The Cottage-Cheese Model of Light

In the essay on the Kirchhoff-Clausius law I described light by analogy with cottage cheese, saying that it consists of curds of energy distributed randomly throughout a whey of electromagnetic fields. More properly I should have said that light consists of curds of electromagnetic field energy distributed throughout a whey of quantum indeterminacy, but that version of the model gets us into quantum electrodynamics, which I want to cover in a later essay.

In 1873 and 1876 respectively James Clerk Maxwell and Adolfo Bartoli hypothesized the existence of radiation pressure based on their studies of light, Maxwell using his electromagnetic theory and Bartoli using the laws of thermodynamics. In this essay I will recapitulate their derivations and calculate the energy density in light by both methods. I will then compare the two results and see what we can infer about the nature of light from the requirement that the two results agree with each other.

First we calculate the energy density of light by the electromagnetic method. For that calculation we must refer to Maxwell’s Equations.

According to Maxwell’s version of Ampere’s law,

(Eq’n 1)

the presence at some point in space of an electric current and/or a time-varying electric field correlates with the presence at that point of a curled magnetic field. If an electric field, time-varying or steady, also exists at the given point, we can calculate the dot product

(Eq’n 2)

The **J**•**E** term in that equation describes the
rate at which the electric field does work upon the electric current. The vector
**J** represents the electric current density (amperes per square meter),
which consists of the electric charge density and the average velocity at which
the charges move (coulombs per cubic meter times meters per second). The
electric field (volts per meter) acts on the electric charge to generate a force
density (newtons per cubic meter), which the motion of the electric charge (with
or against the electric field) works into a power density (watts per cubic
meter).

The second term on the right side of the equality sign in Equation 2 must also represent a power density,

(Eq’n 3)

In that equation E_{E}
represents the energy density that we must associate with the electric field. In
accordance with Maxwell’s derivation of it, the electric-field term in Equation
1 represents a current density, so in the calculation of the energy density the
electric field plays the role of both the charge and the field.

The term on the left side of Equation 2 participates in the vector identity

(Eq’n 4)

With that identity we can rewrite Equation 2 as

(Eq’n 5)

We can replace the curl of the electric field with the corresponding time derivative of the magnetic field by way of Faraday’s law of electromagnetic induction,

(Eq’n 6)

and get, with some algebraic rearrangement,

(Eq’n 7)

The vector **S**=**E**x**H** in that equation represents the
Poynting flux, the rate at which energy flows through a unit of area at the
given point in space.

We may note in passing that in the absence of actual
currents (**J**=0) Equation 7 gives us a continuity equation,

(Eq’n 8)

in which we have the electromagnetic energy density,

(Eq’n 9)

Equation 8 gives us one way of stating the law pertaining to conservation of energy.

By combining the results of certain manipulations of Faraday’s law and Ampere’s law, Maxwell deduced the existence of electromagnetic waves. Sinusoidal waves made of electric and magnetic fields propagate through space, the time-varying aspect of one field supporting the curled aspect of the other (no lumeniferous aether is required: each field is the other’s medium). If a plane wave of wavelength λ and frequency ν propagates in a direction parallel to the x-axis of a Cartesian coordinate frame, then we can describe its fields by writing

(Eq’ns 10)

in which **E**_{0} and **H**_{0} represent the maximum
intensities of the fields. The wave vector (or propagation vector) k=2π/λ
in those equations is related to the angular frequency of the fields’ vibration
ω=2πν
through the relation ω=**k**•**c**,
in which we represent the speed of propagation as a velocity vector.

If we substitute Equations 10 into Faraday’s law, we get, writing down only the magnitudes of the vectors,

(Eq’n 11)

from which we derive

(Eq’n 12)

If we substitute Equations 10 into Ampere’s law, we get

(Eq’n 13)

from which we derive

(Eq’n 14)

Substituting that last equation into Equation 9 tells us that the electromagnetic energy density in the wave conforms to

(Eq’n 15)

in which I have made use of the relation
ε_{0}μ_{0}=1/c^{2}.

A plane wave, with a certain wave number and angular frequency, emerges from a flat-plate emitter and propagates along our x-axis. If the emitter moves in the x-direction, the wave gets Doppler shifted, its wave number and angular frequency being either increased or decreased in the same proportion. We expect that the energy density in the wave will suffer a compression or decompression in the same proportion as the change in the wave number: between any two wave crests the wave contains a certain amount of energy and we expect that energy to remain unchanged when the wave is compressed or decompressed, so we expect the energy density to change in the same way as the wave number does as the distance between the given wave crests changes. Thus we expect that, as long as the wave remains in the same medium, the energy density stands in direct proportion to the wave number.

But if the energy density conforms to that description, the electric-field intensity will not necessarily obey the law pertaining to conservation of linear momentum. Imagine the emitter oscillates along the x-axis, alternately compressing and decompressing the wave as it emits it. An electrically-charged particle lies on the x-axis so that as the wave washes over it the electric field in the wave pushes it alternately in the positive and negative y-direction. Each half wave imposes upon the particle a linear momentum

(Eq’n 16)

In order that the system necessarily obey the conservation law each half wave
must impose the same change in linear momentum upon the particle, alternating
half waves taking alternating algebraic signs. The duration of each half wave
π/ω=π/kc,
so F_{y}=qE must be directly proportional to the wave number. That fact
makes the energy density in the wave proportional to the square of the wave
number,

(Eq’n 17)

In that equation phi represents a coefficient that converts units of the squared wave number into the units of energy density.

If the energy density in light were actually proportional to the wave number, no work would be required to compress or decompress it because the amount of energy between any two wave crests does not change. But with the energy density proportional to the square of the wave number the energy contained between any two wave crests does change, doing so as the distance between the wave crests changes (as, for example, by the wave reflecting off a moving mirror), so work must be done to compress or decompress the light. That fact necessitates that the emitter exert a pressure P upon the light it is emitting (because in all but one inertial frame the emitter is moving and thus carries out the same action as does a moving mirror) and, thus, by Newton’s third law, that the light exert the same pressure on the emitter.

Calculating the rate at which the emitter does work on the light as it emits it necessitates that we select an area A on the emitter and write

(Eq’n 18)

The rate at which energy in the wave passes a given point equals the product
of energy density passing through a given area at the speed of light –
EAc.
Thus we see that P=E_{EM}.

Next we calculate the energy density of light by the thermodynamic method. To set up the calculation we use Adolfo Bartoli’s imaginary experiment.

As I noted in the appended Apologue in "The Kirchhoff-Clausius Law", I had not actually read Adolfo Bartoli’s paper "Radiant Heat and The Second Law of Thermodynamics" when I recreated Bartoli’s imaginary experiment. I had read only a brief description of the thought experiment and had then reconstructed it based on my own knowledge of physics. What follows is more authentic to Bartoli’s actual derivation of the idea of radiation pressure.

We conduct the experiment inside a long
circular-cylindrical tube with a perfectly reflective inner wall. A
super-refrigerator keeps the tube at a temperature close to absolute zero so
that it won’t emit enough radiation to interfere with the experiment. A pair of
pistons, A and B, with perfectly black inner faces close the ends of the tube:
each piston possesses enough heat capacity that its temperature will effectively
not change in the course of the experiment. In front of each piston we mount one
of the two perfectly reflective diaphragms *a* and *b*: we can move
the diaphragms within the tube and light-proof slots in the tube allow us to
remove or insert diaphragms in front of the pistons almost instantaneously.

To prepare the experiment we insert both diaphragms
immediately in front of piston A and then move diaphragm *b* to its
position immediately in front of piston B. That action leaves the interior of
the tube empty of all radiation. Slide diaphragm *a* out of the tube, allow
the radiation coming from piston A to fill the tube and come to equilibrium with
the piston, then slide diaphragm *a* back into the tube: the tube is now
filled with heat radiation carrying an amount of energy Q. Slide diaphragm *b*
out of the tube. This allows heat from piston B to enter the tube but we can
ignore it. Finally slide diaphragm *b* into the position between diaphragm
*a* and piston A, then move diaphragm *a* along the length of the tube
to a position immediately in front of piston B. Thus we push all of the heat
that came from piston B along with the heat that came from piston A into piston
B and leave the tube empty, as it was at the beginning of the procedure. We can
now repeat the procedure as many times as we want, taking heat from piston A and
putting it into piston B.

But, as Bartoli noted in his description of the
experiment, we have made no assumptions about the temperatures of the pistons.
If the absolute temperature of piston A, T_{0}, is less than the
absolute temperature of piston B, T_{1}=T_{0}+ΔT,
then our experiment appears to have violated the second law of thermodynamics.
Heat energy Q has come out of piston A with entropy S_{0}=Q/T_{0}
and gone into piston B with entropy S_{1}=Q/T_{1}, so the
entropy of the system has suffered a net decrease. We know that Reality is so
structured that no system can suffer a net decrease of entropy: at minimum there
can be no change in the entropy and that happens only in regard to reversible
processes.

Our experiment is reversible, so we know that the entropy going into piston B must equal the entropy coming out of piston A, so we must have

(Eq’n 19)

that is, an amount of heat ΔQ must be added to Q as the radiation is pushed into piston B. That fact necessitates that the diaphragm do work on the radiation as it pushes the heat into piston B, so it must exert a force such that

(Eq’n 20)

in which Δl represents the distance between the piston faces and alpha represents the surface area of the diaphragm. By Newton’s third law the radiation must exert the same pressure P on the diaphragm.

If we divide the change in the amount of energy by the associated change in the volume occupied by the heat radiation, we obtain P=E, which means that the pressure exerted by the radiation equals the energy density in the radiation. That’s what we found by the electromagnetic method. In making that calculation I have made a tacit assumption, that the radiation moves in only one direction, which assumption is wrong in this case.

Light does not stand still, so in order for its energy density to exert pressure upon some object the light must ram that object. In the Maxwellian case the light propagates in a beam, so all of the energy density is brought to bear in exerting pressure on whatever the beam strikes; thus, we have P=E. In the Bartolian case we have trapped in a tube radiation that was emitted in all available directions from a black surface. Because we can represent the Poynting vector through its components (as with any other vector), we have one third of the total energy density in the radiation associated with each of the cardinal directions on our Cartesian coordinate grid; thus, on any given surface we have P=E/3. That latter fact leads eventually to the Kirchhoff-Clausius law, but I want to go in a different direction from here.

Imagine observing a pulse of light carrying energy E. The
pulse goes from a medium with index of refraction n_{0} straight into a
medium with index of refraction n_{1}. It changes its speed of
propagation as it goes from one medium into the other, so its volume changes in
accordance with

(Eq’n 21)

The wave number of each of the pulse’s components changes in inverse proportion to the change in the pulse’s volume,

(Eq’n 22)

which means that

(Eq’n 23)

and we have the energy density from Equation 17, so we have the total energy in the pulse as

(Eq’n 24)

The proportionality to k in the last term in that equation reflects the fact that the pulse’s magnetic field must become more or less ponderous to accommodate the changed speed of propagation. Mathematically, though, it comes from absorbing the constant from Equation 23 into the coefficient (phi double prime).

How much pressure does the pulse exert when it hits
something? We note that no work gets done as the pulse crosses the interface
between n_{0} and n_{1}, so the crossing is a perfectly
reversible process (dS=0 and dE=0) and we have, from the master equation of
thermodynamics (TdS=dE+PdV), PV=constant. We thus have

(Eq’n 25)

But that result conflicts with the result in Equation 17. How do we determine which is correct?

Assume that Equation 25 gives us a correct result. The analysis around that equation tells us that E=constant, which keeps our pulse in compliance with the conservation of energy theorem. Of course, the energy content of our light pulse is not strictly constant: we could diminish it by passing the pulse through a filter, which will absorb some of the energy. Nonetheless, the energy remains independent of the volume of space occupied by the pulse; that is, it remains unchanged by any change in the pulse’s volume due to changes in the propagation speed. The energy conforms to that description, ablatable but independent of volume, if it consists of a set of N entities, each carrying a certain fraction of the total energy. We see an analogous phenomenon in a balloon filled with gas: if we chill the gas, the balloon’s volume diminishes but the number of gas particles remains unchanged.

Now we want to calculate the pressure in Equation 25 in a way that’s independent of the index of refraction of the medium through which the pulse propagates. The wave number is not independent of that index but the frequency of the radiation is, so we exploit the fact that ω=kc/n to write

(Eq’n 26)

For the total energy in the pulse we then have

(Eq’n 27)

in which ψ’‘ gives us a number that translates units of reciprocal time (frequency) into units of energy. The volume and its dependence on the index of refraction are absorbed into the number N by way of the number density of the postulated N entities carrying the pulse’s energy.

As our pulse propagates through media of different indices of refraction E remains unchanged and ù remains unchanged. We also expect N to remain unchanged. We certainly know that energy-carrying entities cannot gratuitously appear or disappear: energy would not be conserved. But the entities could conceivably fuse together or undergo fission. If one of these entities were to split apart, either the two fragments must have frequencies lower than that of the original entity (and, yes, this does actually happen under the right circumstances) or the coefficient ψ’‘ must take on lower values for the fragments.

Assume that one of the entities splits in two without
changing its frequency. In that case conservation of energy necessitates that
ψ_{1}'’+ψ_{2}'’=ψ_{0}'’.
Allow the fissioning process to continue so that we describe the energy
contained in the original entity as

(Eq’n 28)

The number M can grow as large as we like, but it can never achieve the state of infinity: the finite-value theorem won’t let that happen. Thus we must infer that ψ’‘ has an absolute minimum finite value. Further, ψ’‘ must have the same value for all values of the frequency: if that proposition did not stand true to Reality, then observers in different inertial frames, seeing each others’ radiation Doppler shifted, would measure different values for ψ’‘ at any given frequency and that fact would invalidate the proposition that ψ’‘ has an absolute minimum finite value. Thus we infer that ψ’‘ represents a universal constant, ħ (Dirac’s constant, Planck’s constant divided by two pi). So for N=1 we write Equation 27 as

(Eq’n 29)

and we have Max Planck’s quantum theorem. We now recognize our postulated energy-carrying entities as the little blobs of energy that we call photons.

The principle of least action also comes into play here. It ensures that Planck’s constant (or Dirac’s constant) represents a maximum realizable value for ψ’‘ as well as a minimum value. If we have an antenna (or other emitter) that vibrates electromagnetically at a certain frequency and if we feed energy into that antenna, then at any given point on the antenna at the very instant that sufficient energy accumulates in a minuscule region around that point that energy will come manifest in a photon, which leaps away from the antenna and propagates through space. The photon represents the least action necessary to radiate the energy in the antenna.

But now we have a problem involving Equation 17. If we have deduced the quantum theory of light from an assumption that energy density varies with the first power of the wave number, then how can we accommodate a statement that the energy density varies with the square of the wave number? In fact, there’s no conflict between Equations 17 and 25: they describe different situations.

In devising Equation 25 we described a pulse of radiation propagating through media with different indices of refraction. The energy in the pulse remains unchanged but the volume of the pulse changes with the speed of propagation. As the pulse slows down or speeds up on entering a new medium, the wave crests move closer together or farther apart, making the wave number (wavelengths per meter) bigger or smaller in reciprocal proportion to the change in the volume occupied by the pulse. Calculating the energy density by, in essence, dividing the energy by the volume (technically, differentiating the energy with respect to the volume) necessitates that we use the ratio

(Eq’n 30)

in the first power, which we see reflected in Equation 25.

In devising Equation 17 we contemplated radiation emerging from a stationary emitter and then from a moving emitter. All features of the experiment being the same except for the relative motion of emitters, the pulse emitted from the moving emitter has a volume different from that of the pulse coming from the stationary emitter. Again the difference in volume corresponds to a difference in wave number in accordance with Equation 30. But the moving emitter does work on the radiation, changing its frequencies and, thus, its wave numbers. When we differentiate the energy with respect to the volume in this case, we get a proportionality equal to the square of the ratio in Equation 30.

So we resolve an apparent dilemma by correlating the mathematical description of two experiments with mental imagery of the experiments, using what Michael Faraday called aids to the imagination. This is how we do our best physics – combining our mathematics with a visualizable model of the phenomenon under study. And now we have a new model of light.

As noted at the beginning of this essay, we have a model in which we make an analogy between light and cottage cheese. In that model light consists of curds of energy (photons) distributed within and propagating with a whey of electromagnetic fields. We thus have a model that connects us to the mathematics of electromagnetism, thermodynamics, and the old quantum theory.

Finally, we have gained a better appreciation of the value of multiple perspectives on a situation or phenomenon. Maxwell’s electromagnetic perspective on radiation pressure is, by itself, an excellent piece of work. Bartoli’s thermodynamic perspective on radiation pressure is also, by itself, an excellent piece of work. Taken together, though, they went beyond excellent to transcendent when they gave us the quantum theory of light. It even seems possible that Max Planck, who knew both pieces of work, devised his quantum hypothesis, at least subconsciously, by the above method.

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