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Feb 25, 2011

Feb 25, 2011. RETARDED POTENTIALS. At point r =(x,y,z), integrate over charges at positions r’. (7). (8). where [ ρ ]  evaluate ρ at retarded time:. Similar for [j]. So potentials at point. and time t are affected by conditions at. Given a charge and current density,

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Feb 25, 2011

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  1. Feb 25, 2011

  2. RETARDED POTENTIALS At point r =(x,y,z), integrate over charges at positions r’ (7) (8) where [ρ] evaluate ρ at retarded time: Similar for [j]

  3. So potentials at point and time t are affected by conditions at Given a charge and current density, find retarded potentials by means of (7) and (8) Then use (1) and (2) to derive E, B Fourier transform  spectrum

  4. Radiation from Moving Charges The Liénard-Wiechart Potentials Retarded potentials of single, moving charges Charge q moves along trajectory velocity at time t is charge density current density delta function

  5. Can integrate over volume d3r to get total charge and current

  6. What is the scalar potential for a moving charge? where [ ] denotes evaluation at retarded time Recall Substitute and integrate

  7. vector Now let scalar then

  8. Now change variables again then Velocity dot both sides  Also define unit vector

  9. so This means evaluate integral at t'‘=0, or t‘=t(retard)

  10. So... beaming factor or, in the bracket notation: Liénard-Wiechart scalar potential Similarly, one can show for the vector potential: Liénard-Wiechart vector potential

  11. Given the potentials one can use to derive E and B. We’ll skip the math and just talk about the result. (see Jackson §14.1)

  12. The Result: The E, B field at point r and time t depends on the retarded position r(ret) and retarded time t(ret) of the charge. Let Field of particle w/ constant velocity Transverse field due to acceleration

  13. Qualitative Picture: transverse “radiation” field propagates at velocity c

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