4.6.2 Electromagnetic Theory Of Light

The electric and magnetic experiences, many of which Faraday explored in his London laboratory, can be summarized in four quantitative laws established by direct measurements involving electric and magnetic objects of experience: Coulomb’s law of the interaction between electrically charged objects, Ampere’s law of the interaction between electric currents, the manifest absence in nature of monopole magnetic charges, and Faraday’s law of the electromotive forces engendered by the motion of magnetic objects. Faraday suggested that the mechanisms through which electric and magnetic objects thus act upon each other resides, not in themselves, but in the space between them; he conceived of this mechanism as being mediated by a field of force acting at each point in space surrounding electric and magnetic objects.

Following this suggestion, as well as Faraday’s discovery that the plane of polarization of polarized light rotates when it passes through a magnetic field, Maxwell proposed an explanation of electric and magnetic experiences in terms of the mechanical properties of the same physical medium that was then believed to explain optical experience. This luminiferous ether would transmit electrical and magnetic influence from one point to its immediate neighbors by way of the elastic properties that it was assumed to possess, much in the same way that it transmitted light.

The object of explanation which Maxwell introduced enabled him to quantify the physical properties of the electric and magnetic fields at any location in space in terms of the currents and charges that characterized the electric and magnetic objects involved. Specifically, Coulomb’s law became:

d i v \vec{D} (\vec{r}) = 4 π ρ (\vec{r})

the absence of monopoles became:

d i v \vec{B} (\vec{r}) = 0

Ampere’s law became:

c u r l \vec{H} (\vec{r}) = \frac{4 π}{c} \vec{j} (\vec{r})

and Faraday’s law of magnetic induction became:

c u r l \vec{E} (\vec{r}) = - \frac{1}{c} \frac{\partial \vec{B} (\vec{r})}{\partial t}

where $\vec{E} (\vec{r})$ represents the electric field vector measuring the force acting on a unit electric charge located at $\vec{r}$ , $\vec{B} (\vec{r})$ represents the magnetic induction field vector measuring the torque acting on a unit magnetic dipole located at $\vec{r}$ , $\vec{j} (\vec{r})$ and $ρ (\vec{r})$ represent the current and charge densities at $\vec{r}$ respectively, and is the speed of light in vacuum. The vector field quantities $D (\vec{r})$ and $\vec{H} (\vec{r})$ are related to the electric field and magnetic induction field vectors by the constitutive relations:

\vec{D} (\vec{r}) = ε \vec{E} (\vec{r})

\vec{H} (\vec{r}) = \frac{1}{μ} \vec{B} (\vec{r})

where $ε$ is the dielectric constant, and $μ$ is the magnetic permeability, of the medium under consideration.

It can easily be shown however, that, while these equations fully comprise the experience contained in the laws they represent, they do not explain optics. Indeed, taking the curl of Ampere’s law in vacuum where the current density is zero everywhere, and using the general mathematical relationship:

c u r l (c u r l (\vec{A})) = g r a d (d i v (\vec{A})) - ∆ \vec{A}

we get:

∆ \vec{H} (\vec{r}) = 0

and similarly, by taking the curl of Faraday’s law in vacuum, we get:

∆ \vec{E} (\vec{r}) = 0

Since these Laplace equations represent propagation with infinite speed, light cannot be represented as a combination of the electric and magnetic fields governed by the equations given above. If Maxwell wished to explain optics with the help of electric and magnetic fields, and he most certainly did, he had to modify his equations in such a way as to obtain disturbances which propagated with finite speed. In other words, he needed to modify his equations in such a way as to obtain, instead of Laplace’s equation for the fields, a wave equation representing propagation with finite speed V:

∆ \vec{E} (\vec{r}) - \frac{1}{V^{2}} \frac{\partial^{2} \vec{E} (\vec{r})}{\partial t^{2}} = 0

and similarly for the magnetic field $\vec{H} (\vec{r})$ . Clearly, the modification needed was the addition to Ampere’s law of a current term equal to $\frac{1}{c} \frac{\partial \vec{D} (\vec{r})}{\partial t}$ for then:

c u r l (c u r l (\vec{H} (\vec{r}))) = c u r l (\frac{1}{c} \frac{\partial \vec{D} (\vec{r})}{\partial t}) = \frac{ε}{c} \frac{\partial}{\partial t} (c u r l (\vec{E} (\vec{r}))) = - \frac{ε μ}{c^{2}} \frac{\partial^{2} \vec{H} (\vec{r})}{\partial t^{2}}

and therefore:

∆ \vec{H} (\vec{r}) - \frac{ε μ}{c^{2}} \frac{\partial^{2} \vec{H} (\vec{r})}{\partial t^{2}} = 0

which represents a wave propagating with speed:

V = \frac{c}{\sqrt{ε μ}}

In vacuum, where, in the Gaussian system of units we have used here, the dielectric constant $ε$ and the permeability $μ$ are both equal to unity, this speed coincides with the speed of light as desired. A similar situation holds for the electric field $\vec{E} (\vec{r})$ as well.

Given the nature of the luminiferous ether that he introduced as an object of explanation for electric and magnetic experiences, Maxwell had little difficulty in understanding what this additional term represented. To him, the ether was a real medium made up of molecules, each containing both positive and negative electric charges, and therefore the effect of an electric field at each point in space would be to separate the positive from the negative charges in the ether molecule residing at that point by displacing them with respect to each other. The resulting current would then have exactly the value needed to modify Ampere’s law in such a way as to impart finite speed to the propagation of electromagnetic disturbances. Maxwell therefore called it the displacement current:

\vec{j_{d}} (\vec{r}) = \frac{1}{4 π} \frac{\partial \vec{D} (\vec{r})}{\partial t}

and replaced Ampere’s law with:

c u r l \vec{H} (\vec{r}) = \frac{4 π}{c} (\vec{j} (\vec{r}) + \vec{j_{d}} (\vec{r}))

Unfortunately, many years of trials and tribulations culminating with the Michelson-Morley experimental result have convinced the scientific community that the Maxwellian ether was causing a lot more trouble than the ontological explanation which it seemed to provide for the displacement current was worth. Hence, at Einstein’s suggestion, the object of explanation which Maxwell called ether was abandoned altogether and a new object of explanation called electromagnetic field was introduced, an object of explanation which was defined by the simple fact that it satisfied the Maxwell equations. The simplicity of this fiat, coupled with the signal success the Maxwell equations were having, compensated for the original need to understand what was it that displaced itself in the displacement current; the formal explanation that the term in Maxwell’s equation which represented the displacement current was in any event needed to render the equations mathematically consistent with charge conservation only served to avoid the issue altogether. The ontological question was thus forgotten at the very moment when it became visible. The prevailing position was well summarized by Richard Feynman in the second volume of his Caltech physics lectures:

Maxwell discussed his ideas in terms of a model in which the vacuum was like an elastic solid. It was not yet customary in his time to think in terms of abstract fields. Today, we understand better that what counts are the equations themselves and not the model used to get them. We may only question whether the equations are true or false. This is answered by doing experiments, and untold numbers of experiments have confirmed Maxwell’s equations. If we take away the scaffolding he used to build it, we find that Maxwell’s beautiful edifice stands on its own.

But for the attitude summarized in Feynman’s statement, the ontological question may have been addressed almost a century earlier than when it eventually was.

The suggestion from philosophy that objects of explanation have a different ontology than do objects of experience thus stands strengthened by the evidence from theory provided in this section. We shall therefore take it from here onward that the two kinds of objects are in fact ontologically different. For whatever reasons, however, either because alternative theoretical positions seemed more reasonable at the time or because it simply decided with Feynman to ignore the issue altogether, the scientific community chose to disregard the signs pointing to this ontological difference. Instead, they extended the process of objectification equally to both the things which are given to us as well as to the concepts which we had ourselves introduced into the theory in order to explain their behavior. Modern science had thus taken the clear position that atomic objects are objects of experience. As we had expected when we set-off on our journey, the transition from the pre-scientific to the modern scientific comportment was indeed ontologically pregnant.