What is the mean free time 𝝉 between collisions for the conduction electrons in copper?

Question

What is the mean free time 𝝉 between collisions for the conduction electrons in copper?  [ρcopper = 1.69 × 10 -8 Ω.m; n = 8.49 × 1028 m -3]

Answer 1

Before doing this problem, it's worth having a mental picture of how conduction electrons behave in a metal. In the absence of an electric field, the conduction electrons can be pictured much like the atoms of an ideal gas in equilibrium. They move around the metal with a certain kinetic energy and a relatively high velocity occasionally scattering and changing direction, but their movements are random and there is no net motion or flow of particles. Their velocity as they move around the conductor is called the random velocity. 

When an electric field is applied to the metal, the randomly moving electrons acquire a net velocity anti-parallel to the electric field. This is known as the drift velocity.

Conduction electrons have a rather high random velocity (~10⁵ m/s), but do to collisions between the electrons and the medium in which they travel, the drift velocity is quite small (~10^-4 m/s). 

For your specific question, resistivity ρ is defined as 1/σ, σ being the conductivity.

Conductivity can be derived in a couple of ways that achieve the same result. For one such derivation, see section 4.4 of M. A. Omar's Elementary Solid State Physics.

Expression for conductivity

σ = n*e²*τ/m_e 

where n is the # of electrons per m³, e is the charge of the electron, τ is the collison time, and m_e is the mass of the electron. 

Rearranging our expression for conductivity to solve for collision time, we obtain: 

τ = σ*m_e/(N*e²)

Subbing in 1/ρ for σ:

τ = m_e/(ρ*N*e²)

Now we can plug in your values and the known mass and charge of the electron to obtain a collision time of

τ = 2.48*10^-14 seconds

It is worth taking a quick intuitive look at this formula to see how it reflects the physical situation: 

1. As n increases, so does σ. More electrons means higher conductivity which checks out. 
2. As e increases, so does σ. A more charged particle means higher conductivity. A more charged particle will experience a greater force under an electric field, which would result in a larger current, so this checks out as well. 
3. As τ increases, so does σ. Longer time between collisions means higher conductivity. This makes sense given the picture we envisioned above. 
4. As m_e increases, σ decreases. A heavier particle means lower conductivity. The electrons are being accelerated by the force of an electric field. F=ma, F/m = a, hence heavier particles will accelerate more slowly under the same field.