Current Density

Using the above diagram, show that the electric current passing through a conductor is given by I=nqAv_d, where v_d is the drift velocity of the charge carriers, A is the perpendicular cross-sectional area and n is the number density of charge carriers. Hence, deduce the equation for current density J, defined as the current per unit area.

$$\begin{aligned} I \, &= \frac{dQ}{dt} \\ I \, &= \frac{d \left( Nq \right)}{dt} \\ I \, &= \frac{d \left( n A l q \right)}{dt} \\ I \, &= nqA \left( \frac{d l}{dt} \right) \\ I \, &= nqA v_{d} \end{aligned}$$

Hence,

$$\begin{aligned} J \, &= \frac{I}{A} \\ J \, &= \frac{n qA v_{d}}{A} \\ J \, &= nq v_{d} \end{aligned}$$

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