Flux

Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. The field lines of a vector field F through surfaces with unit normal n, the angle from n to F is θ. Flux is a measure of how much of the field passes through a given surface. F is decomposed into components perpendicular (⊥) and parallel ( ‖ ) to n. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. Top: Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. Bottom: Field line through a curved surface, showing the setup of the unit normal and surface element to calculate flux. To calculate the flux of a vector field F {\displaystyle \mathbf {F} } (red arrows) through a surface S {\displaystyle S} the surface is divided into small patches d S {\displaystyle dS} . The flux through each patch is equal to the normal (perpendicular) component of the field, the dot product of F ( x ) {\displaystyle \mathbf {F} (\mathbf {x} )} with the unit normal vector n ^ ( x ) {\displaystyle {\hat {\mathbf {n} }}(\mathbf {x} )} (blue arrows) at the point x {\displaystyle \mathbf {x} } multiplied by the area d S {\displaystyle dS} . The sum of F ⋅ n ^ d S {\displaystyle \mathbf {F} \cdot {\hat {\mathbf {n} }}\,dS} for each patch on the surface is the flux through the surface