# Intuitive meaning of the permittivity and the permeability in Electromagnetism

I wonder what the correct way to intuitively understand the concepts of electrical permittivity and magnetic permeability would be.

The electric permittivity $$\varepsilon$$ of a medium is defined as a measure of the electric polarizability of dielectric materials in response to an applied electric field. I usually think of it as a measure of the "resistance" of a medium to the establishment of an electric field in it. According to Coulomb's law, the lower the permittivity, the greater the tolerance of the medium to be penetrated by electric field lines.

$$\mathbf{F}(\mathbf{r})=\frac{q}{4 \pi \varepsilon_{0}} \int \mathrm{d} q^{\prime} \frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}}$$

As for magnetic permeability $$\mu$$ of a medium, it is the measure of magnetization that a material obtains in response to an applied magnetic field. According to Biot and Savart's law, it would have an analogous meaning, but in this case the permeability is proportional to the tolerance of the medium to be crossed by the magnetic field lines.

$$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int_{C} \frac{I d \boldsymbol{\ell} \times \mathbf{r}^{\prime}}{\left|\mathbf{r}^{\prime}\right|^{3}}$$

Are these ideas an appropriate way to intuitively understand these concepts of permittivity and permeability in Electromagnetism?

Yes, you have the right idea. For example, in a linear medium (with electric susceptibility $$\chi$$ and background polarization $$\mathbf{P_0}$$), we can write $$\mathbf{D} \equiv \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon_0(1+\chi) \mathbf{E} + \mathbf{P}_0,$$ we can rearrange the macroscopic Maxwell's equations to say: $$\nabla\cdot\mathbf{E} =\frac{\rho_f}{\varepsilon},$$ where $$\varepsilon = \varepsilon_0 (1 + \chi)$$. For positive susceptibility, oppositely-charged bound charges will accumulate around a free charge and tend to shield it, reducing the macroscopic electric field locally.

However, this shift of charges ultimately causes other charges (now with the same polarity as the free charge) to be exposed elsewhere in the material, and proper application of the macroscopic Maxwell's equations and its boundary conditions will account for this. More generally, $$\mathbf{P}$$ does not have to linearly depend on $$\mathbf{E}$$, but again, as long as you have a model for $$\mathbf{P}$$ for your material, you can still often determine a local expression for effective permittivity that gives some understanding of how charges source electric field lines.

Similarly, we can write $$\mathbf{H} \equiv \frac{1}{\mu_0}(\mathbf{B} - \mathbf{M}),$$ and $$\mathbf{H}$$ features in the macroscopic Maxwell's equations. If we have a model for how the magnetization $$\mathbf{M}$$ in a material varies in response to free current, we can come up with an effective permeability as well, which describes how much $$\mathbf{B}$$-field lines currents generate. Again, if it's a linear material such that $$\mathbf{B} = \mu\mathbf{H} + \mu_0\mathbf{M}_0$$, in the magnetostatic case we will have $$\nabla\times\mathbf{B} = \mu \mathbf{J}_f$$.

I think that for this case it is worthwhile to go back to Maxwell. Even before Maxwell had formulated the set of equations that today we know as 'Maxwell's equations' he had already worked out far reaching ramifications.

Wikisource has a transcript of the 1861 paper On physical lines of force

As we know, Maxwell worked with the supposition of a mediator of electric and magnetic effects.

Maxwell demonstrated something that was analogous to something that Newton had done for propagation of sound.

In the Principia Newton presented a derivation of the speed of sound (in air) from first principles.

The speed of sound arises from two things: the elasticity of air, and the inertial mass of air per unit of volume.

About the elasticity: air with no sound propagating through it is uniform. A force is required to push/pull air away from that uniform state.

About inertia: any sustained oscillation requires an inertia. In the case of air: once a particular volume of air is moving a force is required to stop it again.

That is why Newton was able to derive the speed of sound from first principles. Newton needed only two data points: elasticity of air and inertial mass per unit of volume.

Maxwell recognized that his mediator of the coulomb force and magnetic force had to have a number of physical properties.

In the absence of a source of Coulomb force the mediator is in a uniform state. A source of Coulomb force induces a state away from uniform. When the source of Coulomb force is removed the state reverts to uniform. This acts as a form of elasticity. The stronger the tendency to revert to uniform state, the higher the modulus of elasticity.

In addition Maxwell recognized that in order for the mediator to be the mediator of magnetism it would have to support in some form a persistence of motion. Even without knowing what it is that is moving: there is something with the property that when it is set in motion a force is required to stop it again.

Thus Maxwell recognized that this mediator of electic and magnetic effects possesses the qualities that are necessary to support wave propagation; a form of elasticity and a form of persistence of motion.

These propagating waves, if they existed, had to be transversal waves.

From the 1861 paper On physical lines of force:

Proposition XVI states:

To find the rate of propagation of transverse vibrations through the elastic medium of which the cells are composed, on the supposition that its elasticity is due entirely to forces acting between pairs of particles.