1. Maxwell’s Equations and Electromagnetic Waves
Chapter 29
Maxwell’s Equations and Electromagnetic Waves

2. → Culmination: Electromagnetic waves
Oersted Denmark
Ampere France
Lenz Russia
Faraday England
Coulomb France
Maxwell Scotland
All electric and magnetic phenomena could be described using only 4 equations involving electric and magnetic fields —— Maxwell’s equations
→ Culmination: Electromagnetic waves 2

3. ? Changing E produces B Changing magnetic field electric field
Changing electric field
magnetic field ?
A beautiful symmetry in nature:
Ampere’s law: S2 S1 I ? 3

4. Discontinuity of current
Ampere’s law: I - + E
Contradiction?
→ discontinuity of the current
Which result is right?
An extra term in Ampere’s law → changing 4

5. Electric field between plates:
Displacement current S2 S1
Conduction current: I - + E
Electric field between plates:
Displacement current: 5

6. Generalized Ampere’s law
Ampere’s law in a general form:
1) ID is produced by changing electric field
2) Continuity of total current: 6

7. Continuity of total current:
Charging capacitor
Example1: A parallel-plate capacitor is charging with Determine (a) the conduction current I ; (b) magnetic field.
0.1m I
Solution: (a)
Continuity of total current: 7

8. Generalized Ampere’s law:
(b) magnetic field ? r
Generalized Ampere’s law: 8

9. Capacitor in LC circuit
Question: The voltage on a capacitor is changing as What is the EMF on the square coil inside the capacitor? a d 9

10. Summary of electromagnetism
Electrostatic / induced electric / total electric field:
Magnetic field created by IC or ID : 10

11. Maxwell’s equations (1)
Finally we can state all 4 of Maxwell’s equations:
① source of E
② no magnetic charges / monopoles
③ changing B → E
④ changing E → B 11

12. Maxwell’s equations (2)
Differential form of Maxwell’s equations:
1) Basic equations for all electromagnetism
2) As fundamental as Newton’s laws
3) Important outcome: electromagnetic waves 12

13. Near field & radiation field
Production of EM waves
Maxwell’s prediction
Hertz’s experiment
“Antenna”
Near field & radiation field
→ electromagnetic wave
Also by accelerating charges 13

14. Wave equations for EM wave
Free space: no charges or conduction currents
Wave equation! 14

15. → 1-D (plane) wave equation
Speed of EM wave
3-D wave equation
→ 1-D (plane) wave equation
Compare with standard wave equation: 15

16. Particular solutions:
Properties of EM wave
Particular solutions:
1) Transverse wave
2) In phase: 3) 16

17. Total energy stored per unit volume in EM wave:
Energy in EM wave
Total energy stored per unit volume in EM wave:
Energy transports per unit time per unit area: 17

18. Consider the direction of energy transporting:
Poynting vector
Consider the direction of energy transporting:
→ Poynting vector
Time averaged S is intensity:
Example 2: Show the direction of energy transporting inside the battery and resistor. 18

19. Solution: Rate → time averaged S / intensity
Sunshine
Example3: Radiation from the Sun reaches the Earth at a rate about 1350W/m2. Assume it is a single EM wave, calculate E0 and B0.
Solution: Rate → time averaged S / intensity 19

20. Be absorbed / reflected → radiation pressure
EM waves carry energy
→ also carry momentum
Be absorbed / reflected
→ radiation pressure
Absorbed:
Reflected: 20