# Constants in the Divergence of E

1. Jul 28, 2012

### cpsinkule

why do most modern books claim the divergence of the E field is ρ/ε$_{0}$ but in more classical books, and when you actually derive it mathematically you arrive at 4πρ

2. Jul 28, 2012

Staff Emeritus
MKS vs. CGS units.

3. Jul 29, 2012

### vanhees71

Yes, it's the SI pest. I don't understand, why one has abandoned the good old tradition of presenting electromagnetism in the SI in the experimental course and in the Gaussian units in the theory course. The former is important, because the SI is the system of units used in experimental physics, and the units are well-defined and maintained by the all the national bureaus of standard (NIST in the US, PTB in Germany, etc.).

In theoretical physics, however, it's important to present the inner logic of the mathematical theories and models of our present understanding of nature, and the SI is not well suited for that purpose in regard of classical electrodynamics. The Gaussian system of units has this feature since the components of the electromagnetic field, $\vec{E}$ and $\vec{B}$ as well as the macroscopic auxilliary fields, $\vec{D}$ and $\vec{H}$ (note that these pairings belong together and not the traditional ones!) have the same units as it should be in the most natural setup according to the relativistic formulation of Maxwell's theory, which is the best one according to our present knowledge.

The only remaining "uglyness" of the Gaussian system is the appearance of factors $4 \pi$ in the fundamental equations. This is cured by using the rationalized Gauss units (or Heaviside-Lorentz units), which are the common standard in the theoretical high-energy physics community (who usually also puts $c=\hbar=1$, but that's not a good idea in the introductory course and not within classical physics, where $\hbar$ of course does not appear explicitly).

4. Jul 31, 2012

### PhilDSP

On the other hand, its a strength of SI to present $\epsilon_0$ and $\mu_0$ in the foreground. It reminds you that they are more primitive and lead to decompositions of more derived expressions. And the connection to $\epsilon$ and $\mu$ is obvious where those variables (or tensors) cannot be dispensed with when dealing with media.

Heaviside bemoaned the Gaussian system's hiding of those factors.

If the use of $\epsilon_0$ is too cumbersome then the switch to the $D$ field can be made: $(\nabla \cdot D) = \rho$

Last edited: Jul 31, 2012
5. Jul 31, 2012

### vanhees71

The constants $\epsilon_0$ and $\mu_0$ are not very physical constants they are just conventional factors to define the electric-current unit, Ampere, as a fourth independent basic unit in the SI. The merit is to have handy oders of magnitude for currents in everyday life (where usually within a normal household you deal with currents in the order of some mA to some 10 A).

The only additional physical constant of nature when going from Newtonian mechanics to electromagnetics is the velocity of light in vacuo, $c$, which comes into the game, because electromagnetics is necessarily a relativistic theory since the electromagnetic field is massless (spoken in terms of modern QFT).

To switch to the macroscopic Maxwell equations makes things more complicated. It's much better to derive them from microscopic electromagnetics.

6. Aug 1, 2012

### PhilDSP

It seems to me that the association of any particular combination of fields with either microscopic or macroscopic equations is quite arbitrary. What really makes an approach macroscopic is the averaging or acculmulation procedure of integrating over a spatial region at one time rather than solving the equations for a single point in space. Or is there another consideration?

More importantly, the difference between E and D fields, for instance, is that in Maxwellian terms one is determined on the basis of a force on a line segment while the other is flux through a plane (either of which can vary from point to point).