Theory of Metallic Conduction

Free electrons drift opposite to the electric field's direction while colliding with the crystal's positive ions. Between collisions, the electric force accelerates electrons, but collisions randomly change their direction. The net result is a steady drift velocity superimposed on random thermal motion, creating a directional flow of charge through the conductor.

Mean free time is the average duration between successive collisions an electron experiences as it moves through a conductor. It directly determines the drift velocity and electrical conductivity. A longer mean free time allows electrons to accelerate more between collisions, increasing conductivity, while shorter mean free time reduces the conductor's ability to carry current.

The theory uses Newton's second law to find electron acceleration under an applied electric field. By averaging velocity over collisions and accounting for mean free time, the drift velocity becomes proportional to the electric field. This relationship between current density and electric field directly yields Ohm's law, connecting microscopic electron behavior to macroscopic electrical properties.

Electrical conductivity depends on three factors: the number of free electrons per unit volume, the electron's charge and mass, and the mean free time between collisions. Materials with more free electrons and longer mean free times exhibit higher conductivity. The theory shows that conductivity is inversely proportional to collision frequency, making it sensitive to temperature and material structure.

Higher temperatures cause increased vibrations of ions in the crystal lattice. These vibrations lead to more frequent collisions between electrons and ions, reducing the mean free time. With shorter mean free time, electrons drift more slowly despite the same applied field, resulting in decreased electrical conductivity at elevated temperatures.

Without an applied field, electrons undergo random thermal motion, colliding with ions and changing direction unpredictably. Although individual electrons move constantly, the average velocity over many collisions is zero because there is no net directional bias. Only when an electric field is applied does a preferred drift direction emerge.

The classical model uses Newton's laws and produces results remarkably close to quantum mechanical predictions, including the correct temperature dependence of conductivity. However, quantum mechanics is required to determine the exact mathematical form of how conductivity varies with temperature. The classical approach provides excellent physical insight while remaining computationally simpler for undergraduate understanding.