Final answer:
To derive the conductivity equation σ= ne²/mτ, we utilize the relationship between current density (J), drift velocity (vd), and electric field (E). Drift velocity is influenced by the characteristics of the conductor and electrons, which leads to the conductivity formula being derived from ohmic proportionality between J and E.
Step-by-step explanation:
To derive the conductivity equation σ= ne²/mτ from the drift velocity equation vd = -eE/mτ, we start by acknowledging that drift velocity (vd) is influenced by the electric field (E) in the conductor and the charge of the electrons (e). The negative sign indicates that the electrons move opposite to the electric field due to their negative charge. The symbol τ represents the average time between collisions (mean free time), m is the mass of an electron, and n is the density of free electrons in the conductor.
Ohm's law states that the current density (J) is proportional to the electric field (E), with the constant of proportionality being the conductivity (σ), thus J = σE. We can also express current density (J) as J = nqvd, where q is the electron charge. By substituting the equation for vd into this expression for J, we get J = nqv₄ = nqe(-eE/mτ), which simplifies to J = (ne²/mτ)E. From this, we see that the conductivity (σ) is defined as σ = ne²/mτ.