Question

What is the mean free time between collisions for electrons in an iron wire?

answer in seconds

wrong answers= 3.67e-10s, 1.39e-15s

please help

Answer 1

In each collision with an atom, the electron loses essentially all its forward momentum, so it's velocity after a collision is zero. If the wire there is a current in the wire in response to an imposed potential difference, then between collisions, the electrons are accelerated by the electric field, and a = q*E/m, where q is the charge on an electron, m is the electron mass, and E is the electric field. If the average time between collisions is T, then the average speed of an electron just before it collides is a*T = q*E*T/m, and the average speed of the electrons during the intervals between collisions is (q + q*E*T/m)/2 = q*E*T/(2*m) = v_d. v_d is called the drift velocity.

IThe current in a wire is simply the total amount of charge (Q) that passes through a given cross sectional area (A) of the wire per unit time. N conduction electrons per unit volume(V) in the wire, and these all move with the average drift velocity, then the current in the wire is given by:

I = dQ/dt = q*N*A*vd = (q^2 * N * T *A *E)/(2*m)

The electric field is simply the voltage difference (V) divided by the length of the wire (L), so:

I = V*(q^2 * N * T * A)/(2*m*L).

For an ohmic material, we know phenomenologically that V = I*R, so we can equate R with the reciprocal of the factor that multiplies V in the previous equation:

R = ((2*m)/(q^2 * N * T)) * (L/A)

R = r * L/A

where r = (2*m)/(q^2*N*T) is called the resistivity of the material in the wire. Resistivity has units of ohm*m.

Rearranging this to solve for the mean time between collisions, we have:

T = (2*m)/(q^2 * N * r)

At room temperature, the resistivities of Fe is 9.71*10^-8 ohm*m.

The mass of an electron is 9.109*10^-31 kg and the charge on an electron is 1.602*10^-19 C

The only remaining unknown on the r.h.s. of this equation is the number of free (conduction) electrons per unit volume. To calculate this, we need to multiply the number of conduction atoms per atom (n) by the number density of atoms in the material:

N = (n * Avagadro's number * density)(atomic mass)

Fe has two conduction electrons per atom

The density of Fe is 7.87 gm/cm^3, and atomic masses are 55.845 gm/mol.

This gives:


N_Fe = 5.29*10^23 electrons/m^3

Plugging all these values into the above equation, one finds that:


T_Fe = 1.38*10^-9 s

The mean free time between collisions for electrons in an iron wire is 1.38*10^-9 sec.