chap14 - Chapter Fourteen Radiation by Moving Charges Sir...

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Unformatted text preview: Chapter Fourteen Radiation by Moving Charges Sir Joseph Larmor (1857 - 1942) September 17, 2001 Contents 1 Li´ enard-Wiechert potentials 1 2 Radiation from an Accelerated Charge; the Larmor Formula 9 2.1 Relativistic Larmor Formula . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Example: Synchrotron . . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Example: Linear Acceleration . . . . . . . . . . . . . . . . . . 15 3 Angular distribution of radiation 17 3.1 Example: Parallel acceleration and velocity . . . . . . . . . . . . . . . 17 3.2 Example: Acceleration Perpendicular to Velocity . . . . . . . . . . . 20 3.3 Comparison of Examples . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Radiation of an Ultrarelativistic Charged Particle . . . . . . . . . . . 22 4 Frequency Distribution of the Radiated Energy 25 4.1 Continuous Frequency Distribution . . . . . . . . . . . . . . . . . . . 26 4.2 Discrete Frequency Distribution . . . . . . . . . . . . . . . . . . . . . 30 1 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3.1 A Particle in Instantaneous Circular Motion . . . . . . . . . . 32 4.3.2 A Particle in Circular Motion . . . . . . . . . . . . . . . . . . 37 5 Thomson Scattering; Blue Sky 42 6 Cherenkov Radiation Revisited 46 7 Cherenkov Radiation; Transition Radiation 51 7.1 Cherenkov Radiation in a Dilute Collisionless Plasma . . . . . . . . . 55 8 Example Problems 57 8.1 A Relativistic Particle in a Capacitor . . . . . . . . . . . . . . . . . . 57 8.2 Relativistic Electrons at SLAC . . . . . . . . . . . . . . . . . . . . . . 58 We have already calculated the radiation produced by some known charge and current distribution. Now we are going to do it again. This time, however, we shall consider that the source is a single charge moving in some fairly arbitrary, possibly relativistic, fashion. Here the methods of chapter 9, e.g., multipole expansions, are impractical and there are better ways to approach the problem. 1 Li´ enard-Wiechert potentials The current and charge densities produced by a charge e in motion are ρ ( x , t ) = eδ ( x- x ( t )) J ( x , t ) = e v ( t ) δ ( x- x ( t )) (1) if x ( t ) is the position of the particle at time t and v ( t ) ≡ d x ( t ) /dt ≡ ˙ x ( t ) is its velocity. In four-vector notation, J μ ( x , t ) = ecβ μ δ ( x- x ( t )) (2) 2 where β μ ≡ (1 , β ) is not a four-vector; β = v /c . From J and ρ , one finds A and Φ. We can do this in an infinite space by making use of the retarded Green’s function G ( x , t ; x , t ) = δ ( t- t- | x- x | /c ) / | x- x | . From here, we can evaluate the electromagnetic field as appropriate derivatives of the potentials. All of these manipulations are straightforward. Furthermore, the integrations are relatively easy because there are many delta functions. The problem becomes interesting and unfamiliar, however, for highly relativistic particles which produce large retardation effects....
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This note was uploaded on 01/08/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.

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chap14 - Chapter Fourteen Radiation by Moving Charges Sir...

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