Flux is the amount of a vector field that "flows through a surface." We now discuss the electric flux through a surface (a quantity needed in Gauss's law): ΦE = ∫∫E · dA, where ΦE is the flux through a surface with differential area element dA, and E is the electric field in which the surface lies. There are several important points to consider in this expression: 1. It is an integral over a surface, involving the electric field at the surface. 2. dA is a vector with magnitude equal to the area of an infinitesimal surface element and pointing in a direction normal (and usually outward) to the infinitesimal surface element. 3. The scalar (dot) product E · dA implies that only the component of E normal to the surface contributes to the integral. That is, ΦE = ∫∫E · dA = ∫∫E dA cos(θ), where θ is the angle between E and dA. When you compute flux, try to pick a surface that is either parallel or perpendicular to E, so that the dot product is easy to compute. (Figure 1) Two hemispherical surfaces, 1 and 2, of respective radii ri and r2, are centered at a point charge and are facing each other so that their edges define an annular ring (surface 3), as shown. The field at position due to the point charge is: E(F) = kF/r^2, where k is a constant proportional to the charge, r = 7, and ˆF/r is the unit vector in the radial direction.
