Electrons don’t just magically move through a wire as a current. They move because an electric field inside the wire—a field created by a nonuniform surface charge density on the wire—pushes on the sea of electrons to create the electron current. The field has to keep pushing because the electrons continuously lose energy in collisions with the positive ions that form the structure of the solid. These collisions provide a drag force, much like friction.

We will model the conduction electrons—those electrons that make up the sea of electrons—as free particles moving through the lattice of the metal. In the absence of an electric field, the electrons, like the molecules in a gas, move randomly in all directions with a distribution of speeds. If we assume that the average kinetic energy of the electrons is given by the same $\frac{3}{2}{k}_{\text{B}}T$ that applies to an ideal gas, we can calculate that the average electron speed at room temperature is $\approx {10}^5\text{m/s}$. This estimate turns out, for quantum physics reasons, to be not quite right, but it correctly indicates that the conduction electrons are moving very fast.

However, an individual electron does not travel far before colliding with an ion and being scattered to a new direction. FIGURE 27.11a shows that an electron bounces back and forth between collisions, but its average velocity is zero, and it undergoes no net displacement. This is similar to molecules in a container of gas.

FIGURE 27.11 A microscopic view of a conduction electron moving through a metal.

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Suppose we now turn on an electric field. FIGURE 27.11b shows that the steady electric force causes the electrons to move along parabolic trajectories between collisions. Because of the curvature of the trajectories, the negatively charged electrons begin to drift slowly in the direction opposite the electric field. The motion is similar to a ball moving in a pinball machine with a slight downward tilt. An individual electron ricochets back and forth between the ions at a high rate of speed, but now there is a slow net motion in the “downhill” direction. Even so, this net displacement is a very small effect superimposed on top of the much larger thermal motion. Figure 27.11b has greatly exaggerated the rate at which the drift would occur.

Suppose an electron just had a collision with an ion and has rebounded with velocity $\vec{v}{}_0$. The acceleration of the electron between collisions is

where $E$ is the electric field strength inside the wire and $m$ is the mass of the electron. (We’ll assume that $\vec{E}$ points in the negative $x$-direction.) The field causes the $x$-component of the electron’s velocity to increase linearly with time:

The electron speeds up, with increasing kinetic energy, until its next collision with an ion. The collision transfers much of the electron’s kinetic energy to the ion and thus to the thermal energy of the metal. This energy transfer is the “friction” that raises the temperature of the wire. The electron then rebounds, in a random direction, with a new initial velocity $\vec{v}{}_0$, and starts the process all over.

FIGURE 27.12a shows how the velocity abruptly changes due to a collision. Notice that the acceleration (the slope of the line) is the same before and after the collision. FIGURE 27.12b follows an electron through a series of collisions. You can see that each collision “resets” the velocity. The primary observation we can make from Figure 27.12b is that this repeated process of speeding up and colliding gives the electron a nonzero average velocity. The magnitude of the electron’s average velocity, due to the electric field, is the drift speed $v_d$ of the electron.

FIGURE 27.12 The electron velocity as a function of time.

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If we observe all the electrons in the metal at one instant of time, their average velocity is

where a bar over a quantity indicates an average value. The average value of $v_{0x}$, the velocity with which an electron rebounds after a collision, is zero. We know this because, in the absence of an electric field, the sea of electrons moves neither right nor left.

The quantity $\text{Δ}t$ is the time since the last collision, and—with some statistical calculations that we’ll skip—the average value of $\text{Δ}t$ is the mean time between collisions, which we designate $\tau$. The mean time between collisions, analogous to the mean free path between collisions in the kinetic theory of gases, depends on the metal’s temperature but can be considered a constant in the equations below.

Thus the average speed at which the electrons are pushed along by the electric field is

We can complete our model of conduction by using Equation 27.7 for $v_{\text{d}}$ in the electron-current equation $i_{\text{e}}=n_{\text{e}}A{v}_{\text{d}}$. Upon doing so, we find that an electric field strength $E$ in a wire of cross-section area $A$ causes an electron current

The electron density $n_{\text{e}}$ and the mean time between collisions $\tau$ are properties of the metal.

Equation 27.8 is the main result of this model of conduction. We’ve found that the electron current is directly proportional to the electric field strength. A stronger electric field pushes the electrons faster and thus increases the electron current.