Electric Current

All ordinary matter is built from atoms, each with a positively charged nucleus surrounded by negatively charged electrons. In metals, the outermost one or more electrons of each atom are only loosely bound to their nucleus and can be detached quite easily -- these are the free electrons, and a substance with an abundance of them is called a conductor. At any given temperature, these free electrons move about randomly and continuously throughout the conductor, colliding again and again with the lattice of fixed positive ions. Because this motion is completely random in direction, there is, on average, no net transfer of charge from one end of the conductor to the other, and hence no current, even though the electrons themselves are moving.

Electric current is formally defined as the rate of flow of charge across a given cross-sectional area A of a conductor. If a net charge $\Delta Q$ passes through any cross-section of the conductor in a time interval $\Delta t$, the average current over that interval is

Since the rate of charge flow is not always constant, the more general, instantaneous current is defined as the limit of this average current as $\Delta t \to 0$:

$I = \lim_{\Delta t \to 0}\dfrac{\Delta Q}{\Delta t} = \dfrac{dQ}{dt} \qquad (2.2)$ …