Given the current density, \(\vec j\), one can always determine the current through a surface with area, \(A\), and normal vector, \(\hat n\): \[\begin{aligned} I = A(\vec j\cdot \hat n)\end{aligned}\...Given the current density, \(\vec j\), one can always determine the current through a surface with area, \(A\), and normal vector, \(\hat n\): \[\begin{aligned} I = A(\vec j\cdot \hat n)\end{aligned}\] If the current density changes over the surface, one must take an integral instead: \[\begin{aligned} I=\int \vec j \cdot d\vec A\end{aligned}\] where \(d\vec A\), is a surface element with area, \(A\), and direction given by the normal to the surface at that point.
