Wednesday, April 25, 2012

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.

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