The difference between the equation of Permittivity and Permeability in a medium

Question

I notice that the equations of Permittivity and Permeability in a linear medium are exactly opposite of each other. One is $$\mathbf{D} \equiv \varepsilon \mathbf{E}$$ while the other one is $$\mathbf{B} \equiv \mu \mathbf{H}$$ For the electrostatic, the $$\varepsilon$$ multiples directly to the electric field in free space, but the $$ \mu$$ multiples directly to the magnetic field in a medium. Why is it the case? My current guess is that it is because the electric field is weaker in the present of a medium(because the induced charge partly cancelled out the external electric field), but the magnetic field, depending on the medium, may become stronger than the external magnetic field

Answer 1

This is really a matter of convention. We could have defined $\mu' = 1/\mu$ and defined $\mathbf{H} = \mu' \mathbf{B}$. It turns out that although $\mathbf{E}$ and $\mathbf{B}$ are the "fundamental" electromagnetic quantities, $\mathbf{H}$ is really more like the magnetic counterpart to $\mathbf{E}$, and $\mathbf{B}$ is more like the magnetic counterpart to $\mathbf{D}$. Consequently, it makes sense to keep the definitions in parallel with the $\mathbf{E}$ field.

One way to see this parallel is to notice the symmetry between $\mathbf{D}$ and $\mathbf{B}$ and between $\mathbf{E}$ and $\mathbf{H}$ in the macroscopic Maxwell's equations with magnetic charges and currents:

$\nabla \cdot \mathbf{D} = \rho_{e}$

$\nabla \cdot \mathbf{B} = \rho_{m}$

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} - \mathbf{J}_m$

$\nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}_e$

In an experiment, you also tend control to control $\mathbf{E}$ and $\mathbf{H}$ and not $\mathbf{D}$ and $\mathbf{B}$. e.g. if you use voltages and currents to generate electric and magnetic fields. Voltage directly determines $\mathbf{E}$ and the free current that you supply determines $\mathbf{H}$, not $\mathbf{B}$.