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In the Drude model, we have the equation of motion for electrons in the presence of an EM wave: $$m \frac{\partial\mathbf{v}}{\partial t}=-e(\mathbf{E}+\mathbf{v}\times\mathbf{B})-\frac{m\mathbf{v}}{\tau_c}$$ and we have the current density: $$\mathbf{j}=-n_e e\mathbf{v}.$$

As I understand it, in the current density expression, $\mathbf{v}$ is the drift velocity. But if the first expression is an equation of motion, I would expect $\mathbf{v}$ to be the actual velocity of the electron. I've seen Wikipedia use the first expression with the momentums $\mathbf{p}$ and $\langle\mathbf{p}\rangle$ variously.

Is $\mathbf{v}$ in the first expression actual electron velocity or drift velocity? If it is drift velocity, why can we use that when the equation is apparently dealing with a single electron?

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    $\begingroup$ Your first equation is the equation of motion for the drift velocity (look up the Lorentz equation). So if you have the solution, you can then move on and compute the current density, conductivity and so on. E.g. take the steady-state condition, where $\partial v /\partial t=0$ i.e. the centrifugal force and the magnetic Lorentz force are balancing each other out, then the drift velocity is given by the Eq. $(m/\tau)\mathbf{v}_d=-e\left(\mathbf{E}+\mathbf{v}_d \times \mathbf{B}\right).$ Possible that I misunderstood what you're asking for. $\endgroup$
    – Ellie
    Commented Apr 20, 2016 at 9:28
  • $\begingroup$ @Phonon I've added a clarifying statement. $\endgroup$
    – binaryfunt
    Commented Apr 20, 2016 at 9:33

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This is indeed the drift velocity that appears in the Drude model and not the actual velocity. The reason for that is that we are interested in the electrical current induced by a potential difference in a circuit. This electrical current will measure what changes in the motion of the electrons when compared to the case when there is no applied voltage.

Even in absence of applied voltage across the circuit, electrons are moving all over the place at some crazy speed. But this random aspect of their motion will be unchanged when applied a potential difference. What will change, however, is precisely the drift velocity of the electrons and that's the reason why it is enough to look at this average velocity in the Drude model to derive equations relating current intensity and voltage for instance.

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