Question

Explain the continuity equation for electric current and how does divergence apply for electric current? What is the relaxation time of a material?

Answer 1

A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Integral form
The integral form of the continuity equation states that:

The amount of q in a region increases when additional q flows inward through the surface of the region, and decreases when it flows outward;
The amount of q in a region increases when new q is created inside the region, and decreases when q is destroyed;
Apart from these two processes, there is no other way for the amount of q in a region to change.
Mathematically, the integral form of the continuity equation expressing the rate of increase of q within a volume V is:

Differential form
See also: Conservation law and conservation formBy the divergence theorem, a general continuity equation can also be written in a "differential form":
where

∇⋅ is divergence,
ρ is the amount of the quantity q per unit volume,
j is the flux of q,
t is time,
σ is the generation of q per unit volume per unit time. Terms that generate q (i.e. σ > 0) or remove q (i.e. σ < 0) are referred to as a "sources" and "sinks" respectively.
This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation. Other equations in physics, such as Gauss's law of the electric field and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because j in those cases does not represent the flow of a real physical quantity.

In the case that q is a conserved quantity that cannot be created or destroyed (such as energy), σ = 0 and the equations become: