Different formulas for electric flux

[Alex Schaller]

I noticed that in some textbooks (Physics - Tipler) the electric flux formula is different than in other textbooks (Engineering Electromagnetics -W. Hayt)

Which one should we use?

 

[Doc Al]

Tipler gives the standard definition of flux through a surface. Use that one.

Hayt seems to be talking about the total flux through an enclosed surface. Look up Gauss' law in Tipler for more.

 

[Doc Al]

Read this: Gauss's Law

 

[alan123hk]

I noticed that in some textbooks (Physics - Tipler) the electric flux formula is different than in other textbooks (Engineering Electromagnetics -W. Hayt)
Which one should we use?

Yes, I have also noticed this situation. Personally, I might prefer the definition of $~ ~\Phi_e =Q ~$
https://www.maxwells-equations.com/density/electric-flux.php
https://www.antenna-theory.com/definitions/electricfluxdensity.php

 

[Doc Al]

Be careful to distinguish between electric flux and electric flux density.

 

[Alex Schaller]

Be careful to distinguish between electric flux and electric flux density.

Yes, you are correct.

Hayt conveys "electric flux" as the integration of "electric flux density D" over a surface, whereas Tipler conveys "electric flux" as the integration of "electric field E" over a surface.

 

[alan123hk]

Because electric flux is defined as $~ d \Phi_e = \mathbf {E} \cdot d\mathbf {S}~$, it is very reminiscent that the electric field strength or electric field intensity $~E = \frac {\Phi_e} {S~cos\phi}~$ itself represents the electric flux density, but on the other hand, the electric flux density is defined as $~\mathbf {D}=\epsilon\mathbf {E} ~ ~$(there is an extra symbol$~\epsilon~$ before $\mathbf E$) , which is really a bit confusing to me.

Perhaps we better not call $~\mathbf {D}~$ the electric flux density, the name electric displacement field may be more suitable for it.

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

 

[Alex Schaller]

Thanks for your reply Alan,
I agree with you, perhaps D should be called "displacement flux density" or "displacement density" and not "electric flux density".

On the other hand, flux as per Hayt (Φ = Q) is simpler to understand, but does not yield the same equation as flux according to Tipler (Φ = Q/ε0).

Maybe we should send a suggestion to the editor of Hayt's textbook to look it over?

 

[alan123hk]

Maybe we should send a suggestion to the editor of Hayt's textbook to look it over?

Looking at it from another angle, this matter is actually not a big deal. When we express the same thing, we will use different units. For example, the unit of weight can be mg, kg or pound, etc., and the unit of distance can be meter, kilometer or light-year, etc., so just indicate the unit to avoid misunderstanding.

The unit of electric flux based on $~ \Phi_e = E S~cos\phi~$ is volt meters (Vm).

Because the unit of $~\epsilon~$ is $\frac {C} {Vm} ~$, the unit of electric flux based on $~ \Phi_e =DS~cos\phi = ~\epsilon~E S~cos\phi~$ is $\left(\frac {C} {Vm} \right) \left( Vm \right) = C~$. That's why I mentioned earlier that I prefer the expression $~\Phi_e =Q ~$, which seems to be more concise and beautiful in my opinion.

On the other hand, this is also in line with symmetry.
$~ \Phi_e = D S~cos\phi~$, where D is called the electric flux density
$~ \Phi_m = B S~cos\phi~$, where B is called the magnetic flux density

 

[Alex Schaller]

Thanks Alan!

Talking about symmetry I thought B was more related to E, as H was more related to D (both E and B take into account all the charges present -free and bond-, whereas H and D only consider the free charges).

 

[vanhees71]

Indeed, $\vec{E}$ and $\vec{B}$ belong together and $\vec{D}$ and $\vec{H}$. This becomes very clear in the more consistent manifestly covariant relativistic formulation of electrodynamics, where $\vec{E}$ and $\vec{B}$ are the 6 independent components of an antisymmetric four-tensor $F_{\mu \nu}$ and $\vec{D}$ and $\vec{H}$ of another such tensor $D_{\mu \nu}$.