Dot Product Calculates Perpendicular Current Flux

The flux (or flow) crossing a boundary surface ($S$) is determined exclusively by the component of the current density vector ($\vec{j}$ or $\vec{J}$) that is perpendicular ($\vec{\jmath}_{\perp}$) to the surface. This essential relationship is mathematically quantified by the scalar product (dot product) of the current vector and the surface normal ($\vec{n}$). Consequently, when the current vector is directed parallel or tangential to the surface, the perpendicular component vanishes entirely, resulting in zero net flux. The sign of the scalar product dictates the direction of the flow: a positive scalar product indicates a net outflux (when $\vec{J}$ points predominantly outward), while a negative scalar product indicates a net influx (when $\vec{J}$ points inward).

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