1. W13D1: Displacement Current, Maxwell’s Equations, Wave Equations
Today’s Reading Course Notes: Sections
Class 18

2. Announcements 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 or
Next Reading Assignment W13D2 Course Notes: Sections
Class 28

3. Outline
Maxwell’s Equations
Displacement Current
Class 28

4. Maxwell’s Equations Is there something missing? Week 13, Day 1 4
Class 30 4

5. Maxwell’s Equations One Last Modification: Displacement Current “Displacement Current” has nothing to do with displacement and nothing to do with current
Class 28

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?
Class 28

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
Class 28

8. Displacement Current:
If surface S2 encloses all of the electric flux due to the charged plate then Idis = I
Class 28

9. Maxwell-Ampere’s Law “flow of electric charge”
“changing electric flux”
Class 28

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.
Class 28

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.
Class 28

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.
Class 20

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.
Class 20

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)
Class 28

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.
Class 28

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.
Class 28

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.
Class 28

18. Group Problem: Capacitor
Week 13, Day 2
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
Class 31

19. Week 13, Day 1
Maxwell’s Equations
Class 30

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)
Class 30

21. Electromagnetism Review
Conservation of charge:
E and B fields exert forces on (moving) electric charges:
Energy stored in electric and magnetic fields
Class 30

22. Maxwell’s Equations in Vacua
Class 30

23. Maxwell’s Equations What about free space (no charge or current)?
What about free space (no charge or current)?
Class 30

24. How Do Maxwell’s Equations Lead to EM Waves?
Class 30

25. Wave Equation Start with Ampere-Maxwell Eq and closed oriented loop
Class 30

26. Wave Equation Start with Ampere-Maxwell Eq: Apply it to red rectangle:
So in the limit that dx is very small:
Class 30

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
Class 30

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:
Class 30

29. 1D Wave Equation for Electric Field
Take x-derivative of Eq.(1) and use the Eq. (2)
Class 30

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

31. Definition of Constants and Wave Speed
Week 13, Day 2
Definition of Constants and Wave Speed
Recall exact definitions of
The permittivity of free space is exactly defined by
Class 31

32. Group Problem: 1D Wave Eq. for B
Take appropriate derivatives of the above equations
and show that
Class 30

33. Wave Equations: Summary
Both electric & magnetic fields travel like waves:
with speed
But there are strict relations between them:
Class 30

34. Electromagnetic Waves
Class 30

35. Electromagnetic Radiation: Plane Waves
Class 30