# Electron speed. Time between collisions

A conduction electron travels a metallic conductor of length $l$ in the time $t_c$ when at the ends of the conductor is applied a difference of potential $U$. A free electron travels the same distance in vacuum and under the action of the the same potential difference in the time $t_0$. One can assume that  the mass of the condition electron is equal to the mass of the electron in vacuum.

a) Find that the rapport $t_c/t_0$ does not depend on the potential difference U.

b) Determine the mobility of the electrons in the conductor, if there are given $t_0=2.4*10^{-4} s$, $t_c=10^6 s$, the electron charge $e$ the electron mass $m_e$

a) the conduction electron travels the entire length of the conductor with the transport speed

$v_t=\mu E=\mu*(U/l)$

the time that the electrons needs to travel this length $l$ is

$t_c =l/v_t =l^2/(/mu U)$

The free electron travels in vacuum the same distance in the time

$t_0=\sqrt {\frac{2l}{a}} =l \sqrt {\frac{2m_0}{e U}}$ so that $\frac {t_c}{t_0^2} = \frac{e}{2m_0 \mu}$

b) For mobility from the above expression we get

$\mu = \frac{t_0^2}{t_c}*\frac{e}{2m_0} =5*10^{-3} m^2/(Vs)$

If to a Copper conductor having a length $l=10 m$ and an area of $S=1 mm^2$ one applies a potential difference U=0.62 V, then through this conductor it is established a current of I=4 A. Knowing that each Cu atom gives one electron to conduction, find the average time $\tau$ between two collisions of the electrons with the atoms of the crystal lattice.

The conductivity is the inverse of resistivity

$\sigma =1/\rho= (1/R)*(l/S)=(I/U)(l/S) = (n e^2\tau)/m_0$

Since the concentration of conduction electrons is

$n=\rho N_A/M$

we get

$\tau= \frac{m_0lMI}{\rho e^2 N_A US} =2.71*10^{-14} s$