When you switch on a light and a bulb lights up in an instant, it is as if magic has been done. But there is a marvelous microscopic tale behind this daily miracle—one of minute electrons moving slowly across a conductor, bumping, scattering, and nevertheless producing a smooth, consistent flow of current. This slow creep of electrons is drift, and it is the source of something taken lightly: resistivity.
In this blog, we’ll unpack how electrons drift in a conductor, why resistivity appears, and how these microscopic processes explain Ohm’s law, one of the most fundamental principles of electricity.
Random Motion of Electrons in a Conductor
Picture being in a room full of people running around haphazardly. If you attempted to glance at the entire group, their mean movement would cancel each other out—no one is moving all in one direction. This is what occurs within a conductor.
Inside a piece of metal, electrons are able to move about freely, but they don't travel in a neat stream by themselves. Rather, they constantly crash into the immobile, heavy positive metal lattice ions. Upon each collision, an electron rebounds in a random new direction.
Because of this randomness, the average velocity of all electrons in the absence of an external force is zero. Mathematically:
Here, N is the total number of electrons, and 
is the velocity of the 
electron. This equation tells us that without an electric field, there’s no net flow of electrons across the conductor.
Effect of an Electric Field
Now, what happens if we apply an electric field across the conductor?
This changes everything. An electron, with charge −e, feels a force in the direction opposite to the field (since electrons are negatively charged). According to Newton’s second law, this force produces an acceleration:
where:
- e = charge of an electron,
- E = applied electric field,
- m = mass of the electron.
Suppose an electron has just suffered a collision. After the collision, it starts with some velocity and then accelerates under the influence of the field until the next collision. If the time interval before the next collision is 
, then its velocity after that time becomes:
Here, 
is the velocity immediately after the last collision. This motion is a tug-of-war: random collisions constantly disrupt the path, but the electric field keeps nudging electrons in one direction.
The Concept of Relaxation Time (τ)
Collisions don’t occur at neat intervals like a ticking clock. Instead, they happen randomly. But physicists use a clever trick here—they introduce the concept of relaxation time, denoted by τ.
Relaxation time is the average time interval between two successive collisions for an electron. It acts like a statistical measure of how long, on average, an electron can accelerate before it gets knocked off course.
Using this concept, we can express the average velocity an electron gains from the electric field between collisions. This velocity is known as the drift velocity:
The negative sign reminds us that electrons move opposite to the electric field.
Drift Velocity and Current Flow
Now comes the crucial link between microscopic motion and macroscopic current.
Drift velocity may sound tiny—and it is. In fact, the drift velocity of electrons in household wiring is only a fraction of a millimeter per second! Yet this is enough to produce the current that powers our devices, because the number of electrons moving together is enormous.
Suppose:
- n = number of free electrons per unit volume,
- A = cross-sectional area of the conductor,
- Δt = time interval considered.
Then, the charge transported in time Δt is:
This equation reveals that even though each electron drifts slowly, the combined effect of trillions of them moving together gives us a measurable electric current.
Current Density and Ohm’s Law
Let’s push the math a little further.
The current density, denoted by jjj, is defined as the current per unit area of the conductor:
Substituting the expression for current, we find:
Now, if you compare this with Ohm’s law in microscopic form:
You see that the electrical conductivity σ is:
This is a beautiful result. It tells us that Ohm’s law is not just an empirical rule—it emerges naturally from the microscopic behavior of electrons in a conductor.
The Origin of Resistivity
Conductivity tells us how easily electrons flow; resistivity tells us how much they are hindered.
Since resistivity is simply the reciprocal of conductivity:
This formula gives us the origin of resistivity. It depends on three main factors:
- Number density of free electrons (n)
A higher density of electrons means better conductivity, hence lower resistivity. - Relaxation time (τ)
If collisions are frequent (short τ), electrons don’t get much chance to accelerate, which increases resistivity. - Electron mass and charge
These are universal constants for electrons and remain the same in all conductors.
Thus, resistivity boils down to how often electrons collide and how many electrons are available to carry current.
Visualizing Electron Drift
If you’re still picturing electrons zipping like race cars through a wire, think again. A better analogy is people walking in a crowded hallway. Most bump into others randomly, but if a gentle push is applied in one direction, the whole crowd begins to slowly shift that way. That slow, directed movement is the drift velocity.
Meanwhile, the “bumping into others” represents the collisions causing resistivity. Without collisions, electrons would just keep accelerating under the electric field, and resistivity would not exist.
Real-World Connection: Why Wires Heat Up
This microscopic theory explains everyday experiences too. When electrons collide with ions in the lattice, they transfer energy to them. This energy shows up as heat. That’s why resistivity leads to wires heating up when current passes through them.
Materials with long relaxation times (few collisions) conduct better, like copper and silver. Materials with shorter relaxation times have higher resistivity and waste more energy as heat.
Limitations of the Model
Of course, the picture we’ve drawn is simplified. We assumed:
- n (number of free electrons per unit volume) is constant,
- τ (relaxation time) is constant,
- Temperature effects and lattice vibrations are ignored.
In actuality, resistivity is temperature-dependent, and not all materials meet this simple model exactly. Semiconductors, insulators, and superconductors require different theoretical models to account for. But for regular metals at moderate temperatures, this model is gorgeous.
Conclusion
The manner in which the electrons move in an electric field may appear like a tiny trifle, but this is what breaks open the mysteries of resistivity and which is the foundation of the Ohm law. In essence, this is nothing more than millions of mini-collisions occurring millions of times in an electrical conduction.
They are painfully slow, keep on bumping into things and yet they are able to generate the continuous currents that power our homes, our laptops and the entire modern world too. The next time you put in your charger, consider it: it is not some hectic flow of fast electric particles slipping through the wire, but a slow, steady movement which actually gives it all it can be.