# Current density dependence

1. Aug 4, 2014

### almarpa

Hello all.

As far as I know, the equations to obtain current density is:

J=∑(qi ni vi)

, where qi is the charge of each class of charge carrier in the material, ni is the volumen density of that class of charge carrier, and vi is the average drift speed of those charge carriers. Current density is a vector quantity that may depend on time and point coordinates (position).

My question is, where does this position and time dependence come from?

Can volume density of charge carriers depend both on the position and time considered (ni=ni(r, t))? Why?

Can drift speed depend both on position and time (vi=vi(r, t))? Why?

Thank you so much.

2. Aug 4, 2014

### ShayanJ

Yes, both of them can be functions of time and spatial coordinates.
For drift velocity, the reason is the time and spatial dependence of the electric field inside the conductor.
In fact the electric field causing the current isn't simply a constant field across the wire. The motion of every electron in any point of the wire changes the field in different places of the wire. Its very complicated.
For density, charges are in motion, both drift and thermal motion. Also there are impurities and other kinds of imperfections.

3. Aug 4, 2014

### almarpa

Thank you. That is what I supposed, but I needed your expert advice.

Could I ask you one more question?

When current density and electric field across a conductor are related by Ohm's law:

JE

, I guess that both current density and electric field can be a function of position and time, as well. But, conductivity is only a function of position (and temperature), but not of time. Is it right?

Thanks!!

PS: By the way, if you do not mind, I will forward you a question that I asked in the forum, but nobody answered:

Let me explain my doubt. Currently I am studying electromagnetism (with Griffiths and Purcell books). I have already studied Electrostatics (constant in time electric fields produced by stationary charges), and magnetostatics (constant in time magnetic fields produced by steady currents). Now I am studyig the unit about electric currents. In this unit continuity equation is introduced, as well as Ohm's law and electromotive force.

However, although they do not say it explicitly, I have the feel that in this unit all the books consider steady currents when talking about Ohm's law and emf. Is it so? Is Ohm's law an empirical result only valid for steady currents, or is it also valid for, say, non steady currents driven through a metal?

if so, are all the equations derived from it (V=IR, R=L/σA) valid for non steady currents?

Regards.