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Maxwell's Equations and Electromagnetic Waves: Uniting Electric and Magnetic Phenomena

Explore the culmination of electromagnetism through Maxwell's equations, revealing the beautiful symmetry in nature, the role of changing electric and magnetic fields, and how electromagnetic waves are produced. Understand the generalized Ampere's law and its applications, including the conduction current in a charging capacitor and the prediction and properties of electromagnetic waves. Delve into the wave equations for electromagnetic waves, their speed, energy, and direction of energy transport with the Poynting vector, and learn about radiation pressure. Discover the profound implications of Maxwell's equations in describing all electric and magnetic phenomena using only four fundamental equations.

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Maxwell's Equations and Electromagnetic Waves: Uniting Electric and Magnetic Phenomena

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  1. Chapter 29 Maxwell’s Equations and Electromagnetic Waves

  2. Oersted Denmark Ampere France Lenz Russia Faraday England Maxwell Scotland Coulomb France 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. S2 S1 I Changing E produces B Changing magnetic field electric field Changing electric field magnetic field ? A beautiful symmetry in nature: Ampere’s law: ? 3

  4. S2 S1 + + + + I An extra term in Ampere’s law → changing E - - - - Discontinuity of current Ampere’s law: Contradiction? → discontinuity of the current Which result is right? 4

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

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

  8. r I 0.1m (b) magnetic field ? Generalized Ampere’s law: 8

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

  10. Summary of electromagnetism Electrostatic / induced electric / total electricfield: Magneticfield 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. 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. Speed of EM wave 3-D wave equation → 1-D (plane) wave equation Compare with standard wave equation: 15

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

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

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

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