Manjunath.R
#16/1, 8th Main Road, Shivanagar, Rajajinagar, Bangalore560010, Karnataka, India
*Corresponding Author Email: [email protected]
*Website: http://www.myw3schools.com/
Abstract
Cherenkov radiation is the electromagnetic radiation emitted when a charged particle (such as an electron) passes through an insulator at a constant speed greater than the speed of light in that medium. In this article, we provide a simple, concise discussion about "Cherenkov radiation" which demonstrates the characteristic blue glow of an underwater nuclear reactor.
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Pavel Alekseyevich Cherenkov was a Soviet physicist who shared the Nobel Prize in physics in 1958 with Ilya Frank and Igor Tamm for the discovery of Cherenkov radiation, made in 1934.
In some situations, photon behaves like a wave, while in others, it behaves like a particle. The photons can be thought of as both waves and particles. In 1924 a French physicist Louis de Broglie developed a formula to relate this dual wave and particle behavior:
hc E = hυ, c = λυ, E = = mc2, λ
where E and m are the energy and mass of the photon, υ and λ are the frequency and wavelength of the photon, h is the Planck constant, c is the speed of light. From this we obtain the definition of the photon wavelength through the Planck constant and the momentum of the photon: h λ = mc This equation is used to describe the wave properties of matter, specifically, the wave nature of the electron:
2 h λe = m푒v
m0 where λe is wavelength, h is Planck's constant, me = is the relativistic mass of the v2 √1− c2 electron, moving at a velocity v.
h pe = λ푒
From this it follows that,
dp푒 dλ푒 h = − × 2 dt dt λ푒
2 dp푒 p푒 dλ푒 = × − dt h dt Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis" in 1686. His second law defines a force exerted on the electron to be equal to the rate of change in momentum of the electron:
dp푒 F = dt 2 p푒 dλ푒 F = × − h dt
According to the law that nothing may travel faster than the speed of light – i.e., according to the Albert Einstein's law of variation of mass with velocity (the most famous formula in the world. In the minds of hundreds of millions of people it is firmly associated with the menace of atomic weapons. Millions perceive it as a symbol of relativity theory):
m0 me = v2 √1− c2
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3 That the electron's mass me in motion at speed v is the mass m0 at rest divided by the factor
v2 √1 − implies: the mass of the electron is not constant; it varies with changes in its velocity. c2
Differentiating the above equation, we get:
2 2 mev dv + v dme = c dme
2 2 dme (c – v ) = mev dv
In relativistic mechanics (the arguably most famous cult of modern physics, which has a highly interesting history which dates back mainly to Albert Einstein and may be a little earlier to H.
2 2 Poincaré), we define the energy mec which a moving electron possess to be = m0c + Ek.
2 2 mec = m0c + Ek
2 dm푒c dEk = = Fv dt dt
dp푒 dm푒 dv F = = v + me dt dt dt v2 F = F + me a c2
m푒a F = v2 1− c2 So as v approaches c, the bottom term approaches zero and therefore the force approaches infinity. It requires an infinite amount of force to accelerate the electron to the speed of light.
Because:
m0 me = v2 √1− c2
Therefore:
3 m푒a F = 2 m0
For non-relativistic case (v << c), the above equation reduces to F = m0a.
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4 2 3 p푒 dλ푒 m푒a × − = 2 h dt m0
From this it follows that,
2 2 m v dλ푒 a = 0 × − hm푒 dt
Thus, we have the formula for the calculation of the acceleration of the electron.
Cherenkov radiation is the electromagnetic radiation emitted when a charged particle (such as an electron) travels in a medium with speed v such that:
c < v < c n where c is speed of light in vacuum, and n is the refractive index of the medium. We define the ratio between the speed of the particle and the speed of light as:
v β = c
Using simple trigonometric relation one can determine the Cherenkov angle:
1 cosθ = nβ
c cosθ = nv
v2 1 = c2 n2cos2θ
Since the charged particle is relativistic, we can use the relation:
5 2 2 m v dλ푒 a = 0 × − hm푒 dt
The heavier the charged particle, the higher kinetic energy it must 2 2 2 v √c −v dλ푒 possess to be able to emit a = 2 × − c λC dt Cherenkov radiation.
h where λC = is the Compton wavelength of the electron. m0c
2 υC √1−β dλ푒 a = × − n2cos2θ dt
where υC is the Compton frequency of the electron.
The emission of Cherenkov radiation depends on the refractive index n of c cosθ = nv the medium or the velocity v of the particle in that medium.
2 2 2 2 2 2 Ee – E0 = (mec + m0c ) (mec − m0c )
Since:
2 2 (mec − m0c ) = Ek
2 2 2 Ee = √E0 + p푒c Therefore:
6 2 2 2 p푒 m푒v Ek = = (m푒+ m0) (m푒+ m0)
2 2 m푒c Ek = 2 2 n cos θ (m푒+ m0)
m0 The mass me of an electron moving with a velocity v is given by me = where: m0 = rest v2 √1− c2 mass of electron and c = speed of light.
m0c √c2 − v2 = m푒
2 2 √c −v m0c = v m푒v
2 c λ푒 √ 2 − 1 = v λC
λC λC λe = = c2 √n2cos2θ−1 √ −1 v2
References:
Light – The Physics of the Photon, by Ole Keller.
University Physics with Modern Physics by Hugh D. Young.
Isaac Newton and the Laws of Motion by Andrea Gianopoulos.
An Introduction to Cherenkov Radiation by H Alaeian.
Relativity by Albert Einstein.
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