3.5. DRIFT OF ELECTRONS AND THE ORIGIN OF RESISTIVITY

As remarked before, an electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Thus, if there are N electrons and the velocity of the ith electron (i = 1, 2, 3, ... N ) at a given time is v, then

$\frac{1}{N} \sum_{i=1}^{N} v_i = 0$

Electrons in a conductor are constantly colliding with fixed ions, which scatter their motion in random directions. After such a collision, even though the speed of the electron remains the same, the direction may change unpredictably. As a result, when you calculate the average velocity of a large number of electrons, their directions cancel out, leading to an average velocity of zero. This means that, although individual electrons are moving, their collective motion does not result in a net flow without the influence of an external electric field.

Think of a crowded room filled with people moving around randomly. If you were to average the direction in which everyone is walking, you would find that there's no overall movement towards any door; people are just bumping into each other and changing directions without contributing to a mass exit.

Consider now the situation when an electric field is present. Electrons will be accelerated due to this field by

$a = \frac{-eE}{m}$ where -e is the charge and m is the mass of an electron.

Consider again the ith electron at a given time t. This electron would have had its last collision some time before t, and let $t_i$ be the time elapsed after its last collision. Thus, the average velocity of the electrons at time $t$ is the average of all the velocities. The average of $v_i$ is zero [since immediately after any collision, the direction of the velocity of an electron is completely random]. Let us denote by $\tau$, the average time between successive collisions.

When an electric field is applied to a conductor, due to the electric force acting on electrons, they experience acceleration in the direction of the field. The force is given by the formula above, where the charge of the electron and its mass determines how quickly it responds. While electrons do accelerate, their motion is interrupted by frequent collisions with the fixed ions in the conductor. This collision process averages out the acceleration over time, leading to a steady average drift velocity rather than continuous acceleration.

Imagine you are driving a car on a straight road. Your speed increases as you press the accelerator, similar to how electrons accelerate due to an electric field. However, every time you hit a pothole (representing collisions), you are jolted and lose some of your speed. Even though you are pushing the pedal to accelerate, the frequent hits from potholes keep you from going faster than a certain speed; similarly, the random collisions keep electrons from continually accelerating.

Thus, averaging gives us for the average velocity $v_d$:

$v_{d} = \frac{-eE\tau}{m}$

The drift velocity, represented by $v_d$, is the average velocity of electrons in a conductor when an electric field is applied. It can be derived from considering the accumulated effects of electron acceleration over the average time between collisions. The negative sign indicates that electrons move in the opposite direction of the electric field, as they are negatively charged. The expression shows that the drift velocity is directly proportional to the strength of the electric field and the average relaxation time between collisions.

Think about water flowing through a pipe. While the pressure from a pump pushes the water along, the water molecules may bump into each other and the sides of the pipe. Similar to how more pressure leads to higher flow rates, a stronger electric field causes electrons to drift faster. The 'drift velocity' is like the average speed of water flowing despite its turbulent path.

Because of the drift, there will be net transport of charges across any area perpendicular to E. Consider a planar area A, located inside the conductor such that the normal to the area is parallel to E. Then the total charge transported across this area A in time $\Delta t$ is given by:

$I = -neA v_{d}$ where n is the number of free electrons per unit volume in the metal.

This equation shows how drift velocity leads to a measurable current. The drift of electrons results in a net flow of charge across any cross-sectional area that is perpendicular to their motion. 'n' represents the density of charge carriers (electrons), and 'A' is the area through which they are moving. The product of these values with the drift velocity gives the total current, which is a measure of how much charge is passing through that area over time.

Imagine a group of people trying to exit a theater through a single door. The number of people trying to leave (analogous to 'n') and the width of the door (the area 'A') both influence how many can exit per second (the current). If they all move towards the exit at the same steady pace (like the drift velocity), you can easily calculate how many people will exit each second.

From the relations for current density $j$ and drift velocity, we can derive Ohm's Law. As found, $j = \sigma E$, where $\sigma$ is the conductivity of the material. Therefore, combining all these relations leads us to the conclusion that

$\sigma = \frac{ne^2}{m\tau}.$

This final relationship connects all the previous concepts into Ohm's Law. From understanding how electron drift produces a measurable current density, we can see how the performance of conductors can be characterized by conductivity, which incorporates the charge density and the average time between collisions. The result shows that higher charge density or lower collision time leads to better conductivity, explaining why different materials have varying electrical properties.

Consider a crowded mall during a sale. If there's a good flow of shoppers (analogous to current density), the rate at which shoppers can make purchases (conductivity) depends on how many entrance/exit points are available and how quickly people move through the aisles (representing charge density and collision time). If more exits are available and crowd control is effective, shopping can proceed smoothly—as conductivity improves in materials with favorable properties.

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