Speed of Light in Vacuum and in Solids; Cherenkov radiation

I'll start near the end (speed of light), and then work my way backwards, then jump to the finish (Cherenkov radiation).  The speed of light in a vacuum is defined based on Maxwell's equations, which can be re-arranged to give a wave equation, yielding a velocity of the wave that is:
c = 1 / sqrt(εo * μo)

εo  is the permittivity of free space.  This constant is used to calculate the electric field at some distance (r) from a charge.  It basically says if you have this much charge (Q), you get this much electric field (E).

μo is permeability of free space.  Similar to εo, this describes how if you have this much current (I) you get this much magnetic field (B)

The speed of light in a linear dielectric material is determined by the permittivity of the material (ε) , and the permeability of the material (μ):
v = 1 / sqrt(ε * μ)

These have the same meanings as above, but apply within the material.  For example, ε tells you if you have this much charge (Q) within your material, you can find this much electric field (E).

But the speed of light is not about charges or currents.  It is about electric and magnetic fields oscillating, and that oscillation propagating far away (and long after) the original charges' and currents' motions stopped.  So how do these constants come to define the speed of light?  The short answer is Maxwell's equations.  Basically, the infinitesimal story can be written as:

1. electric charge undergoes a small acceleration
2. perpendicular to the direction of motion, an electric field increases in magnitude
3. Maxwell's third equation states that the change of the magnetic field in time is the negative curl of the electric field.  Leaving the mathematical details aside, the end result is that the increasing electric field from (2) causes an increasing magnetic field
4. Maxwell's fourth equation states an analogous relationship to (3):  the change of the electric field in time is the curl of the magnetic field.  Again, the increasing magnetic field causes an increasing electric field.
5. (3) and (4) then set up the propagation through empty space / vacuum.

In Maxwell's equations, the constant of proportionality that determines how much change in electric field (dE / dt) in time you get for some curl in the magnetic field (div B) is εo * μo.  So the story is that the time change / response of one type of field (electric or magnetic) to the other type determines the speed of propagation.  Now, this applies the same in materials - except that instead of having εo * μo describe that time response, we have ε * μ.

What about these material constants?  Well, the short answer is that the microscopic charges / structures of the material determines how much electric field (E) you get for a given charge (Q).  Generally these numbers (ε, μ) are greater than their vacuum counterparts.  A way to think about this is to imagine a "test" charge within a material.  This test charge will cause the microscopic charges within the material to be attracted / repelled.  This rearrangement of charge mimics and amplifies the presence of the test charge, causing it to appear like the test charge is larger than it is, causing the electric field (E) to be larger.

We can apply the same story to understand the slower speed of light within the material.  The adjusting electric field (of the wave) in the material now has to push on the microscopic charges and they have to re-arrange before the field can affect the magnetic field, and vice-versa, thus slowing down the propagation.

Cherenkov radiation

Cherenkov radiation occurs when a particle traveling near the speed of light (in a vacuum) enters a material.  Radiation is emitted as the particle slows down to a speed less than the speed of light in that material.  Note that a particle traveling with a constant velocity does not normally emit radiation; it is only in the case where the speed of the particle exceeds the speed of light in that material.

Thinking about the above, and Cherenkov radiation leads to this story:  The Cherenkov particle is traveling near the speed of light in vacuum (c) and approaches a material where the speed of light is slower (v).  Zooming way in to look at the particle, so that it is well separated in our field of view from the microscopic charges of the material, we see electric and magnetic fields behaving as they do in a vacuum in the immediate vicinity.  However, further out, we see these electric and magnetic fields interacting with the microscopic charges of the material.  The electric and magnetic fields do not (approximately) propagate beyond the nearby microscopic charges until the microscopic charges have time to re-arrange / respond (this is the from the discussion above of the difference between ε and εo).  In fact, the time to rearrange is so slow that once it is within the material the Cherenkov particle is going to catch up to the electric / magnetic fields within the material.  The Cherenkov particle is now "driving" in a concerted manner the electric / magnetic fields - leading the attack from the position of the van / wedge!  

Consider in contrast a regular, non-Cherenkov particle, traveling at less than the speed of light in the material (v).  The electric / magnetic fields in the material from this particle's motion propagate faster than the particle is traveling, so they outpace the particle.  The microscopic charges have time to "equilibrate" / rearrange around the particle's motion.  The difference here is that as some microscopic charges get pushed one way, others fill in the opposite way.  There is no uniform motion of the microscopic charges, and hence no net emission of radiation.

Application to faster than light motion in a vacuum

Thanks to Joel for pointing me towards this via discussion about the incident where, at the OPERA experiment, they thought they had observed neutrinos traveling faster than the speed of light.  A key paper (provided by Joel) was a discussion about how Cherenkov radiation should cause any neutrinos traveling faster than the speed of light to emit radiation (and/or particles, electron-positron pairs) and thus lose energy rapidly.  But that mechanism then allows for supra-luminal velocities - and leads me to imagine stories like the above applied to vacuum conditions.  A particle traveling faster than c has local, microscopic fields that propagate faster than c, and when they spread further out from the particle interact with the vacuum fields to cause Cherenkov radiation?  What would the scale of these microscopic fields be? Planck?  Some other characteristic wavelength of the vacuum radiation?  Interesting to think about.

Photons and their relation to the waves in the electric / magnetic fields

When I was in college and I first heard about the concept of photons in physics, I initially guessed that a photon would correspond to the field (electric / magnetic) between 2 nodes of the wave:

I was told this was not correct, and I left it at that, but recently I figured I would investigate why it was wrong.  That is what this post is about.

The energy of a photon is given by:

E = h * ν

h is Planck's constant 6.626e-34 [J*s]

ν is frequency [Hz]

I looked up the energy carried by electromagnetic waves in my go-to book:  "Introduction to Electrodynamics" by David J. Griffiths:

S = c * εo * Eo * cos^2(k*z - ω*t + δ)

c is the speed of light

εo is the permittivity of space

Eo is the (rms? max?) magnitude of the electric field

cos^2 is cosine squared

k is the wavevector, defined by the relationship between frequency and the speed of light

ω is the frequency of the radiation

δ is the phase

We can simplify this by assuming z = 0,  δ = 0 - this just says we are looking at what happens at z = 0, and there is no phase offset..

S = c * εo * Eo * cos^2(ω*t)

This equation defines the energy per unit time, per unit area.  So for the above, we would choose as our unit of time one cycle, or one half cycle of the wave.  But that still leaves the problem of the area.  Also, there is no classical restriction on the magnitude of the electric field (Eo).  The question then becomes, for a given area, is it possible to have Eo be so low that the energy of the photon spans more than 1 cycle?  There is nothing in the equations to prevent this.  Is there experimental evidence of it?

This essentially comes down to "single photon" experiments - experiments in which photons are measured one at a time.  I start by reading the wikipedia entry on the double slit experiment:

http://en.wikipedia.org/wiki/Double-slit_experiment

and will post my thoughts of this and other reading separately.

Update:

Single photon experiments with red photons from a He-Ne laser are not too hard to do:

http://ophelia.princeton.edu/~page/single_photon.html

The separation between individual photons is 2 km, which is much longer than the wavelength of the radiation (~700 nm) therefore, given the above framework the photon would be spanning billions of nodes!!

Other related links:

lowest measured forces (and by extension, electric fields):

http://www.economist.com/node/15949079

http://www.technologyreview.com/blog/arxiv/25018/

review of some single photon experiments:

http://www.bourbaphy.fr/grangier.pdf

single photon and complementarity:

http://www.ptep-online.com/index_files/2007/PP-09-19

proof of single photon existence - single photon hitting beam splitter, arriving at only one detector:

http://prd.aps.org/abstract/PRD/v9/i4/p853_1

Moving faster than the speed of light

I've been thinking about physics a lot lately, and I'm starting to jot things down, so I don't keep going over the same ground again but also to help me iron out the logical inconsistencies that can creep in when you do a problem in Caput.  A lot of this is really me just thinking about physics that I've read, and doing thought experiments so I can understand it.

This post is thoughts about what would happen if you moved faster than the speed of light.  Start with these premises:

• The only way we know about a particle (anything really) is the effect / force that particle exerts on other particles.
• You push a block with your hand:  the electric / magnetic forces from  the electrons in the atoms in the proteins / molecules in the cells in your hand interact with the electrons in the atoms in the molecules (cellulose) in the block of wood
• The above example is for electricity and magnetism, but (I've read) applies equally well to other, more exotic forces - e.g. the strong nuclear force between quarks in an atom's nucleus
• The forces between particles can be represented as fields.  Fields are vectors that exist through space that indicate the force (magnitude, direction) that a "test" particle would experience if it were at that location
• Imagine 2 charged particles.  From Coulomb's law, we can calculate the force between them.  Or, for each particle we could determine the field it generates throughout space.  Then, the force on each particle is determined by the field generated by the other particle.
• Movement of particles causes changes in the fields
• As the location of 2 particles gets closer together, the force they exert on each other increases.  Similarly, the field strength increases.  
• The changes in the fields propagates at the speed of light

The above might sound crazy, but they are well established physics, with tons of experimental evidence.  Given the above it is almost nonsensical to talk about a particle moving faster than the speed of light.  Which is somewhat expected - the above description of reality is based on the tenet that nothing travels faster than light.  But the exercise of investigating what would happen if something moved faster than light helps me understand the relationships.  So, 2 scenarios to imagine:

Particle approaches at faster than light

The particle will arrive at a location before the effect of the particle being at the location does. This is just logically inconsistent.

Particle moves away faster than light

This situation is harder to rule out.  As the particle recedes, it is not arriving before its effect.  The problem with this one occurs for two situations I can think of:

1.  Imagine another particle, chasing this one.  The "effective" location, based on the fields, is only moving at the speed of light.  In this case, the particle has effectively "disappeared".  The chasing particle sees only the location represented by the field

2. Imagine instead of a single particle, an atom moving faster than the speed of light.  Background:  for a stationary atom emitting radiation, the frequency is intrinsic to the motion of the oscillation of the electron(s) within the atom.  The radiation, regardless of the relative velocity between the emitting atom and the observer, propagates at the speed of light.  The wavelength is determined by the frequency and the speed of light.

Now, for the atom moving faster than the speed of light:  Take the period of oscillation, imagine the first cycle has occurred.  Now, in this period of time, the atom has traveled a distance greater than the wavelength of the radiation, and a new cycle occurs.  So the separation in peaks / troughs between the first and second cycle is greater than the wavelength (as it would be defined for regular sub-luminal speeds).  Furthermore, for the third cycle, the discrepancy is even greater.  So, even though the atom is travelling at constant velocity, the radiation is continuously increasingly red-shifted (chirped down!).   Effectively, as time goes on, the emission of radiation is red-shifted until it would disappear completely.  Now, this description is discrete, but it could be made continuous.

Why would the above be impossible or inconsistent?  Well, the particle, in this case, has effectively disappeared from the universe, since internally it is emitting radiation, but this vanishes / does not appear anywhere else.

The reverse of this is also possible to imagine, in which an atom emitting radiation approaches at faster than the speed of light, and the radiation is continuously increasingly blue shifted.  In this case, leaving aside the issue from above of the particle arriving before its effect, the radiation observed would be increasingly blue shifted over time (chirped up!).  Where is the increased power / energy coming from?  Again, the internal state of the atom is disconnected from the rest of the universe.