Free Electron Theory
FREE ELECTRON THEORY
Classical free electron theory of metals
This theory was developed by Drude and Lorentz and hence is also known as Drude-Lorentz theory. According to this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a container. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy.
Drift velocity
If no electric field is applied on a conductor, the free electrons move in random directions. They collide with each other and also with the positive ions. Since the motion is completely random, average velocity in any direction is zero. If a constant electric field is established inside a conductor, the electrons experience a force F = -eE due to which they move in the direction opposite to direction of the field. These electrons undergo frequent collisions with positive ions. In each such collision, direction of motion of electrons undergoes random changes. As a result, in addition to the random motion, the electrons are subjected to a very slow directional motion. This motion is called drift and the average velocity of this motion is called drift velocity vd.
Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric field = -eE.
By Newton’s law, -eE = mdvd/dt dvd = -eEdt/m
Integrating,
Vd = -eEt/m + Constant
When t = 0, vd = 0 Therefore Constant = 0
Vd = -eEt/m --------------- (1)
Electrical conductivity Consider a wire of length ‘dl’ and area of cross section ‘A’ subjected to an electric field E. If ‘n’ is the concentration of the electrons, the number of electrons flowing through the wire in dt seconds = nAvddt.
The quantity of charge flowing in time dt = nAvddt.e
Therefore I = dq/dt = neAvd
Current density J = I/A = nevd
Subsittuting the value of vd from (1),
J = nee Et/m = ne2Et/m --------------- (2)
Classical free electron theory of metals
This theory was developed by Drude and Lorentz and hence is also known as Drude-Lorentz theory. According to this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a container. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy.
Drift velocity
If no electric field is applied on a conductor, the free electrons move in random directions. They collide with each other and also with the positive ions. Since the motion is completely random, average velocity in any direction is zero. If a constant electric field is established inside a conductor, the electrons experience a force F = -eE due to which they move in the direction opposite to direction of the field. These electrons undergo frequent collisions with positive ions. In each such collision, direction of motion of electrons undergoes random changes. As a result, in addition to the random motion, the electrons are subjected to a very slow directional motion. This motion is called drift and the average velocity of this motion is called drift velocity vd.
Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric field = -eE.
By Newton’s law, -eE = mdvd/dt dvd = -eEdt/m
Integrating,
Vd = -eEt/m + Constant
When t = 0, vd = 0 Therefore Constant = 0
Vd = -eEt/m --------------- (1)
Electrical conductivity Consider a wire of length ‘dl’ and area of cross section ‘A’ subjected to an electric field E. If ‘n’ is the concentration of the electrons, the number of electrons flowing through the wire in dt seconds = nAvddt.
The quantity of charge flowing in time dt = nAvddt.e
Therefore I = dq/dt = neAvd
Current density J = I/A = nevd
Subsittuting the value of vd from (1),
J = nee Et/m = ne2Et/m --------------- (2)