I have come upon a paradox, to me, in the vector characteristics of electric flux. I wonder if others have fallen into - and resolved - this problem and might be able to resolve this.

An electric field is a vector and hence when it contacts, and passes through a surface of area dA, it would seem that it is still a vector, ie. the electric flux dψ in the equation dψ=E n dA (where E and the unit vector n are both vectors) must also be a vector.

When integrated, ψ, is regarded as a scalar. It would seem that the flux ψ must "flow" in a direction and my problem is that it would still seem to have the characteristics of a vector.

Wikipedia states (http://en.wikipedia.org/wiki/Flux)
In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the amount that flows through a unit area per unit time. Flux in this definition is a vector.
In the field of electromagnetism and mathematics, flux is usually the integral of a vector quantity over a finite surface. It is an integral operator and acts on a vector field as do the gradient, divergence and curl found in vector analysis. The result of this integration is a scalar quantity.

WHY IS THE INTEGRATED ELECTRIC FLUX NOT A VECTOR?

 

WHY IS THE INTEGRATED ELECTRIC FLUX NOT A VECTOR?

The quantity that is integrated \( d\psi\) is considered a scalar quantity, not a vector, because it is the dot product of the field vector with the differential area vector. The infinitesimal area (\(dA\) or \(dS\))is given the direction of a vector that is normal to the surface.

With magnetic fields, we define a magnetic flux density which is a vector field. Perhaps this is more similar to the flux you are familiar with in fluid mechanics. I'm not sure, but maybe you are just reading too much into the terminology which could be a little misleading when comparing different subjects.

The basic idea makes intuitive sense if we think about fluid flowing in a pipe. If the flux density is integrated over area then we get the total flow rate through the cross section of the pipe. The flow needs to be perpendicular to the cross sectional area to truly flow in the pipe. Hence, the dot product is used to capture only the flow that is leaving the surface. If it is parallel to the cross section, then the fluid can't flow through. The flow rate is not a vector, but a value (scalar) despite the fact that we know the direction of flow is with the pipe. So the flux density has a direction for each of the vectors in the vector field, but the total flux does not have direction and is a scalar.

 

Thank you steveb. The step that will get me over this hurdle is the definition of dψ as a dot product and thus it is a scalar value - but the olympics have 10 hurdles per race - I suspect there are others ahead in this topic!