Physics

Electric current is really just counting how many electrons flow past a point per second, weighted by their charge. The key insight is that in time $t$, all electrons within a distance $v_d \cdot t$ from a cross-section will pass through it—so the volume of conductor you're sampling is $A \cdot v_d \cdot t$. Multiply that volume by electron density $n$ to get the total number of electrons, then by charge $e$ to get total charge, and divide by time to find the current flowing through.

Electric current is really just counting how many electrons flow past a point per second, weighted by their charge. The key insight is that in time $t$, all electrons within a distance $v_d \cdot t$ from a cross-section will pass through it—so the volume of conductor you're sampling is $A \cdot v_d \cdot t$. Multiply that volume by electron density $n$ to get the total number of electrons, then by charge $e$ to get total charge, and divide by time to find the current flowing through.