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Chapter 29 Maxwell’s Equations and Electromagnetic Waves Coulomb Oersted France Denmark Ampere France Faraday England Lenz Russia 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: B dl 0 I S1 0 S2 E B S2 S1 ? I 3 Discontinuity of current S2 Ampere’s law: B dl S1 0 I S1 I 0 S2 + + + + - E Contradiction? → discontinuity of the current j dS I 0 Which result is right? S1 S2 An extra term in Ampere’s law → changing E 4 Displacement current Conduction current: S1 dQ d I S dt dt I I d dE j 0 S dt dt Electric field between plates: Displacement current: Q S2 + + + + - Q E E 0 dE dE jD 0 , ID 0 dt dt 5 Generalized Ampere’s law ID I dE dE jD 0 , ID 0 dt dt Ampere’s law in a general form: dE B dl 0 (I I D )in 0 Iin 0 0 dt 1) ID is produced by changing electric field 2) Continuity of total current: (j j D ) dS 0 6 Charging capacitor Example1: A parallel-plate capacitor is charging with dE / dt 2 109V / m s. Determine (a) the conduction current I ; (b) magnetic field. Solution: (a) dE jD 0 dt Continuity of total current: I 0.1m dE I I D jD S R 0 dt 2 5.56 104 A 7 dE / dt 2 10 V / m s 9 0.1m (b) magnetic field ? Generalized Ampere’s law: B dl (I I 0 I r ) D in 0 0 dE dE 2 r 0.1m : B 2 r 0 0 r B r dt 2 dt 0 I r 0.1m : B 2 r 0 I B 2 r 8 Capacitor in LC circuit Question: The voltage on a capacitor is changing as V V0 cos t . What is the EMF on the square coil inside the capacitor? dE 0 dV jD 0 dt d dt μ0 jD B r B 2 d a a dB dt 9 Summary of electromagnetism Electrostatic / induced electric / total electric field: E s E s dS Qin 0 dl 0; ; E dS 0 i dB Ei dl dt Magnetic field created by IC or ID : Q E dS 0 dB E dl dt B dS 0 dE B dl 0 (I I D )in 0 I 0 0 dt 10 Maxwell’s equations (1) Finally we can state all 4 of Maxwell’s equations: E dS Q 0 ① source of E B dS 0 ② no magnetic charges/monopoles d B E dl dt ③ changing B → E dE B dl 0 I 0 0 dt ④ changing E → B 11 Maxwell’s equations (2) Differential form of Maxwell’s equations: E 0 B 0 B E t E B 0 j 0 0 t 1) Basic equations for all electromagnetism 2) As fundamental as Newton’s laws 3) Important outcome: electromagnetic waves 12 Production of EM waves Maxwell’s prediction Hertz’s experiment “Antenna” B Near field & radiation field E → electromagnetic wave (t ) Also by accelerating charges E Antenna Ground 13 Wave equations for EM wave Free space: no charges or conduction currents E 0 B 0 E B t B 0 0 E t 2 E 2 E 0 0 2 t ( E ) ( B) t 2 E 0 0 2 t ( E) ( E) 2 E 2 E Wave equation! 14 Speed of EM wave 2 E 2 E 0 0 2 t 2 B 2 B 0 0 2 t 3-D wave equation → 1-D (plane) wave equation 2 E 2 E 0 0 2 , 2 x t 2 B 2 B 0 0 2 2 x t Compare with standard wave equation: 2 2 y y 1 2 8 v v 3.0 10 m/s c ! 2 2 t x 0 0 15 Properties of EM wave 2 2 E E 2 c 2 , 2 x t Particular solutions: 1) Transverse wave 2) In phase: 2 2 B B 2 c 2 2 x t x E E0 cos (t ) c x B B0 cos (t ) c E c B 3) E B 16 Energy in EM wave Total energy stored per unit volume in EM wave: 2 2 1 1 B B u 0 E2 0 E2 2 2 0 0 E ( c B 1 0 0 ) Energy transports per unit time per unit area: dU udV u A cdt EB dU S uc 0 A dt 17 Poynting vector Consider the direction of energy transporting: S 1 0 ( E B) → Poynting vector 1 E0 B0 Time averaged S is intensity: S 2 0 Example 2: Show the direction of energy transporting inside the battery and resistor. I B E S 18 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 2S 1 E0 B0 1 2 3 E 1.01 10 V /m S 0 cE0 0 0c 2 0 2 1 u c um c 2 E0 B0 3.37 106 T c 19 *Radiation pressure EM waves carry energy → also carry momentum Be absorbed / reflected → radiation pressure Absorbed: P S c Reflected: P 2S c 20