week6 - 1 Monday October 24 Special Relativity Re view...

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Unformatted text preview: 1 Monday, October 24: Special Relativity Re- view Personally, I’m very fond of classical physics. As we’ve seen recently, you can derive some very useful electromagnetic formulae without taking into ac- count quantum mechanics or special relativity. However, just as Max Planck demonstrated that you sometimes have to take quantization into effect, Al- bert Einstein (just a century ago) demonstrated that you sometimes have to take relativistic effects into account. There are times when classical New- tonian physics is an inadequate approximation to reality. In this course, I’m trying to minimize the use of quantum mechanics, for fear of intruding into the domain of your other radiation course (Astronomy 823: Theoretical Spectroscopy). However, I am going to plunge into special relativity, to see how our previously derived results vary in the limit that the speed of charged particles approaches the speed of light. One of the more entertaining aspects of the universe, at present, is its great variety. There’s a huge range of densities, temperatures, and electric and magnetic field strengths. In a few regions of the universe, the tempera- ture is high enough for electrons to be relativistic ( v e ∼ c ). For electrons to have thermal velocities near the speed of light, the thermal energy must be comparable to or greater than the rest energy of the electron: kT ≥ m e c 2 ∼ . 5 MeV , (1) which requires a temperature 1 T ≥ m e c 2 /k ∼ 6 × 10 9 K . (2) In a magnetic field of flux density B , an electron on an orbit of radius r will be relativistic if ω cyc r = fl fl fl fl q e B m e c fl fl fl fl r ∼ c . (3) This requires | B | r ∼ m e c 2 | q e | ∼ 2000 gauss cm . (4) Near a magnetized neutron star with B ∼ 10 9 gauss, even electrons moving in tiny orbits r ∼ 20 nm require relativistic treatment. 1 Generally useful approximation: 1 MeV → 10 10 K. 1 The special theory of relativity (alias special relativity ) is based on two simple postulates. The first postulate states: (1) The laws of physics are the same in all inertial frames of reference. An inertial frame of reference is one in which Newton’s Laws of Motion hold true. Thus, in an inertial frame, m ¨ ~ r = ~ F , (5) where m is a particle’s mass and ~ F is the net force on the particle. A rotating frame of reference is an example of a frame that is not inertial. In a frame rotating with a constant angular velocity ~ Ω, the equation of motion is m ¨ ~ r = ~ F- m (2 ~ Ω × ˙ ~ r )- m ( ~ Ω × [ ~ Ω × ~ r ]) . (6) Thus, in a rotating frame of reference, there are two fictitious forces; the “Coriolis force” (proportional to ~ Ω × ˙ ~ r ) and the “centrifugal force” (propor- tional to ~ Ω × [ ~ Ω × ~ r ]). An inertial frame may also be defined as a frame in which there are no fictitious forces....
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week6 - 1 Monday October 24 Special Relativity Re view...

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