Drude Model of Conduction

The Drude model of conduction connects Current and Resistance to the atomic model, which helps define Conductivity & Resistivity of materials.

Conduction, a property of Conductors (mostly metals), can be explained using the kinetic energy from electrons and the net movement of them.

In a metal, electrons randomly move around with kinetic energy and collide with themselves and other particles, changing direction, but because this movement is so random, the net movement is 0, so the charge is also always zero (because charge is when excess electrons are in a location). Generally, this movement is called the thermal speed of electrons

$$v_{th}\approx 10^{6}m/s$$

However, what we can do is use the force generated by an Electric Field to cause some net movement on these electrons. The individual velocities are still fairly random, but the net movement, called the drift speed is positive, and in the direction opposite to the electrical field

$${\vec{v}}_{drift}=-10^{-4}\hat{u}m/s$$

where $\hat{u}$ is the direction of the electric field.

Net Electron Movement §

Assume we have a cross sectional volume through some conductor, and we apply an electric field through the conductor:

$${\vec{v}}_{drift}=\frac{x}{t}\to x={\vec{v}}_{drift}\times t$$

The total number of electrons that travel through the volume can be given through the volume density of the electrons:

${\rho }_e=\frac{Electrons}{V}$ (in $m^{-3}$) (NOT CHARGE DENSITY!)

$$N_{electronsthroughV}={\rho }_{e}V={\rho }_{e}Ax={\rho }_{e}A{v}_{drift}t$$

Relation to Current & Electron-Current §

Now, we can use the formula of Current:

$$I=\frac{q}{t}$$

$q=N_e\times q_e$ (Total charge is number of electrics x charge per electron)

$$I=\frac{N_e{q}_e}{t}=\frac{{\rho }_{e}A{v}_{drift}{q}_{e}t}{t}$$

$$I={\rho }_{e}A{v}_{drift}{q}_e$$

We can define a new unit, the electron current, $i$, to show how many electrons travel through the volume:

$$i=\frac{I}{q_e}(in{q}_e/s)$$

$$i={\rho }_e{v}_{drift}\times A$$