By the divergence theorem the surface integral can be written as a volume integral Is E da=h,V ed Thus we have A,v. edr'=-/p(ra ∈∩ Since the volume is arbitrary V·E(r)=-p(r) i.e. at any point of space, the divergence of e is equal to the charge density divided by the permittivity The gauss law in differential form
Remarks: The integral form of the gauss law is very useful to get the E, especially when the charge distribution has some symmetry Examples (1)a point charge (2 )a infinite line of charge (3 a infinite sheet of charge a spherical charge The differential form of the gauss law is of funda mental importance, as it tells how the charge density determines the electric field e
3.3 Conductors A conductor is a material inside which charges can flow freely. (no resistance) For electrostatics,- equilibrium,- fixed in space, zero electric field -all points are at the same potential.(otherwise, charges will move. If a conductor is placed in an electric field charges flow within it, so as to produce a second electric field that cancels the first one in the conductor Applying Gauss'law V. E=p/Eo inside conductors, SInce E=0,→p=0 Conclusion the charge density is 0 inside a conductor Corollary any net charge on a conductor must reside on its surface