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Unformatted text preview: 1 Monday, November 7: Synchrotron Radia- tion for Beginners An accelerated electron emits electromagnetic radiation. The most effective way to accelerate an electron is to use electromagnetic forces. Since electrons have mass, they can also be accelerated gravitationally. However, electrons are seldom in freefall for long. In addition, gravitational accelerations are frequently small. An acceleration g = 980 cm s- 2 makes an electron radiate with a power P = 2 e 2 g 2 3 c 3 = 5 . 5 × 10- 45 erg s- 1 . (1) This amount of power would be produced if the electron passed a proton at a distance b = ˆ e 2 m e g ! 1 / 2 = 510 cm . (2) Thus, if a free electron is falling near the Earth’s surface, if there’s a proton within five meters, the electrostatic acceleration will be greater than the gravitational acceleration. 1 An electron experiences an electromagnetic force if it passes a positively charged ion; the light it emits in this case is bremsstrahlung. In the limit that the relative velocity ~v of the electron and ion is small ( v ¿ c ), the electron experiences a pure electric force directed toward the ion. In the case of relativistic bremsstrahlung, the electron experiences a mixture of electric and magnetic fields in its instantaneous rest frame as the ion zips past. There exist circumstances in which the electron experiences a purely mag- netic force. Suppose, in some inertial frame of reference, there exists a uni- form magnetic flux density ~ B , with a negligibly small electric field strength ~ E . In the universe, it is not difficult to find magnetic fields, ranging in strength from the microgauss fields of intergalactic space to the teragauss fields near the most highly magnetized neutron stars. They aren’t usually uniform in strength, but the constant ~ B approximation is a useful place to start. If the velocity of the electron is small, then (as we’ve seen in Problem Set 2) the electron has a circular orbit in a plane perpendicular to ~ B , with 1 This is just another way of stating that gravity is a pathetically feeble force. 1 angular frequency ω cyc = Be m e c = 1 . 8 × 10 7 s- 1 ˆ B 1 gauss ! . (3) The radiation produced in this non-relativistic case is called cyclotron radia- tion . In the limit v → c , the cyclotron radiation is monochromatic, with fre- quency ν = ω cyc / (2 π ). Thus, the Earth’s magnetic field, with B ∼ . 5 gauss, would produce radio waves with ν ∼ 1 MHz. Producing visible light by the cyclotron process would require a much higher magnetic flux density: B ∼ 2 × 10 8 gauss. The power produced by cyclotron radiation is P cyc = 2 e 2 ω 4 cyc r 2 3 c 3 = 2 3 r 2 c µ v cyc c ¶ 2 B 2 (4) = 1 . 6 × 10- 15 erg s- 1 β 2 cyc ˆ B 1 gauss !...
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- Fall '05
- Electron, Magnetic Field, magnetic ﬂux density, Lightman