Maxwell equations: how to know the behavior of charge and current?

Question

In school-level tasks, when (almost) all substances are linear, homogeneous and isotropic, we have $D=\epsilon E$, $H=B/\mu$ and thus Maxwell "in material" equations (1) say how $E$ and $B$ depend on time given known dependence of $\rho$ (free charge density) and $j$ (free current density). Here they are in CGS unit system: $$\left\{\begin{aligned} \text{div} D=4\pi\rho\\ \text{div} B=0\\ \text{rot} E = -\frac{1}{c}\frac{\partial B}{\partial t}\\ \text{rot} H = \frac{4\pi}{c}j+\frac{1}{c}\frac{\partial D}{\partial t} \end{aligned}\right.$$ Also we know continuity equation $\partial \rho/\partial t + \text{div} j=0$. But this is not enough to determine, how j will change over time or in statical case, how $j$ is distributed in the conductor. What are other equations for $j$? Are there any for some "ideal case"?

For example, I don't know actually, is the following task correct or under determined:

Electric current I flows along infinite cylindrical conductor. Inner radius is $r$, outer is $R$, magnetic constants of all substances are given ($\mu_1,\mu_2, \mu_3$ from inside out). Find magnetic field ($B$ and $H$) and current distribution in a conductor.

The question: Is there any "standard" equations for $j$? Particularly, is the task above well-determined?

Answer 1

Maxwell equation need to be accompanied by Lorentz equation $$\vec{a} = \frac{q}{m} \vec{E} + \frac{q}{m} \vec{v} \times \vec{B}.$$

As there are different masses for charged particles, one cannot form one general equation for behaviour of charge (or current).

In complex scenarios (like in a conductor), there are effective relations like $$\vec{j} = \sigma \vec{E}$$ (see post by Tobias Kienzler). In case of magnetic field in a conductor, current 'generates' electrical field not only parallel, but also perpendicular, to its vector. See - Hall effect.

Answer 2

In general, Ohm's law

$$\vec j = \sigma \vec E$$

is used, where $\sigma$ is the conductivity.

In your example however, you'll use $j = I / A$ where $A$ is the cross section of the conductor, in general electric current through an area $A$ is defined as

$$I := \iint_{A} \vec j\cdot d\vec n.$$