W13D1: Displacement Current, Maxwell ’ s Equations, Wave Equations

1. W13D1:Displacement Current, Maxwell’s Equations,Wave Equations Today’s Reading Course Notes: Sections 13.1-13.4

2. Math Review Week 13 Tuesday 9pm-11 pm in 32-082 PS 9 due Week 13 Tuesday April 30 at 9 pm in boxes outside 32-082 or 26-152 Next Reading Assignment W13D2 Course Notes: Sections 13.5-13.7 Announcements

3. Outline Maxwell’s Equations Displacement Current

4. Maxwell’s Equations Is there something missing? 4

5. Maxwell’s EquationsOne Last Modification: Displacement Current “Displacement Current”has nothing to do with displacement and nothing to do with current

6. Ampere’s Law: Capacitor Consider a charging capacitor: Use Ampere’s Law to calculate the magnetic field just above the top plate 1) Surface S1: Ienc= I 2) Surface S2: Ienc = 0 What’s Going On?

7. Displacement Current We don’t have current between the capacitor plates but we do have a changing E field. Can we “make” a current out of that? This is called the “displacement current”. It is not a flow of charge but proportional to changing electric flux

8. Displacement Current: If surface S2 encloses all of the electric flux due to the charged plate then Idis = I

9. Maxwell-Ampere’s Law “flow of electric charge” “changing electric flux”

10. Concept Question: Capacitor If instead of integrating the magnetic field around the pictured Amperian circular loop of radius r we were to integrate around an Amperian loop of the same radius R as the plates (b) then the integral of the magnetic field around the closed path would be • the same. • larger. • smaller.

11. Concept Q. Answer: Capacitor Answer 2. The line integral of B is larger for larger r As we increase the radius of our Amperian loop we enclose more flux and hence the magnitude of the integral will increase.

12. Sign Conventions: Right Hand Rule Integration direction clockwise for line integral requires that unit normal points into page for surface integral. Current positive into the page. Negative out of page. Electric flux positive into page, negative out of page.

13. Sign Conventions: Right Hand Rule Integration direction counter clockwise for line integral requires that unit normal points out page for surface integral. Current positive out of page. Negative into page. Electric flux positive out of page, negative into page.

14. Concept Question: Capacitor Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is • Zero (No current through loop) • Positive • Negative • Can’t tell (need to know direction of E)

15. Concept Q. Answer: Capacitor Answer 2. The line integral of B is positive. There is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line line integral is positive.

16. Concept Question: Capacitor The figures above show a side and top view of a capacitor with charge Q and electric and magnetic fields E and B at time t. At this time the charge Q is: • Increasing in time • Constant in time. • Decreasing in time.

17. Concept Q. Answer: Capacitor Answer 1. The charge Q is increasing in time The B field is counterclockwise, which means that the if we choose counterclockwise circulation direction, the electric flux must be increasing in time. So positive charge is increasing on the bottom plate.

18. Group Problem: Capacitor A circular capacitor of spacing d and radius R is in a circuit carrying the steady current i shown. At time t = 0 , the plates are uncharged • Find the electric field E(t) at P vs. time t (mag. & dir.) • Find the magnetic field B(t) at P

19. Maxwell’s Equations

20. Electromagnetism Review E fields are associated with: (1) electric charges (Gauss’s Law ) (2) time changing B fields (Faraday’s Law) B fields are associated with (3a) moving electric charges (Ampere-Maxwell Law) (3b) time changing E fields (Maxwell’s Addition (Ampere-Maxwell Law) Conservation of magnetic flux (4) No magnetic charge (Gauss’s Law for Magnetism)

21. Electromagnetism Review • Conservation of charge: • E and B fields exert forces on (moving) electric charges: • Energy stored in electric and magnetic fields

22. Maxwell’s Equationsin Vacua

23. 0 0 Maxwell’s Equations What about free space (no charge or current)?

24. How Do Maxwell’s Equations Lead to EM Waves?

25. Wave Equation Start with Ampere-Maxwell Eq and closed oriented loop

26. Wave Equation Start with Ampere-Maxwell Eq: Apply it to red rectangle: So in the limit that dx is very small:

27. Group Problem: Wave Equation Use Faraday’s Law and apply it to red rectangle to find the partial differential equation in order to find a relationship between

28. Group Problem: Wave Equation Sol. Use Faraday’s Law: and apply it to red rectangle: So in the limit that dx is very small:

29. 1D Wave Equation for Electric Field Take x-derivative of Eq.(1) and use the Eq. (2)

30. 1D Wave Equation for E This is an equation for a wave. Let

31. Definition of Constants and Wave Speed Recall exact definitions of The permittivity of free space is exactly defined by

32. Group Problem: 1D Wave Eq. for B Take appropriate derivatives of the above equations and show that

33. Wave Equations: Summary Both electric & magnetic fields travel like waves: with speed But there are strict relations between them:

34. Electromagnetic Waves

35. Electromagnetic Radiation: Plane Waves http://youtu.be/3IvZF_LXzcc