The effects
in PHYSICS

Lorentz Transformations
Cerenkov effect
Doppler effect
Auger effect
Photoelectric effect
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Compton effect
Pair production effect
X rays
Sagnac Effect
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Lasers

CERENKOV EFFECT



Let' consider a charged particle of velocity Vp
incident  in a transparent medium.
This particle emits photons in the "y" direction. Furthermore,
the incident charged particle recoils.


The emitted photon propagates with a speed Vw through the medium. If the refraction index medium is n then the velocity Vw is not equal c ( speed of light in the vacuum) but tp c/n; that is the speed of light in that medium. Let's write: Vw = c/n (1)
We consider a high-energy charged particle, that has a velocity greater than Vw. The related Cerenkov radiation is a chok wave.
At first, we can quickly write the following relationship, by solving for the angle y, we have cos y = Vw/Vp (2) or cos y = c/n.Vp We know that "y" could not exist if its cosine is greater than 1 ( or less than -1) : cos y = 1 gives Vp = Vw = c/n cos y greater than 1 : Vw greater than Vp : No radiation any more. The threishold to have this emitted radiation is then: Vparticule = c/n
Now, let's find the true relationship relating to cos a. Consider the following reaction:

Where the electron is traveling through the medium and gives a photon and a recoil electron. The following figure shows the mechanism:

Let's do the following calculations: Write that the liear momentum and the energy of this system are both concerved in the relativistic context. We will use the following factor: with g1 = g(V1) and g2 = g(V2) And the relation: E0= m0c2 for the rest mass (3)

Before the electron enters the medium

The incident electron had: P1 : for the liear momentum. E1: for the total energy, with E1 = m 1c2 where m1 = g1m0 Or E12 = P12 c2 + m02c4(4)

While the electron is traveling inside the medium:

The electron has: P2 : for the liear momentum. E2: for the total energy, with E2 = m 2c2 where m2 = g2m0 Or E22 = P22 c2 + m02c4 (5) The photon has: Pg : for the liear momentum. Its magnitude is p g = hN/c ( h : Planck constant, N the frequency of the emitted radiation, and c the speed of light in the vaccum) Because the emitted photon is moving through the medium of refraction index equal to n, we have: pg= hN/(c/n) (6) Eg = hN: for its total energy. The conservation of the momentum is: P1 + P2 = Pg (7) hence: P2 = Pg - P1 Or: p2 2 = pg2 + p12 - 2.p1 pg cos y (8) The conservation of energy is: E1 = Eg + E2 That is: P12 c2 + m02c4 = hN + P22 c2 + m02c4 Using the relation (8) : (P12 c2 + m02c4) 1/2 - hN = (g2 + p12 - 2.p1 pg cos y + m02c4 )1/2 (9) Hence, using (6) cos y = (2hN.(P12 c2 + m02c4) 1/2 + ( n2 - 1)h2N2)/2c2p1p2 (10) We have : p1 = g1m0V1 Substituting this relation into (10), we get: cos y = c/nV1 + (n2- 1) hN/2nc2g1m0V1 Or : cos y = c/nV1 + ((n2 -1 ). hN (c2 - V2)1/2 )/(2ncm02V1)








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