1. Electromagnetism Ch.7 Methods of Math. Physics, Friday 8 April 2011, EJZ Inductors and inductance Waves and wave equations Electromagnetism & Maxwell’s eqns Derive EM wave equation and speed of light

4. Faraday’s Law

9. Break time…

10. Waves

11. Wave equation 1. Differentiate d D/ d t  d 2 D/ d t 2 2. Differentiate d D/ d x  d 2 D/ d x 2 3. Find the speed from

12. Causes and effects of E Gauss: E fields diverge from charges Lorentz force: E fields can move charges F = q E

13. Causes and effects of B Ampere: B fields curl around currents Lorentz force: B fields can bend moving charges F = q v x B = IL x B

14. Changing fields create new fields! Faraday: Changing magnetic flux induces circulating electric field Guess what a changing E field induces?

15. Changing E field creates B field! Current piles charge onto capacitor Magnetic field doesn’t stop Changing electric flux →“displacement current” → magnetic circulation

16. Partial Maxwell’s equations Charge → E fieldCurrent → B field Faraday Changing B → E Ampere Changing E → B

17. Maxwell eqns → electromagnetic waves Consider waves traveling in the x direction with frequency f =   and wavelength =  /k E(x,t)=E 0 sin (kx-  t) and B(x,t)=B 0 sin (kx-  t) Do these solve Faraday and Ampere’s laws?

18. Faraday + Ampere

19. Sub in: E=E 0 sin (kx-  t) and B=B 0 sin (kx-  t)

20. Speed of Maxwellian waves? Faraday:   = k E 0 Ampere:      E 0 =kB 0 Eliminate B 0 /E 0 and solve for v=  /k    x 10 -7 Tm/A   = 8.85 x 10 -12 C 2 N/m 2

21. Maxwell equations → Light E(x,t)=E 0 sin (kx-  t) and B(x,t)=B 0 sin (kx-  t) solve Faraday’s and Ampere’s laws. Electromagnetic waves in vacuum have speed c and energy/volume = 1/2  0 E 2 = B 2 /(2  0 )

22. Full Maxwell equations in integral form

23. Integral to differential form Gauss’ Law: apply Divergence Thm: and the Definition of charge density: to find the Differential form:

24. Integral to differential form Ampere’s Law: apply Curl Thm: and the Definition of current density: to find the Differential form:

25. Integral to differential form Faraday’s Law: apply Curl Thm: to find the Differential form:

26. Finish integral to differential form…

28. Maxwell eqns in differential form