Exploring Scalar and Vector Flux in Electromagnetism

[Greg Bernhardt]

Definition/Summary

Flux sometimes means total flow through a surface (a scalar), and sometimes means flow per unit area (a vector).

In electromagnetism, flux always means total flow through a surface (a scalar), and is measured in webers (magnetic flux) or volt-metres (electric flux).

Scalar flux is the amount of a vector field going through a surface: it is the integral (over the surface) of the normal component of the field: $\Phi\ =\ \oint_S\mathbf{E}\cdot d\mathbf{A}$

For a closed surface, this equals (Gauss' theorem, or the divergence theorem) the integral (over the interior) of the divergence of the field: $\Phi\ =\ \int\int\int_V \mathbf{\nabla}\cdot\mathbf{E}\,dxdydz$.

Therefore the scalar flux, through a closed surface, of an electric field is proportional to the enclosed charge (Gauss' law: $\Phi_{E}\ =\ Q_{total}/\varepsilon_0,\ \ \Phi_{D}\ =\ Q_{free}/\varepsilon_0,\ \ \Phi_{P}\ =\ -Q_{bound}/\varepsilon_0$), and of a magnetic field is zero (Gauss' law for magnetism: $\Phi_{B}\ =\ \Phi_{H}\ =\ \Phi_{M}\ =\ 0$).

Equations

FLUX THROUGH A CLOSED SURFACE, S:

Gauss' Law:

$$\Phi_\mathbf{E}(S)\ =\ \oint_S\mathbf{E}\cdot d\mathbf{A}\ =\ Q/\varepsilon_0$$

Gauss' Law for Magnetism:

$$\Phi_\mathbf{B}(S)\ =\ \oint_S\mathbf{B}\cdot d\mathbf{A}\ =\ 0$$

RATE OF CHANGE OF FLUX THROUGH A CLOSED CURVE, C:

Ampère-Maxwell Law:

$$\mu_0\varepsilon_0\frac{\partial\Phi_\mathbf{E}(S)}{\partial t}\ =\ \mu_0\varepsilon_0\frac{\partial}{\partial t}\int_S\mathbf{E}\cdot d\mathbf{A}\ =\ \oint_C\mathbf{B}\cdot d\mathbf{\ell}\ -\ \mu_0I$$

Faraday's law:

$$\frac{\partial\Phi_\mathbf{B}(S)}{\partial t}\ =\ \frac{\partial}{\partial t}\int_S \mathbf{B}\cdot d\mathbf{A}\ =\ -\oint_C\mathbf{E}\cdot d\mathbf{\ell}$$

E and B are the electric and magnetic fields; a closed surface is the boundary of a volume, and Q is the charge within that volume; in the last two laws, S is any surface whose boundary is the closed curve C; I is the current passing through C or S; the symbol $\oint$ indicates that the integral is over a closed surface or curve

those are the flux (or integral) versions of the total-charge versions of Maxwell's equations; there are also free-charge versions of Gauss' law and the Ampère-Maxwell law which use D H free charge and free current:

Gauss' Law:

$$\Phi_\mathbf{D}(S)\ =\ \oint_S\mathbf{D}\cdot d\mathbf{A}\ =\ Q_{free}$$

Ampère-Maxwell Law:

$$\frac{\partial\Phi_\mathbf{D}(S)}{\partial t}\ =\ \frac{\partial}{\partial t}\int_S\mathbf{D}\cdot d\mathbf{A}\ =\ \oint_C\mathbf{H}\cdot d\mathbf{\ell}\ -\ I_{free}$$

Extended explanation

Scalar flux vs vector flux:

The vector form of flux is the density (per area, not the usual density per volume ) of the scalar form of flux.

In electromagnetism, it is called the flux density …

ie, in electromagnetism, flux is flow across a surface, and flux density is the density (per area) of that flow;

flux in other topics, is the same as flux density in electromagnetism.​

Flux density in electromagnetism:

Magnetic flux, $\Phi_m$, is a scalar, measured in webers (or volt-seconds), and is a total amount measured across a surface (ie, you don't have flux at a point).

Magnetic flux density, $\mathbf{B}$, is a vector, measured in webers per square metre (or teslas), and exists at each point.

The flux across a surface S is the integral of the magnetic flux density over that surface:

$\Phi_m\ =\ \int\int_S\ \mathbf{B}\cdot d\mathbf{S}$​

(and is zero across a closed surface)

Magnetic flux density is what physicists more commonly call the magnetic field.

It is a density per area, rather than the usual density per volume.

Similarly, electric flux, $\Phi_e$, is a scalar, measured in volt-metres, and electric flux density (also a density per area), $\mathbf{E}$, is a vector, measured in volts per metre (and is more commonly called the electric field).​

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[AaronS]

Thanks for the overview on Flux