Question

describe what Gauss’s Law is about and when is it useful?

Answer 1

Gauss's law, in integral form, relates the flux of the electric field through some closed surface S to the charge enclosed within the volume bounded by S. Precisely, it is the statement that given an electric field E(r) defined over space, the flux integral over any closed surface Swill always yield

∮SE⋅da=Qenc/ϵ0

Normally surface integrals over vector fields involve parametrizing the surface

E⋅da=E⋅n^da,E⋅da=E⋅n^da,

where n^n^ is the unit normal to the surface and can be calculated from the parametrization. This quantity can assume different values everywhere along the surface.

So far I've only talked about the difficulties in computing the flux integral of a vector field over a general surface. When using Gauss's law, we have the added problem of not knowing the electric field (this is the quantity we're trying to find!). We now are tasked with computing an integral over an undefined function! This is where symmetry comes in and saves the troubled physicist.

Essentially, symmetric charge distributions allow one to choose a convenient surface (which preserves the symmetry) to remove EE from the integral. For example, consider a uniformly charged spherical volume of radius R Due to symmetry, one can argue that the electric field generated from this distribution must be radially symmetric. If we take our surface S to be a sphere of radius r, then we find that the normal to the sphere and the direction of the electric field coincide, so

E⋅da=|E|∮Sda=4πr2|E|,E⋅da=|E|∮Sda=4πr2|E|,

since we are now simply computing the surface area of a sphere. We can now simply divide to find the answer:

|E|=14πϵ0Qencr2.|E|=14πϵ0Qencr2.

To summarize, Gauss's (integral) law relates the flux integral of the electric field to the charge contained within a surface. Because we do not know the electric field, Gauss's law is only useful when we can remove the electric field from within the integral, which happens when the charge distribution displays certain spatial symmetries (spherical, cylindrical, planar)