# I Electrostatics/Magnetostatics - E,D,P and B,H,M

1. May 11, 2017

### tarkin

I keep getting confused with E and D, in electrostatics, and B and H, in magnetostatics.
I've done some reading around, but some people seem to say the opposite of others. I was hoping someone could clarify for me once and for all!

So, my thinking was that, E is the total electric field, due to both the free charges, and bound charges in the material. D is the external electric field, due to only the free charges, and P is the polarisation, only due to bound charges, but defined as going in the opposite direction.

Is this correct? It seems to make sense, but I've heard D called "induced field", which doesn't agree with what I've written above.
Also, since the equation is written D = εE + P, would this not imply that D is the total field, rather than external field?

In Magnetostatics, my thinking was that B is the total field, H is applied (external) field, and M is magnetisation.

Basically, can someone clarify for me, which fields are the external fields (due to free charges),
and which are the total fields (due to free + bound charges) ?

Thanks

2. May 11, 2017

The electric field is $E$ and has contributions from both polarization charge and free electric charge. The two forms of electric charge are physically indistinguishable as far as the electric field which they generate. The electric displacement vector $D$ is thereby a mathematical construction, and it is not something that can be physically measured. Assuming you have a polarization $P$, and a polarization charge density $- \nabla \cdot P=\rho_p$, the electric displacement vector $D=\epsilon_o E+P$ satisfies $\nabla \cdot D=\rho_{free}$, but it is physically impossible to have any kind of meter that simply measures $D$. The electric displacement vector $D$ is not an observable physical entity.
$\\$ A similar thing applies to the "magnetic field" $H$. The $H$ field results from two sources in the "pole" theory of magnetism: $\\$ 1) Magnetic poles (fictitious) where $\rho_m=-\nabla \cdot M$ is the magnetic pole (magnetic charge) density, and a field $H$ results from these poles in an inverse square law manner analogous to the electrostatic $E$ with $\rho_{total}$ (where $\rho_{total}=\rho_{free}+\rho_p$ ) replaced by $\rho_m$, and $\epsilon_o$ is replaced by $\mu_o$. $\\$ and $\\$ 2) Free currents=where $H_{free}=B_{free}/\mu_o$, and $B_{free}$ is computed from Biot-Savart's law (or Ampere's law.) $\\$ The magnetic field $H$ in this theory, (where the equation $B=\mu_o H+M$ is analogous to the electrostatic $D=\epsilon_o E+P$) , is a mathematical construction and is (similar to $D$), not a physically observable entity. The (real) magnetic field $B$, just like $E$, is a physically observable entity. The "pole" theory of magnetism can actually be shown to have its origins in the magnetic fields from magnetic surface currents. There was a homework post about a year ago on Physics Forums that gives a good introduction to magnetic surface currents and the connection to the magnetic "poles".. Here is a "link" to that post: Magnetic field of a ferromagnetic cylinder $\\$ Additional item: $E$ and $P$, and $B$ and $M$ are the physically measurable quantities in these problems that have real physical meaning. $D$ and $H$ are both simply mathematical constructions that are very useful for computational purposes in order to arrive at solutions for $E$ and $B$. ( Both $P$ and $M$ might be somewhat difficult to measure, but with the proper instrumentation would, in fact, be measurable. That is not the case though with $D$ and $H$.) $\\$ One more additional item: The magnetic field $B$ satisfies Maxwell's equation: $\nabla \times B=\mu_o J_{total} +\mu_o \epsilon_o \frac{dE}{dt}$ where $J_{total}=J_{free}+J_{m}+J_p$. The magnetic current density $J_m=\nabla \times M/\mu_o$ consists of surface currents by Stokes law, where surface current per unit length $K_m=M \times \hat{n}/\mu_o$. (A polarization charge density $J_p=\frac{dP}{dt}$ arises from changing polarization charges, but does not occur in the steady state.) The magnetic field $B$ is calculated from the $J_m$ and $K_m$ by Biot-Savart's law (or Ampere's law). When working with magnetic surface currents, any (fictitious) magnetic "poles" are ignored. $\\$ The exact same magnetic field $B$ that is computed in the above manner (with surface currents) is also computed by the "pole" method, where surface currents are ignored, and the equation $B=\mu_o H+M$ is used along with magnetic pole density $\rho_m=-\nabla \cdot M$.

Last edited: May 11, 2017