11.1 Electric Current

Electric current is the rate at which charge passes through a cross section of a conductor. If an amount of charge dq passes through in a small time dt, then the instantaneous current is

I = \frac{dq}{dt}

This definition is naturally calculus-based because current is a rate. If the current is constant, then this reduces to

I = \frac{\Delta q}{\Delta t}

Conventional current points in the direction positive charge would move, even though in metallic conductors the actual mobile charges are electrons moving the opposite way. The current is the same at every point in a steady single-path conductor because charge is conserved.

This topic becomes much more powerful when you connect current to microscopic charge motion. If charge carriers of number density n, charge q, and drift speed v_d move through a wire of cross-sectional area A, then in time dt the carriers in a cylinder of length v_d dt move through the cross section. The total charge passing through is ${dq = nqA v_d dt}$ , so

I = \frac{dq}{dt} = nqAv_d

This derivation explains why a thicker wire or a larger drift speed leads to more current. It also shows why current is not about one charge moving fast, but about very many charges moving together through material.

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