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

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

• H is not the magnetic field in a medium. It is just the magnetic field. In vacuum too.
– nasu
Oct 31, 2021 at 14:57
• But like D, H is something that we cannot measure directly as field and stay the same even if I change the medium. Am I right? Oct 31, 2021 at 16:40

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 $$\mathbf{B}$$.