Gauss' Law is one of the fundamental pieces of electromagnetics for finding the electric electric field around an arbitrary charge distribution. In essence, Gauss' Law states that the flux of the electric field through a closed imaginary surface is proportional to the sum of the charge enclosed by the surface.
The flux φ is defined in terms of the dot product of the electric field line and an elemental vector perpendicular to the surface, ds. This vector is by convention pointing outward from the surface. As shown in Figure 1.
The rea son why Gauss' law is useful is because the closed surface can be any closed surface so we can often choose a surface has a particular symmetry which makes the problem easier to solve. The flux is a measure of the number of electric field lines passing through the surface.
The value of the flux depends on the angle, θ between the electric field vector E and the ds. Thus if E and ds are parallel, the flux is 1, if E and ds are perpendicular then the flux is 0.
∫ SE.ds = ∑q/ε0(1)
Where E is the electric field, ds is a vector normal to the surface and area ds. ρ is the charge density of the
As an example, consider a point charge Q at the origin. Calculate the flux of E passing through a spherical Gaussian surface of radius r
φ = ∫ SE(r).ds = r Q/(4πε0)∫ Sr^/r2 . (r2 sinθ dθ dφ r^)
since ds = r2 sinθ dθ dφ r^ in spherical polar coordinates.
φ = Q/(4πε0)∫ θ=0 θ=π∫ φ=0 φ=2π sin θ dθ dφ(r^.r^) = Q/ε0∫ θ=0 θ=πsin θ dθ = Q/ε0
There is another way of expressing the same law which is known as the differential form of Gauss' law. To arrive at the new differential form, we use a theory known as the divergence theorem which converts the surface integral to a volume integral.
φ = ∫ SE(r).ds = ∫ v∇.E(r).dτ = 1/ε0∫ vρ(r) dτ
Thus we have,
∇.E(r) = &rho(r)/ε0(2)