# Similarities and analogies between the $E, P$ and $D$ fields with the $B, M$ and $H$ fields and their limitations

In the course of my learning electromagnetism, I’ve noticed there are a striking amount of symmetries in electrostatics and magnetostatics, almost down to replacing divergence operators with curl operators. For instance,

$$\vec P = \epsilon \vec E$$ $$\vec H = \mu \vec B$$

For linear media, and

$$\vec D = \epsilon_0 \vec E + \vec P$$ $$\vec B = \mu_0 (\vec H + \vec M)$$

And where $$\vec H$$ is defined, at least in Griffiths, in a completely analogous way to electric displacement, save for the typical curl operator that is usual for magnetostatics and a current density instead of charge density.

I am tempted to say that the effects of polarization and magnetization are totally analogous, that “$$\vec H$$ is basically the magnetostatic equivalent of $$\vec D$$” since I have far more trouble visualizing magnetization than I do polarization so if I can get away with thinking this way it’d make my learning easier I think. Do I have it wrong? Am I justified in thinking things in terms of analogizing from polarization? Am I oversimplifying things massively? Be as pedantic as you’d like.

• I always thought that $\mathbf{B}$ was equivalent of $\mathbf{D}$ since, $\boldsymbol{\nabla}.\mathbf{D}=0$ (dielectric, no free charges) is equivalent to $\boldsymbol{\nabla}.\mathbf{B}=0$ (no magnetic charges). In that respect $\mu$ seems to be somewhat unfortunatelly defined. – Cryo Mar 25 '19 at 13:35
• Also do not forget that magnetic fields and magnetization transform as axial vectors (psdeudo-vectors), whilst electric field and polarization density transform as polar vectors (i.e. "normal" vectors). – Cryo Mar 25 '19 at 13:38
• Polarization = density of electric dipoles per volume, magnetization = density of magnetic dipoles per volume. – Cryo Mar 25 '19 at 13:39