Physics

$τ=ne_{2}σm $

$l=(σn)_{(−1)}$

Where $l$ is the mean free path, $n$ is the number of target particles per unit volume, and $σ$ is the effective cross-sectional area of collision.

- An electron will suffer collisions with the heavy fixed ions, but after collision, it will emerge with the same speed but in random directions. If we consider all the electrons, their average velocity will be zero since their directions are random. Hence, $N1 ∑_{i=1}v_{i}=0$
- Now, if an electric field is present. Electrons will be accelerated due to this field by $a=m−eE $ where $−e$ is the charge and $m$ is the mass of electron.
- Consider the $ith$ electron at a given time $t$. This electron would have had its last collision some time before $t$, and let $t_{i}$ be the time elapsed after its last collision. If $v_{i}$ was its velocity immediately after the last collision, then its velocity $V_{i}$ at time t is $V_{i}=v_{i}+(m−eE t_{i})$.
- The average velocity of the electrons at time t is the average of all the $V_{i}s$.
- Hence $(V_{i})_{average}=(v_{i})_{average}+meE (t_{i})_{average}$
- The average of $v_{i}s$ is zero as seen above.
- The average of $t_{i}s$ is called relaxation time denoted by $τ$.
- The average of $V_{i}s$ is called drift velocity $V_{d}$
- Hence $V_{d}=0+m−eE τ$
- The last result show that electron move with average velocity which is independent of time.

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