho }{\epsilon _o}\:dV

Set the equation to 0

\\\oint_{V}(\nabla E)\:dV-\oint_{V}\frac{\rho }{\epsilon _o}\:dV=0\\\oint_{V}((\nabla E)-\frac{\rho }{\epsilon _o})\:dV

The above equation says that the integral of a quantity is 0. Because the only quantity for which the integral is 0, is 0 itself, the expression in the integrand can be set to 0.

\nabla E-\frac{\rho }{\epsilon _o}=0

This leads to Gauss' law in differential form

\nabla E=\frac{\rho }{\epsilon _o}

Eels — EdS= F ds (VF) dv qE)dV ((VE) - L) dv
Posted by

himanshu.meshram

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@Demo User

What is that inverted triangle??? What is the meaning of inverted delta E??? What is divergence???

Answer:

 The divergence of A is the sum of how fast the vector function is changing

divergence mathematically defined

The symbol  is the partial derivative symbol, which means rate of change with respect to x. 

Posted by

sudhir.kumar

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