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A very simple relation exists between current density and drift speed, namely, that:

$$J = nev_d$$

…where $n$ is the number density of conduction electrons, $e$ is the electronic charge, and $v_d$ is the drift speed of the electrons in the medium.

The actual speed of electrons in a conductor is very large, of the order of $v_e \approx 10^6 \; \rm{\frac{m}{s}}$. The electron drift speed, on the other hand, just reflects the tendency of electrons, on average to move in one direction or another in response to an electric field. It is much, much smaller! Typically, in fact, the drift speeds of electrons in everyday wiring are of the order of $v_d \approx 10^{-4} \text{ m}$.

The number density of electrons in copper is:

$$n = 8.5 \times 10^{28} \; \rm{\frac{electrons}{m^3}}$$

…and the electronic charge is:

$$e = 1.602 \times 10^{-19} \text{ C}$$

Standard 10 gauge copper wire, such as you might find in your house or apartment wiring, has a radius of about:

$$r = 1.3 \text{ mm}$$

Suppose, instead of the electrons in such a wire having a typical drift velocity of $v_d \approx 10^{-4} \; \rm{\frac{m}{s}}$, they had a drift velocity more like that of the electron random speed, so that:

$$v_d = 1.0 \times 10^6 \; \rm{\frac{m}{s}}$$

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