Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

up vote 2 down vote accepted

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.

share|improve this answer

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

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

share|improve this answer
    
Does Ohm's law holds in the presents of magnetic field? I thought that there should be a Lorentz force, acting on electrons in the substance, which depends on magnetic field. So it should affect the equation for j. Doesn't it? –  Fiktor Nov 10 '10 at 9:43
    
Or probably it just affects a distribution of the charge so that there will be negative charge on inner surface and positive on the outer. Doesn't it? –  Fiktor Nov 10 '10 at 9:47
    
@Fiktor: This is called "conduction current" and applies in materials. For example it does not apply for "convection current" where real charge displacement is involved like in a particle accelerator or in a cathodic television. In the second case, your objection about magnetic field is valid and this is completely different story. –  Cedric H. Nov 10 '10 at 10:10

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.