# On the microscopic description of a steady electric current

In Purcell's Electricity and magnetism, page 137, the author derived a formula describing the average velocity $$\overline{\textbf{u}}$$ of positive ions inside a conductor, a gas made of neutral molecules as well as the ions, when subject to an electric field $$\textbf{E}$$, and it is given by

$$\overline{\textbf{u}}= \frac{e\textbf{E}\overline{t}}{M}$$

where $$M$$ is the mass of the ion, $$e$$ the elementary charge, and $$\overline{t}$$ is the mean time before a collision occurs between an ion an a neutral molecule within the gas.

This shows that the average velocity of a charge carrier is proportional to the force applied to it. If we observe only the average velocity, it looks as if the medium were resisting the motion with a force proportional to the velocity. That is the kind of frictional drag you feel if you try to stir thick syrup with a spoon, a "viscous" drag. Whenever charge carriers behave like this, we can expect something like Ohm's law.

How did the author infer from the formula the existence of a frictional force? The relation, as far as I can see, only says the resulting average velocity is proportional to the applied field, it doesn't inform us about the existence or non-existence of a frictional force that opposes this very field.

You have missed out an important point.

The equation $$\overline{\textbf{u}}= \dfrac{e\textbf{E}\overline{t}}{M}$$ can be rewritten as $$e\textbf{E} - \dfrac{M}{\overline{t}} \overline{\textbf{u}}= 0$$

So you have the force due to an electric field minus a "frictional" force which is proportional to the (terminal) velocity,$$\overline{\textbf{u}}$$, equal to zero, with $$\dfrac{M}{\overline{t}}$$ the constant of proportionality.

Compare this with the terminal velocity motion of a sphere in a viscous fluid under the action of the gravitational force with $$m\textbf{g} - k\textbf{u}_{\rm terminal}= 0$$ where $$k\textbf{u}_{\rm terminal}$$ is Stokes' drag force..

You could write the total current as

$$J=ne\bar{\bf u}$$

where $$n$$ is the number density of particles per volume of material and $$e$$ is the charge on each particle.

So you have

$$J=ne\frac{e{\bf E}\bar{t}}{M}$$

If you simplify things by defining $$\sigma={ne^2\bar t}/{M}$$, you have the microscopic form of Ohm's Law

$$J=\sigma{\bf E}$$

Somehow, something is applying a force to the particles that is counter to the one produced by the electric field, resulting in an equilibrium at a certain average velocity (or current density).

Whether it's appropriate to call that force "fictional" or not is above my pay grade, but you are not seeing the particles accelerating indefinitely as you would if you applied a constant force to them and nothing was happening to accelerate them in the opposite direction.