**Branch :**First Year-Engineering Syllabus

**Subject :**Physics-II

## Classical free Electron theory

**Introduction:**

The fact that electricity can ﬂow through a substance was discovered by 17^{th} century German physicist Otto von Guericke. Conduction was rediscovered independently by Englishman Stephen Gray during the early 1700s. Gray also noted that some substances are good conductors while others are insulators. The electron theory, which is the basis of modern electrical theory, was ﬁrst advanced by Dutch physicist Hendrik Antoon Lorentz in 1892. The widespread use of electricity as a source of power is largely due to the work of pioneering American engineers and inventors such as Thomas Alva Edison, Nikola Tesla, and Charles Proteus Steinmetz during the late 19^{th} and early 20^{th} centuries.

In this, we discuss the theory of conduction, speciﬁcally the theory of classical conduction whose defects were explained by the quantum theory. The modiﬁcations that the quantum theory adds to classical conduction not only explains the ﬂaws that arose in the classical theory, but also adds a new dimension to conduction that is currently leading to new developments in the physics world.

**Theory of classical free Electron:**

Around 1900, Paul Drude improved the theory of classical conduction given by Lorentz. He reasoned that since metals conduct electricity so well, they must contain free electrons that move through a lattice of positive ions. This motion of electrons led to the formation of Ohm’s law. The free-moving electrons act just as a gas would; moving in every direction throughout the lattice. These electrons collide with the lattice ions as they move about, which is key in understanding thermal equilibrium.

The average velocity due to the thermal energy is zero since the electrons are going in every direction. There is a way of aﬀecting this free motion of electrons, which is by use of an electric ﬁeld. This process is known as electrical conduction and theory is called Drude-Lorentz theory.

The assumptions of the Drude-Lorentz classical theory of free-electrons are the following.

- Metals contain free electrons that move through a lattice of positive ions. These free electrons are responsible for electrical conduction when an electric potential is maintained across the conductor.

- Electric ﬁeld produced by lattice ions is considered to be uniform throughout the solid and hence neglected.

- The force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected.

- Free electrons in a metal resemble molecules of a gas and therefore the laws of kinetic theory of gases are applicable to free electrons. The motion of an electron is completely random. In the absence of electric ﬁeld, number of electrons crossing any cross section of a conductor in one direction is equal to number of electrons crossing the same cross section in opposite direction. Therefore net electric current is Zero. This random motion of the electrons is due to thermalenergy. Hence, the average kinetic energy of the electron is given by

where, m_{e} is the mass of the electron, v_{th} is the thermal velocity, k is the Boltzmann constant and T is the absolute temperature. Therefore, the thermal velocity of free-electrons in a metal at given temperature is given by

- Electric current in the conductor is due to the drift velocity acquired by the electrons in the presence of the applied electric ﬁeld.