# Strange nature of Electric Field

1. Aug 22, 2014

### Maharshi Roy

Divergence of electric field is 0 for the field produced by a point charge at origin at points other than origin is 0; But on applying divergence theorem it is found that the flux is not 0 which is contradictory. Hence we introduce the dirac delta function as:-
∇.E=4πδ^3(r)
such that other than 0; it is 0 but at zero, it assumes a value such that ∫(∇.E)d(vol)=4π;
But on the subsequent chapters,
The differential form of Gauss law states that:
∇.E=ρ/ε
even for points other than 0;

Last edited: Aug 22, 2014
2. Aug 22, 2014

### Delta²

The differential form of Gauss law ∇.E=ρ/ε is always correct. For the case of a point charge the volume charge density ρ=4πδ^3(r) that is it is equal to the 3D dirac function. In this special case the charge density is zero everywhere except in the origin where the point charge sits and it creates a "special infinity" in the charge density the dirac infinity (such that the proper volume integral is not zero).

But for other cases we might have different formulas for ρ, for example if we have a non conducting sphere of radius R and uniform volume charge density C then the expression for ρ would be

ρ(r)=C if r<=R
ρ(r)=0 if r>R.