[Physics] Question regarding Drift velocity in general

electric-circuitselectricityelectrostaticskinetic-theory

The derivation of drift velocity in case of electrons is equivalent to the case of an charged ionic gas and therefore all the arguments also apply there. Now for an ideal "ionic" gas which interacts with each others only as perfect spheres would (for arguments sake), when we apply an external electric field, they would just show drift, that means that they "favour" one direction more than the others where drift velocity is given by $v_d=(eE/m)\tau$ where $\tau$ is the average relaxation time. Hence the net velocities of the electrons wont vectorially add to zero, and moreover the speed as given by kinetic theory of matter would not depend only on the temperature but also depend on electric field. Does this not contradict two fundamental postulates of kinetic theory?

Moreover in case of electrons in a circuit with many series resistors, the relaxation time would be different. The resistivity as defined by $E=\rho J$ and $\rho=m/(ne^2\tau)$ where j=current density, n=number of electrons per unit volume and e=electronic charge, should be different for all the series resistors. But then will not the drift velocity be different for all the resistors if the top equation is correct since the resistivities might defer, causing the relaxation time to differ. Doesn't this contradict the postulation of a steady current which requires constant drift velocity. Moreover suppose we use the same material of different cross section, since there is no area dependence in the drift velocity, it shouldn't change but then again we don't get a steady current as J remains a constant and I is simply J times A.

Best Answer

Alternate method to derive drift velocity:

Consider a field $\vec{E}$ inside the conductor. Using equations of motion we can say that for every charge inside the conductor, $$\vec{v_1}=\vec{u_1}+\frac{\vec{E}e}{m}t_1$$ $$\vec{v_2}=\vec{u_2}+\frac{\vec{E}e}{m}t_2$$ $$.$$ $$.$$ $$.$$ $$\vec{v_n}=\vec{u_n}+\frac{\vec{E}e}{m}t_n$$
where $t_1,t_2,...t_n$ are the times until each of them collide with another particle respectively.

Summing them and dividing by the number of charge particles($N$) we get, $$\sum_{i=1}^n\frac{\vec{v_i}}{N}=\sum_{i=1}^n\frac{\vec{u_i}}{N}+\frac{\vec{E}e}{m}\sum_{i=1}^n \frac {t_i}{N}$$

We can say that $$\sum_{i=1}^n\vec{u_i}=0$$ as the charges initially are in perfect random motion. Thus substituting $\sum_{i=1}^n\frac{\vec{v_i}}{N}$ as $\vec{v_d}$, the drift velocity and $\sum_{i=1}^n \frac {t_i}{N}$ as $\tau$, the relaxation time for each particle we get $$\vec{v_d} = \frac{\vec{E}e}{m}\tau$$

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