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29. Maxwell’s Equations Topics Laws of Electric & Magnetic Fields James Clerk Maxwell Maxwell’s Equations 2 Laws of Electric & Magnetic Fields 3 James Clerk Maxwell 1831 – 1879 In 1865, Maxwell published a paper entitled: A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). This is one of the greatest scientific papers ever written. 4 Maxwell’s Equations E dA Qinside Closed Surface 0 B dA 0 Closed Surface d m E dr dt Closed Loop 5 de B dr 0 I 0 0 dt Closed Loop Displacement Current Maxwell realized that Ampere’s law is not valid when the current is discontinuous as is true of the current through a parallel plate capacitor: B dr 0 I Encircled Closed Loop wikimedia.org 6 Displacement Current He concluded that when the charge within an enclosed surface is changing it is necessary to add to Ampere’s law another current called the displacement current: ID dQinside de ID 0 dt dt B dr 0 ( I I D ) Closed Loop 7 wikimedia.org The nd 2 Unification of Forces The 2nd Unification of Forces 0 is the electric 0 8.854 10 C /(N m ) 12 0 is the magnetic constant 2 2 constant 0 4 107 N/A 2 0 0 4 10 [ N/A ] 7 2 12 8.854 10 [C /(N m )] 17 2 2 1.113 10 [s /m ] 9 2 2 The 2nd Unification of Forces From 0 0 1.113 1017 [s 2 /m 2 ] we can write 1 0 0 2.998 10 m/s 8 which is the speed of light in vacuum! 10 Light “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” (1866) 11 2nd Unification 3rd Unification 4th Unification? 5th Unification? 1st Unification 12 Summary 13 Maxwell’s Equations Gauss’s Law for E Gauss’s Law for B Faraday’s Law Ampere’s Generalized Law Electromagnetic Waves Topics Maxwell’s Wave Equations Waves – Recap Electromagnetic Waves Electromagnetic Radiation 15 Maxwell’s Wave Equations E 1 E Wave equation for E 2 2 2 x c t 2 2 1 B Wave equation B 2 2 for B 2 x c t 2 2 These equations describe electric and magnetic waves traveling in the x direction 16 Maxwell’s Wave Equations Maxwell showed that the different components of the electric and magnetic fields are related: Ez By x t Relationship between Bz and Ey 17 Relationship between Ez and By E y Bz t x Waves – Recap Wave number Stationary wave y ( x) A sin(kx) k 2 Wave traveling in x direction y ( x) A sin k ( x vt ) 18 2 kv T Electromagnetic Waves Consider an electric wave, traveling in the positive x direction, but oscillating in the y direction: E y ( x, t ) Ep sin(kx t ) We can find Bz from Ep 19 E y Bz t x Electromagnetic Waves This leads to the result Bz ( x, t ) Bp sin(kx t ) where z Bp (k / ) Ep that is, Bp 20 Ep cBp Electromagnetic Waves Electromagnetic waves always travel in the direction of the ˆ E Ep sin(kx t ) j Poynting vector: B Bp sin(kx t ) kˆ S y EB 0 Units: W/m2 x z 21 Electromagnetic Waves But the direction of the electric and magnetic fields themselves, that is, their polarization, can change Linear polarization y x z 22 Polarizers Only a component Epcos of the electric field along the polarization axis can get through 1 Law of Malus S S0 cos 2 23 2 3 90o Electromagnetic Radiation The Electromagnetic Spectrum 25 Spectral Response human bee butterfly http://landsat.gsfc.nasa.gov/education/compositor 26 Himalyan balsam Electromagnetic Radiation An electromagnetic wave carries energy and momentum. The average power per unit area is called the intensity of the wave The momentum per unit time (that is, force) per unit area is called the radiation pressure 27 Electromagnetic Radiation The radiation pressure, Prad, is given by S Prad c where the average intensity is given by 1 Ep Bp S 0 2 0 EB 28 which can be written in terms of energy density: S cuE cuB The Pressure of Sunshine Solar Luminosity L = 3.8 x 1026 W Astronomical Unit r = 1.5 x 1011 m Intensity S = L / 4 r2 Pressure P =S/c 29 The Pressure of Sunshine Intensity S = L / 4 r2 = 1370 W/m2 Pressure =S/c = 4.6 N/m2 30 P Interstellar Travel 31 Credit: Michael Carroll, The Planetary Society Summary 32 Maxwell’s Equations 2nd Unification of forces Electromagnetic waves Universal speed c = 3 x 108 m/s Electromagnetic Waves Gamma rays to radio waves Carry energy and momentum Exert pressure