1. 02/18/2015PHY 712 Spring 2015 -- Lecture 151 PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 Plan for Lecture 15: Finish reading Chapter 6 1.Some details of Liénard-Wiechert results 2.Energy density and flux associated with electromagnetic fields 3.Time harmonic fields

2. 02/18/2015PHY 712 Spring 2015 -- Lecture 152

3. 02/18/2015PHY 712 Spring 2015 -- Lecture 153

4. 02/18/2015PHY 712 Spring 2015 -- Lecture 154

5. 02/18/2015PHY 712 Spring 2015 -- Lecture 155 Solution of Maxwell’s equations in the Lorentz gauge -- continued Liénard-Wiechert potentials and fields -- Determination of the scalar and vector potentials for a moving point particle (also see Landau and Lifshitz The Classical Theory of Fields, Chapter 8.) Consider the fields produced by the following source: a point charge q moving on a trajectory R q (t). q R q (t)

6. 02/18/2015PHY 712 Spring 2015 -- Lecture 156 Solution of Maxwell’s equations in the Lorentz gauge -- continued We performing the integrations over first d 3 r’ and then dt’ making use of the fact that for any function of t’, where the ``retarded time'' is defined to be

7. 02/18/2015PHY 712 Spring 2015 -- Lecture 157 Solution of Maxwell’s equations in the Lorentz gauge -- continued Resulting scalar and vector potentials: Notation:

8. 02/18/2015PHY 712 Spring 2015 -- Lecture 158 Solution of Maxwell’s equations in the Lorentz gauge -- continued In order to find the electric and magnetic fields, we need to evaluate The trick of evaluating these derivatives is that the retarded time t r depends on position r and on itself. We can show the following results using the shorthand notation: and

9. 02/18/2015PHY 712 Spring 2015 -- Lecture 159 Solution of Maxwell’s equations in the Lorentz gauge -- continued

10. 02/18/2015PHY 712 Spring 2015 -- Lecture 1510 Energy analysis of electromagnetic fields and sources

11. 02/18/2015PHY 712 Spring 2015 -- Lecture 1511 Energy analysis of electromagnetic fields and sources - - continued

12. 02/18/2015PHY 712 Spring 2015 -- Lecture 1512 Energy analysis of electromagnetic fields and sources - - continued

13. 02/18/2015PHY 712 Spring 2015 -- Lecture 1513 Momentum analysis of electromagnetic fields and sources

14. 02/18/2015PHY 712 Spring 2015 -- Lecture 1514 Comment on treatment of time-harmonic fields

15. 02/18/2015PHY 712 Spring 2015 -- Lecture 1515 Comment on treatment of time-harmonic fields -- continued Note -- in all of these, the real part is taken at the end of the calculation. Equations in time domain in frequency domain

16. 02/18/2015PHY 712 Spring 2015 -- Lecture 1516 Comment on treatment of time-harmonic fields -- continued

17. 02/18/2015PHY 712 Spring 2015 -- Lecture 1517 Summary and review

18. 02/18/2015PHY 712 Spring 2015 -- Lecture 1518

19. 02/18/2015PHY 712 Spring 2015 -- Lecture 1519 Analysis of Maxwell’s equations without sources -- continued:

20. 02/18/2015PHY 712 Spring 2015 -- Lecture 1520 Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:

21. 02/18/2015PHY 712 Spring 2015 -- Lecture 1521 Analysis of Maxwell’s equations without sources -- continued: Note: , n, k can all be complex; for the moment we will assume that they are all real (no dissipation).

22. 02/18/2015PHY 712 Spring 2015 -- Lecture 1522 Analysis of Maxwell’s equations without sources -- continued:

23. 02/18/2015PHY 712 Spring 2015 -- Lecture 1523 Analysis of Maxwell’s equations without sources -- continued: Transverse Electric and Magnetic (TEM) waves

24. 02/18/2015PHY 712 Spring 2015 -- Lecture 1524 More general result (for homework)