# Electric flux (Φe) vs electric flux density (D)

1. Sep 8, 2010

### black7ack

As the title states i would like to know the definitions of the above two terms. Also i would like to know the differences and a relation (if there is one) between them. I would be really grateful to anyone who takes the time to answer my probably silly questions.

2. Sep 8, 2010

### cmos

This is a great example of why engineers should stop calling D the "electric flux density." The other name for D is the "electric displacement;" maybe it's not a great name, but for reasons you'll see below, it's at least less confusing.

Anyway, D is defined as
$$\textbf{D}=\epsilon_{0}\textbf{E} + \textbf{P}$$
where P is the electric polarization (i.e. dipole moment per unit volume). For a linear isotropic material, this simplifies to
$$\textbf{D}=\epsilon\textbf{E}$$.

Now, what is commonly called "electric flux" is simply the surface integral of E (not D!).

3. Sep 8, 2010

### Born2bwire

I have seen one exception to this though. I recall that the CRC handbook defines electric flux as the surface integral of the D field. I have posted about this in the past but the Google Books reference that I could link to is gone now I think. It makes me wonder if historically it once made more sense but I never looked into it. Maybe a peek into an older text like say Stratton may reveal something.

4. Sep 9, 2010

### black7ack

Truly thanks for your quick answer. I must say you made it clear enough. I think i can take it up from here :D

5. Sep 9, 2010

### cmos

First consider my statement that it's primarily engineers who use the term electric/magnetic "flux density" for D and B. Then consider that is was primarily engineers (or as Stratton put it, "a subversive group of engineers" ) that pushed for the jump from Gaussian to MKSA units. Now we can continue the discussion assuming MKSA (i.e. SI) units.

Now the units of D are C/m2; the units of B are T=Wb/m2. From this I can somewhat understand why they would call it a flux density. Density due to the 1/m2; flux due to the fact that a surface integral over a "flowing quantity" is a "mathematician's flux" (that being separate from the other common definition of a flux - 1/s/cm2/sr).

I guess at this point we can ask, what are useful calculations? Obviously the integral of B over an open surface is useful (Faraday induction). But what is more useful, the surface integral of E or of D? Over a closed surface of open surface?

Well over a closed surface is the usual application of Gauss' law. But forgetting homework sets, does anybody actually use this that often for practical purposes? If yes, then do we prefer working with E or with D? My immediate response (open to debate) is that we prefer using E. In that case, it makes sense to call the surface integral of E the "electric flux," simply because that is what is useful.

The only other open question is in regards to using a open surface with either E or D. Does anybody actually make use of Maxwellian induction for anything? Again, if so, E or D?

One last note, the only reference I have in my office for "electric flux" is Griffiths; he says it's the surface integral of E. But then again, he only mentions it in passing as a stepping stone while introducing Gauss' law. I'll try to remember to take a look at Stratton and Jackson when I get home tonight; or maybe somebody else could look into it....

6. Sep 9, 2010

### black7ack

In (Φe)'s case it's fairly clear that the "flowing quantity" is E, but if you don't mind me asking what is this "flowing quantity" in D's case?

Also could you suggest to me a textbook upon this matter ( electromagnetic fields) that contains a sufficient number of examples. Bear in mind that the only relevant textbook i have read is Giancolli's physics for scientists and engineers with modern physics(4th ed), which i found extremely easy to read (the first textbook of physics that i would read just for fun :P).

7. Sep 10, 2010

### cmos

Well, the hijacked discussion that I was having with Born2bwire was on whether ΦE should use E or D; i.e. should it be the surface integral of E or of D. Regardless, whichever you choose, E or D, that is the "flowing quantity."

In my opinion, the best, upper-level book for E&M is "Introduction to Electrodynamics," by David Griffiths. I think most people will agree with me. It's the standard for use in junior/senior-level physics classes. He writes with a very informal tone, so it's great for self-study, especially if you are a "younger" student. You could also supplement this with an engineering text for chapters on transmission-line theory and antennas, if you care about such things. In that regard, I prefer Cheng's "Field and Wave Electromagnetics," but I think there is less general consensus on what is the best engineering text at this level.

Be warned though, for studying E&M beyond the freshman level, it's important to have a good grasp of vector calculus beforehand. Knowledge of differential equations (for boundary-value problems) and complex algebra (for wave analysis) wouldn't hurt either, but can usually be picked up along the way.

8. Sep 10, 2010

### cmos

So, based on the indecies; Stratton, Jackson, and Van Bladel never even mention "electric flux." Smythe "mentions" it as being the surface integral of E, but as with Griffiths, this is really only done in passing as he introduces Gauss' law.

It seems to me that both Smythe and Griffiths don't really make a strict definition of "electric flux" but rather say something along the lines of, 'this type of integral can be thought of as a flux integral.... and we have Gauss' law!'