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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.

sangstar
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  • 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
  • You need to look at the Maxwell equations and some example physical problems. Then you will find that the more apt pairing is $H$ with $E$ and $B$ with $D$, because they have similar equations. Think of the defn as ${\bf H} = \mu_0^{-1} {\bf B} - {\bf M}$. See here for further info: https://physics.stackexchange.com/questions/300741/whats-the-difference-between-magnetic-fields-h-and-b/611551#611551 – Andrew Steane Mar 27 '21 at 23:09

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$\vec{H}$ 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?

All these analogies are limited and context-dependent.

Sometimes $\mathbf H$ is analogous to $\mathbf E$, since in static scenario they both obey Coulomb's law, have similar equations, similar "lines of force"; bar magnet' H field is similar to electrically polarized bar's E field.

But sometimes $\mathbf B$ is analogous to $\mathbf E$, for example in diamagnetics (copper, bismuth), the magnetized medium decreases $\mathbf B$ field inside, just as polarized dielectric decreases $\mathbf E$ inside.

There is no generally valid analogy, these are 4 different quantities with unique properties.

Ján Lalinský
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