Understanding Gauss' law: diff b/w E and D flux?

In summary, The definitions of electric and magnetic flux are based on different mathematical constructions, which can be confusing at first glance. However, the reason for this is that the quantities ## D ## and ## H ## are non-physical and are simply mathematical constructions used for convenient calculations. The physical quantities are the electric field ## E ## and the magnetic field ## B ##, which can be measured and are tied to the total charge density. Additionally, the magnetic pole method and the electric displacement field also rely on mathematical constructions, making them non-physical. This may be confusing at first, but understanding the difference between these mathematical constructs and physical quantities can help clarify their use and importance in the study of electromagnetism.
  • #1
tim9000
867
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I noticed the other day something odd in how we use Electric and Magnetic flux.
The definitions I refer to are magnetic flux density (B), magnetic flux intensity (H), electric displacement field (D) or Electric field density (D) and electric field (E):
B = μH
ΦB = B*Area

&

D = εE
ΦE = E*Area

for magnetic and electric fields, respectively.

Is the electric displacement field the same thing as electric field density?

So this seems like a lack of symmetry of the flux quantities at first glance as the magnetic field flux is based off a magnetic field density but electric flux is based off electric field intensity. I find this odd, is there a reason for this/why do we do it this way?

My other question is along the same lines but regarding Gauss' law:

According to wiki:
"Equation involving the E field
Gauss's law can be stated using either the electric field E or the electric displacement field D. This section shows some of the forms with E; the form with D is below, as are other forms with E."

Site:
https://en.wikipedia.org/wiki/Gauss'_law

It further says:
"Equation involving the D field
Free, bound, and total charge
Main article: Electric polarization
The electric charge that arises in the simplest textbook situations would be classified as "free charge"..."

I also tried reading:

https://en.wikipedia.org/wiki/Electric_displacement_field

&

https://en.wikipedia.org/wiki/Electric_displacement_field

Which states:
"The displacement field satisfies Gauss's law in a dielectric:

∇ ⋅ D = ρ − ρ b = ρ f
30ef917ea58f457f5538dce26b6100ec34cbb596
.
Proof:
[show]
"

But I'm confused, is Maxwell's differential equation of Gauss' law
(https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0076e721a4b485bda8ff427f00e73c6efb6006)
the total charge density (both bound and free charge) and why is it not the same as ∇⋅D = ρfree ?
Basically, what's the difference between and why do we use one and not the other?

I'm struggling to get this intuitively.

Much appreciate your assistance.

Cheers
 
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  • #2
I think the answer to your question is somewhat simple, but it took me lots and lots of effort to figure this out. Both the quantities ## D ## and ## H ## are non-physical, unlike ## E ## and ## B ##. The parameters ## D ## and ## H ## can not be measured. They are very useful mathematical constructions, but don't represent real physical entities. For a lot of detailed mathematics where I was able to tie the "pole method" of magnetism to the "surface current" method, see https://www.overleaf.com/read/kdhnbkpypxfk Because the "pole model" of electrostatics also enters into the discussion, I believe I also discussed in the write-up how ## D ## like ## H ## is also simply a mathematical construction. ## \\ ## Additional item of interest: For the electric field ## E ##, we have, in c.g.s. units, ## \nabla \cdot E=4 \pi \rho_{total} ## where ## \rho_{total}=\rho_{free}+\rho_{p} ##. It is impossible for any physical device to distinguish the electric field ## E ## that comes from ## \rho_{free} ## vs. that which comes from ## \rho_p ##. Thereby ## D ##, which distinguishes the two with the equation ## \nabla \cdot D=4 \pi \rho_{free} ##, is non-physical, and simply a mathematical construction. (Note: Polarization charge density comes from gradients in the polarization vector ## P ##. ## \rho_p=-\nabla \cdot P ##). ## \\ ## Editing: And similarly for the magnetic field ## B ##: The ## B ## field caused by currents in conductors can not be physically distinguished from that caused by bound magnetic surface currents. The magnetic pole method does a mathematical construction that generates a well-defined ## H ##, but it is non-physical, and the magnetic field that is measured in all cases is the ## B ## field.
 
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Related to Understanding Gauss' law: diff b/w E and D flux?

1. What is Gauss' law and how does it relate to electric and displacement flux?

Gauss' law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the total charge enclosed by that surface. It states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space. This law also applies to displacement flux, which is the flux of electric displacement field (D) through a surface. The main difference between electric flux (E) and displacement flux (D) is that E takes into account both free and bound charges, while D only takes into account free charges.

2. How can Gauss' law be used to solve practical problems in electromagnetism?

Gauss' law is a powerful tool that can be used to solve a variety of problems in electromagnetism. By applying the law, we can determine the electric field at any point in space, as well as the net charge enclosed by a closed surface. This makes it useful for calculating the electric field and charge distributions around conductors, capacitors, and other systems.

3. What are the assumptions made in Gauss' law?

Gauss' law makes several assumptions in order to be applicable. These include assuming that the electric field is continuous and that it follows the inverse square law, that the charge distribution is symmetric, and that the permittivity of free space is constant.

4. How does Gauss' law relate to other laws and equations in electromagnetism?

Gauss' law is one of four Maxwell's equations, which are a set of fundamental laws that describe the behavior of electric and magnetic fields. It is closely related to Coulomb's law, which describes the force between two point charges, and it can also be derived from the divergence theorem.

5. Can Gauss' law be applied to non-uniform electric fields?

Yes, Gauss' law can be applied to non-uniform electric fields. In this case, the surface integral of the electric flux must be evaluated over a closed surface that surrounds the charge distribution of interest. This allows us to calculate the electric field at a specific point in space, even if the field is not uniform.

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