Electrical currents and how they behave

Conductivity




The electrical conductors

Moving a charge thru a material from one region to another constitutes a current. The ability of a material to allow the movement of a charge is the material's conductivity, and it is a material specific property.

In regards to conductivity, materials are divided into conductors, semiconductors and insulators. From a material point of view, the three types differ in the way, that
Here you should be aware that this is not either/or. It is a gradual change from conductors over semicoductors to insulators. Also one should be aware of the common misunderstanding that insulators cannot conduct a current. This is not correct, and the difference between conducting poorly and not at all, is particularly relevant for e.g. power pylons and transformer stations. Insulator only means that the resistivity, is above a certain level. Apparently there is no clear definition of when the meaterial goes from being a semiconductor to being an insulator, but at resistivities of 1012 ohm · m (which is the resistivity for glass) or higher, you refer to the materials as insulators.

The difference between conductors and semiconductors is important in regards to the conductivity as a function of temperature. The common rule for conductors is, that the conductivity increases with decreasing temperature, and in some cases become a superconductor, when the temperature is sufficiently low. The reason for this is that heat will affect the movement of the electrons, making them more and more erratic (or less organized) in their movement as the temperature increases. The more organized the electrons move, the better the conductivity. Semiconductors on the other hand, requires that the electrons are torn free of the atoms, and as the temperature decreases, the movement of the atoms also decreases, making it increasingly harder to tear an electron free, so for semiconductors, the conductivity normally decreases with decreasing temperatures.



The electrical current

Getting the charge to move along in the conductor requires a steady force on the mobile charge, e.g. an electric field
E
. The force of the charge q is

F
= q ·
E


In a conductor, a charge moves along, accelerating until it hits an obstacle e.g. an atom and looses some kinetic energy, after which it is accelerated again. The result is an erratic back and forth motion and a gradual drift in the direction of the electrical field. Inelastic collisions with stationary charges transfer all energy to these, resulting in an increase in vibrational energy i.e. the temperature in the conductor increases.

As the charges move back and forth, there is a general flow in one direction, which is the measurable current. The practical way of working with conductivity is therefore looking at the number of charges (net charge), ΔQ, flowing thru an area per time unit, Δt. This is the current I.

I = ΔQ
Δt

The flow of charges may change over time, so instead the derivative instantaneous current is used:

I = dQ
dt

The SI unit of current is Ampere, A, equivalent to coulomb per second, C/s.

When the charges move thru the conducting material, the current can be measured as drift velocity v. If we look at a cross-section of a conducting material, through which there is an electrical field,
E
, directed from left to right, free positively charged particles will move in the same direction as the field while negatively charged particles will move in the opposite direction. Even though the positive and negatively charged particles moves in opposite directions, the resulting current is always in the direction of the positive charges. The reason for this, is that it is the buildup of positive charges, that drives the current. This is done both by moving the positive charges in that direction, and by removing negative charges.

For n charged particles, with the charge q, moving thru the cross-section A in the timespan Δt, the charge ΔQ, moving thru the cross-section is

ΔQ = n · q · v · A · Δt

It is noteworthy, that the charge q in these calculations is actually the numeric value of the charge i.e. the particles e and H+ both have the charge 1, Ca2+ and SO42− both have the charge 2 etc. The reason for this is that velocity contrary to speed has an inherent directional vector from being calculated from start position to end position, in physics classes usually written as x-x0. If the direction of the positive charges is positive, the direction of the negative charges will be negative. The negative charge times the negative distance gives a positive product i.e. q · v is always ≥ 0.

The resulting current thru the cross-section is:

I = ΔQ = n · q · v · A
Δt

For a solid where only the electrons are moving, this is sufficient, however if more particles with different charges are moving, the actual current is

I = ΔQ = A ·
i
Σ
1
ni · qi · vi
Δt

In some cases, the density of the current, i.e. current per area unit, makes more sense. The current density is designated J:

J = I =
i
Σ
1
ni · qi · vi
A

The current density can also be described as a vector:

J
=
I =
i
Σ
1
ni · qi ·
vi
A

Here the result of the negatively charged elements having a negative drift velocity vector comes into effect, making all vector contributions from n · q ·
v
always having the same direction as
J
, and
J
will always have the same direction as
E
.