What is the true meaning of the fields $\mathbf D (\mathbf r, t)$ and $\mathbf H (\mathbf r, t)$?

Question

The meaning of the electric field is

$$ \mathbf E = \frac{\text{force}}{\text{unit charge}} $$ the meaning of the polarization field is $$ \mathbf P = \frac{\text{electric dipole moments}}{\text{unit volume}} $$ For the magnetic field is $$ \mathbf B = \frac{\text{force}}{\text{unit charge moving with unit velocity such that the force is maximum}} $$ The magnetization field is the $$ \mathbf M = \frac{\text{magnetic dipole moments}}{\text{unit volume}} $$

Question:

I know the definitions of the fields $\mathbf D$ and $\mathbf H$ as $$ \mathbf D = \varepsilon_0 \mathbf E + \mathbf P \\ \mathbf H = \frac{1}{\mu_0} \mathbf B - \mathbf M $$

Is there such a simple meaning for the fields $\mathbf D$ and $\mathbf H$ similar to those of $\mathbf E, \mathbf P, \mathbf B$ and $\mathbf M$ ?

Answer 1

Think flux-density (from Gauss's law): $$\mathbf D = \frac{\text{free charge on one capacitor plate}}{\text{area of capacitor plate}}$$

Think circulation-density (from Ampere's Law): $$\mathbf H = \frac{\text{free current in coil}}{\text{length of coil}}$$