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

→ 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

? 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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