Kirchhoff’s Law Revisited

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    Gustav Robert Kirchhoff’s radiation law expresses the fact that at each and every wavelength of electromagnetic radiation any given body has an emissivity equal to its absorbtivity. That law originates in our knowledge that we can create filters that will allow only light with a narrow band of wavelengths to pass through. Kirchhoff’s law comes from our understanding that any system in which we use such a filter to shield one component from the others must, nonetheless, conform to the fundamental requirement of the second law of thermodynamics: even in a system with monochromatic filters net heat energy cannot, of itself, go from a cold body to a hot body.

    But the analysis that led to that result involves a tacit assumption. We have assumed that a monochromatic filter allows light of a single wavelength to pass through it with equal efficiency in both directions. But a device called a Faraday rotator brings that assumption and its consequence into question.

    In 1845 Michael Faraday discovered that light propagating through glass in the direction of an imposed magnetic field got its electric field rotated about a line parallel to the direction of propagation. In addition he discovered that the direction in which the rotation occurs comes out independent of the direction in which the light propagates. Imagine winding a wire in a solenoidal coil around a glass cylinder and then making an electric current flow through the wire to generate a magnetic field in the glass with the field lines running parallel to the cylinder’s axis. If light then passes through that cylinder, its electric and magnetic fields rotate in the same sense in which the electric current flows in the solenoid, regardless of the direction in which the light propagates. That phenomenon yields the possibility of producing an optical diode, a device that allows light to pass one way but not in the other.

    An optical diode gives us the possibility of putting Kirchhoff’s law to the test. It seems to enable us to create a purely passive system in which net heat flows from a cold body to a hot body. In order to analyze that possibility we need to use polarized light, so we get two possible set-ups for our imaginary experiment.

    Those two possibilities reflect the means we use to polarize the light passing through our apparatus. If we use Glan-Foucault polarizers, as physicists and engineers do when working with powerful lasers, to polarize and analyze the light that we send through our system, then we have the choice of making the polarizers either absorb the radiation that they don’t allow to pass through them or reflect it back whence it came. Let’s consider the second of those options in the first version of our imaginary experiment.

    We have a cold body and a hot body, each enclosed within a spherical shell made of a perfectly reflecting material that neither absorbs nor radiates heat. We connect those two spherical shells with a tube made of the same material. Where each of the spherical shells faces the interior of the tube the material of the shell becomes perfectly transparent to light of only one wavelength: light of that wavelength can pass freely through the tube and thus go between the hot and cold bodies. That situation obeys Kirchhoff’s law and fulfills the requirement of the second law of thermodynamics: heat radiates from each body to the other, but more heat goes from hot to cold, so we have a net movement of thermal energy from hot to cold.

    Next we tightly fit two Glan-Foucault polarizers and a Faraday rotator inside the tube. Inside the Faraday rotator we choose, arbitrarily, to establish the magnetic field pointing from the cold side to the hot side of the apparatus; thus, as seen from the perspective of the cold body, any ray of light will appear to rotate its plane of electric vibration clockwise. If we make the magnetic field sufficiently intense, and we assume that we do, then any light passing through the Faraday rotator will get its plane of vibration rotated by forty-five degrees. We so orient the cold-side Glan-Foucault polarizer that the polarization of electric vibration makes a perfectly vertical line and we so orient the hot-side polarizer that the line marking the direction of polarization tilts forty-five degrees clockwise from the vertical as seen from the cold-side polarizer.

    Assume that the polarizers reflect the radiation that they don’t pass back whence it came. That means that the cold-side polarizer reflects half of the radiation coming to it from the cold body and passes the other half, which then goes through the Faraday rotator, goes through the hot-side polarizer, and gets absorbed by the hot body. Likewise, the hot-side polarizer reflects half of the radiation coming to it from the hot body and passes the other half, which passes through the Faraday rotator and reflects off the cold-side polarizer, goes back through the Faraday rotator and reflects off the hot-side polarizer, then goes through the Faraday rotator again, passes through the cold-side polarizer, and gets absorbed by the cold body. Though the radiation from the hot body takes a temporal detour, passing through the Faraday rotator two more times than the cold-body radiation does, the radiation flow through the device does, in fact, conform to the requirements of Kirchhoff’s law.

    Now assume that the polarizers absorb the radiation that they don’t pass. If we use Glan-Foucault polarizers, we need only replace the reflective coating on their sides with absorbent coatings, such as black paint. In this case each polarizer absorbs half of the heat radiating through the filter from the hot body. Meanwhile half of the heat coming from the cold body gets absorbed by the cold-side polarizer and the other half passes through the whole optical diode and gets absorbed by the hot body. In this case we seem to have net heat flowing from the cold body to the hot body, thereby violating the second law of thermodynamics.

    In this case the illusion of violating Clausius’ law dissolves when we notice that the system incorporates two other thermally active bodies – the polarizers. As they absorb radiation, the polarizers heat up and emit their own heat radiation. We assume that the four bodies – the cold body, the hot body, and the two polarizers – eventually reach the same temperature, but would they remain in that state?

    We have tacitly assumed that, at the chosen wavelength, our bodies act as black bodies with respect to the absorbed radiation. The radiation of appropriate polarization, of course, simply passes through the polarizers. And, as described in the appendix, the heat radiating from each of the polarizers comes out as radiation polarized in the plane perpendicular to the polarizer’s plane of polarization.

    In Figure 1 I show a diagrammatic representation of the system with the energy flowing from each component. We assume that the thermally active bodies emit and absorb radiation from surfaces of equal area, so we need only concern ourselves with the total radiation passing into or out of those surfaces, rather than the radiation per unit area. We take the heat emanating from the formerly-cold body (CB) at the filtered wavelength as I joules per second.

 

Fig. 1

(created with GeoGebra; see note below)

    We see that each of the four thermally active elements radiates the same amount of heat at the filtered wavelength. The formerly-cold body and the formerly-hot body (HB) each radiates all of its output toward the other three elements. The cold-side polarizer (CSP) and the hot-side polarizer (HSP), separated from each other by the Faraday rotater (FR), each radiates the same amount of heat, sending I/2 in opposite directions. The rays carrying that energy come out of the polarizers with polarizations oriented perpendicular to each polarizer’s plane of polarization. When we follow the emitted radiation to where it gets absorbed and add up the contributions body by body, we find that each of the bodies receives heat at exactly the same rate at which it emits heat. Thus the system occupies a state of thermal equilibrium and remains in that state until some outside factor either draws heat out of the system or adds heat to it or some heat engine redistributes heat within the system.

    Thus, we prove and verify the proposition that even the use of a Faraday rotator does not enable violations of Kirchhoff’s law or of the Second Law of Thermodynamics.

Appendix: The Glan-Foucault Polarizer

    In the imaginary experiment described above I used polarizers based on the phenomenon of birefringence. In certain transparent crystalline substances we find anisotropy in the forces that bind the atoms together, so the material responds differently to radiation whose electric fields vibrate in different directions; it acts as if it has two different indices of refraction, so we call it birefringent, which means that it displays double refraction. When a ray of light passes through a birefringent crystal in the right way it splits into two rays – the ordinary ray (or o-ray) and the extraordinary ray (or e-ray) – both rays consisting of linearly polarized light, the planes of polarization making a right angle with each other. That fact allows us to use birefringent crystals to create and manipulate polarized light.

    Consider a Glan-Foucault polarizer, as shown in Figure A-1. In that figure we understand that the direction that opticians call the optic axis points straight into or out of the page. In the configuration shown a ray of unpolarized light strikes the left input/output face at normal incidence and gets split into an o-ray and an e-ray. At the crystal/air-gap interface the e-ray (with its electric field oriented parallel to the optic axis) gets transmitted across the gap and into the opposite crystal and the o-ray (with its electric field oriented perpendicular to the optic axis) gets reflected off the interface.

Fig. A-1

(created with GeoGebra; see note below)

For our imaginary experiment we want the internal angle of incidence, the angle φ at which light strikes the prism/air-gap interface, to equal forty-five degrees; specifically, we want the o-ray to suffer total internal reflection at that interface while the e-ray gets transmitted when incident on the interface at that angle. To satisfy that requirement we need a birefringent material whose indices of refraction conform to the requirement of Snell’s law in the form

(Inequality A-1)

In this case the reciprocal of the sine equals 1.414 (for φ=45˚), so we can meet our twin criteria with crystals made of a substance such as sodium nitrate (ne=1.3369, no=1.5854).

    Our understanding of Snell’s law tells us that, while the e-ray has 100% polarization, the o-ray does not. But we assume into our premises the proposition that with an appropriate crystalline material and suitable optical manipulation we can make the polarization of the o-ray come as close to 100% as we choose.

    The silvering of the upper and lower sides of the prism reflects the o-ray back to the prism/air-gap interface, which reflects it back whence it came. That makes the polarizer act as a half mirror, which polarizes the light striking it by allowing the e-ray to pass through while reflecting the o-ray back to its source. Thus we get the active optical element that we need for our first imaginary experiment.

    If we blacken the upper and lower sides of the prism, as shown in Figure A-2, then the polarizer absorbs the o-ray and heats up. For convenience we assume that the crystal itself has zero heat capacity, so the heat resides only in the black layers. The tube in which we mount the polarizers also acts as a thermal insulator that prevents heat from leaving the black layers through their outer sides, so that they can lose heat only by radiating it into the crystal.

Fig. A-2

(created with GeoGebra; see note below)

    Each black layer radiates heat as unpolarized radiation in all directions available to it within the prism. But we want the radiation, at one wavelength, to strike the prism/air-gap interface at an angle of forty-five degrees plus or minus some minuscule deflection, so between each black layer and the crystal we put a wavelength filter and a pseudo-collimator, a thin, tightly packed array of silvered tubes that only allow radiation to pass in the desired direction, submicroscopic serrations in their walls reflecting errant rays back into the black layer.

    Heat radiates from each black surface and propagates to the prism/air-gap interface. There the radiation gets split into its o-ray and e-ray components. The e-ray crosses the interface and gets absorbed by the opposite black surface. The o-ray reflects off the interface and propagates parallel to the axis of the experimental apparatus. Thus we get the active optical element that we need for our second imaginary experiment.

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    I created the three figures in this essay with GeoGebra, a free program that you can find on the Internet at www.geogebra.org. The program was created by Judith Hohenwarter and Markus Hohenwarter and posted on the Internet for public use. The program consists of a manual, an exercise book, and the interactive mathematics program itself, all of which you can copy into your computer (indeed, the creators encourage you to do so). This is a great program; it’s easy to use; and I recommend it to anyone who needs to create mathematical diagrams for their Internet files.

 

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