Electric Currents||Electromagnetism

2021-02-24 11:12 作者:湮灭的末影狐 0人读过 | 我要投稿

//Well, our Italiano professor's spoken English is... a little hard to understand...

//In this chapter we discuss charge in motion.

4.1 Electric current and current density

An electric current is charge in motion. The electric current in a wire is the amount of charge passing through it per unit time.

$I%3D%5Cfrac%7B%5CDelta%20Q%7D%7B%5CDelta%20t%7D$

It can also be expressed as the flux of electric current density:

$I%3D%5Ciint_S%20%5Cvec%20j%20%5Ccdot%20%5Cmathrm%20d%5Cvec%20S$

and electric current density is determined by the density of charge carriers $n$ , the amount of charge on each carrier $q$ , and the average speed of the carriers $%5Cvec%20v$ .

$%5Cvec%20j%20%3D%20nq%5Cvec%20v$

4.2 Steady currents and charge conservation

We've mentioned conservation of charge:

$%5Cnabla%5Ccdot%20%5Cvec%20j%20%2B%5Cfrac%7B%5Cpartial%20%5Crho%7D%7B%5Cpartial%20t%7D%3D0$

And in a steady or stationary current system, we know the charge density can't grow infinite,

$%5Cnabla%5Ccdot%5Cvec%20j%3D0$

4.3 Electrical conductivity and Ohm's law

We know in electrostatic cases $%5Cvec%20E$ in a conductor must be 0. But when an electric field $%5Cvec%20E$ exists inside a conductor and pushes free charges (electrons) inside, the result is an electric current in the direction of $%5Cvec%20E$ . (And that's not electrostatic case.)

$%5Cvec%20j%20%3D%20%5Csigma%20%5Cvec%20E$

That's the differential form of Ohm's law. $%5Csigma%20$ is called the conductivity of the material.

We're more familiar to its original form:

$V%3DIR$

Generally the rule above should be written as

$%5Cleft(%5Cbegin%7Barray%7D%7Bc%7D%0Aj_%7Bx%7D%20%5C%5C%0Aj_%7By%7D%20%5C%5C%0Aj_%7Bz%7D%0A%5Cend%7Barray%7D%5Cright)%3D%5Cleft(%5Cbegin%7Barray%7D%7Bccc%7D%0A%5Csigma_%7Bx%20x%7D%20%26%20%5Csigma_%7Bx%20y%7D%20%26%20%5Csigma_%7Bx%20z%7D%20%5C%5C%0A%5Csigma_%7By%20x%7D%20%26%20%5Csigma_%7By%20y%7D%20%26%20%5Csigma_%7By%20z%7D%20%5C%5C%0A%5Csigma_%7Bz%20x%7D%20%26%20%5Csigma_%7Bz%20y%7D%20%26%20%5Csigma_%7Bz%20z%7D%0A%5Cend%7Barray%7D%5Cright)%5Cleft(%5Cbegin%7Barray%7D%7Bc%7D%0AE_%7Bx%7D%20%5C%5C%0AE_%7By%7D%20%5C%5C%0AE_%7Bz%7D%0A%5Cend%7Barray%7D%5Cright)$