Gauss' law is a fundamental law in electromagnetism that relates the electric field at a point to the charge enclosed by a surface surrounding that point. It states that the flux of the electric field through any closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space.
The 1/r^3 relationship in Gauss' law is a consequence of the inverse square law for the electric field. The electric field is proportional to the inverse of the distance squared, but when considering a spherical surface, the surface area increases with the square of the distance. Therefore, the electric flux through the surface decreases with the inverse of the distance cubed.
Gauss' law can be derived from the Maxwell's equations, specifically the divergence theorem, which relates the flux of a vector field through a surface to the divergence of that field within the enclosed volume. The divergence theorem is then applied to the electric field and the charge density to obtain Gauss' law.
Gauss' law has many practical applications in electromagnetism, such as calculating the electric field and potential of charged particles and conducting objects, determining the capacitance of a system, and solving boundary value problems in electrostatics.
While Gauss' law holds true in most cases, there are some exceptions, such as when dealing with time-varying electric fields or in the presence of magnetic fields. In these cases, the modified forms of Gauss' law, known as the Ampere-Maxwell law and the Faraday's law of induction, must be used.