Exploring Scalar and Vector Flux in Electromagnetism

• Greg Bernhardt
In summary, Flux always means total flow through a surface in electromagnetism, 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, and is proportional to the enclosed charge (Gauss' law: \Phi_{E}\ =\ Q_{total}/\varepsilon_0). Ampère-Maxwell law: \mu_0\varepsilon_0\frac{\partial\Phi_\mathbf{E}(S)}{\partial t}\ =\ \mu_0\varepsilon_0\frac{\partial}{\partial t}\
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:

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

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

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!

Jared Edington
Thanks for the overview on Flux

1. What is the difference between scalar and vector quantities in electromagnetism?

Scalar quantities in electromagnetism have only magnitude and no direction, such as electric charge and potential energy. Vector quantities have both magnitude and direction, such as electric field and magnetic force.

2. How is scalar flux different from vector flux in electromagnetism?

Scalar flux is the total amount of scalar quantity passing through a given surface, while vector flux is the total amount of vector quantity passing through a given surface in a specific direction.

3. What is the importance of understanding scalar and vector flux in electromagnetism?

Understanding scalar and vector flux allows scientists to accurately describe and analyze electromagnetic phenomena, such as electric and magnetic fields, and their interactions with charged particles and currents. It also helps in the development of new technologies, such as electric motors and generators.

4. How do you calculate scalar and vector flux in electromagnetism?

Scalar flux can be calculated by integrating the scalar quantity over a given surface, while vector flux can be calculated by taking the dot product of the vector quantity and the surface area vector. This can be represented mathematically as ∫S scalar quantity dS for scalar flux and ∫S vector quantity · dS for vector flux.

5. Can scalar and vector flux be measured experimentally?

Yes, both scalar and vector flux can be measured experimentally using various instruments such as electric field meters, magnetic field sensors, and current probes. These measurements are crucial for validating theoretical predictions and for understanding the behavior of electromagnetic fields in real-world scenarios.

• Electromagnetism
Replies
2
Views
727
• Electromagnetism
Replies
5
Views
5K
• Electromagnetism
Replies
2
Views
634
• Electromagnetism
Replies
3
Views
798
• Electromagnetism
Replies
2
Views
1K
• Electromagnetism
Replies
13
Views
2K
• Electromagnetism
Replies
1
Views
1K
• Electromagnetism
Replies
5
Views
1K
• Electromagnetism
Replies
1
Views
1K
• Electromagnetism
Replies
2
Views
480