Gauss's Law is one of the four Maxwell equations that form the foundation of classical electromagnetism. It relates electric fields to their sources—electric charges—through a simple yet profound mathematical statement. Physically, it tells us that the net electric flux through a closed surface is proportional to the total electric charge enclosed within that surface.
It is expressed in two forms: the integral form, which relates the total electric flux to enclosed charge, and the differential form, which relates the divergence of the electric field to the local charge density.
Consider a single point charge q located at the origin. The electric field produced by this charge, from Coulomb's law, is:
E(r)=4πε01r2qr^
To compute the total electric flux through a spherical surface of radius r centered on the charge, we calculate:
∮E⋅dA=4πε01r2q∮r^⋅dA
But dA points radially outward, and on a sphere r^⋅dA=dA, so:
∮E⋅dA=4πε0r2q⋅4πr2=ε0q
This is the integral form of Gauss’s Law for a point charge. For a general charge distribution, we integrate over the volume to get the total enclosed charge Qenc=∫VρdV:
Positive charges are sources of electric field lines, while negative charges are sinks.
The electric field diverges from a region of space where positive charge exists.
In regions with no charge (ρ=0), the divergence of the electric field is zero—field lines may bend but do not start or end.
The differential form gives a local description of how charge creates electric fields, while the integral form gives a global description relating total flux to total enclosed charge.
Example: Electric Field Around a Sphericaly Symmetric Charged Star with Radial Charge Profile
Problem:
Consider an isolated, non-rotating, spherically symmetric star of radius R whose internal charge distribution varies with radius according to:
ρ(r)=ρ0(1−R2r2)for r≤R
Outside the star (r>R), the space is vacuum.
Find the electric field E(r) both inside and outside the star.
This solution reflects how Gauss’s Law can handle non-uniform charge distributions and shows the natural transition between interior and exterior fields. It's relevant in modeling charged fluids or test-field behavior in highly symmetric astrophysical objects.
Gauss's Law provides a deep connection between electric fields and charge. It offers both a global viewpoint (via flux through surfaces) and a local viewpoint (via field divergence). In symmetrical situations, it allows for simple, elegant solutions that bypass the need for direct integration from Coulomb's Law.
The transition from the integral to the differential form introduces the idea of field theory—replacing forces between particles with field equations valid throughout space.