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Electromagnetic Waves Maxwell’s Equations Qenc E dA B dA 0 d E B d l ( I ) c dt d M E d l dt K o K m o p202c33: 1 “Sourceless” Maxwell’s Equations E dA 0 B dA 0 d E d B dl dt dt E dA d M d E dl dt dt B dA p202c33: 2 A simple Electromagnetic Wave Pulse: E and B constant within a “sheet” moving at velocity v y E v B x z ... need to verify consistency with Maxwell’s Equations p202c33: 3 E dA 0 B dA 0 Field lines continue forever: each field line which enters (exits) a closed surface must also enter (exit), so net number of field lines entering (exiting) a closed surface must be zero. p202c33: 4 B dl BL d ( E dA) ELvdt 0 d E dA ELv dt dA dl L v dt d B dl dt E dA BL ELv B Ev p202c33: 5 E dl EL dl d ( B dA) BLvdt 0 d B dA BLv dt dA L v dt d E dl dt B dA EL BLv E Bv p202c33: 6 Pulse is consistent with Maxwell' s Equations iff B vE E Bv v 1 c speed of light in vacuum 1 o o 2.99792458 108 m s other important features v||E B E Bv p202c33: 7 y General relations between (crossed) E and B fields creating EM waves. d E dl dt B dA E dl ( E y ( x x) E y ( x))a B dA Bz ax a x x take lim x0 E y B z x t p202c33: 8 y d B dl dt B dA B dl ( Bz ( x x) Bz ( x))a E dA E y ax take x x a lim x 0 E y B z x t p202c33: 9 E y Bz x t Start with E y B z x t 2 E Ey B z y x x x 2 x t 2 E y E y 1 Bz 2 t t t t x 2Ey 2Ey 2 x t 2 2Ey 1 2Ey 1 2 ;v 2 2 x v t Classical Wave Equation! p202c33: 10 Sinusoidal Electromagnetic Waves Ey Bz E yˆ Emax sin( kx t ) B zˆBmax sin( kx t ) k 2 f Emax x 2f v k cBmax p202c33: 11 Energy in an electromagnetic wave 1 2 1 2 u E B 2 2 E but B E v so u E 2 For a Harmonic Wave u E 2 Emax sin 2 (kx t ) 2 uav Emax 2 2 p202c33: 12 Energy in an electromagnetic wave Energy : dU udV E 2 ( Avdt ) Energy Flow : dU 2 EB 2 S A vE E dt 1 S EB S Poynting Vector S = Intensity (instantan eous) p202c33: 13 Energy Flow and Harmonic Waves 1 S EB xˆ Emax Bmax sin 2 (kx t ) Emax Bmax Emax cEmax S av I Intensity 2 2 c 2 2 2 p202c33: 14 Example: A radio station the surface of the earth emits 50 kW sinusoidal waves. Determine the intensity, and the Electric and Magnetic field amplitudes for an orbiting satellite at a distance of 100 km from the station. p202c33: 15 Momentum in an electromagnetic wave Momentum Density p S = 2 V c pavg S avg 2 V c Radiation Pressure : F S av F 2 S av Pabs Pref A c A c p202c33: 16 Example: Satellite in previous example has a 2m diameter antenna. What is the force of the radiation on the antenna assuming perfect reflection? p202c33: 17 Standing Waves: Superposition of equal amplitude traveling waves of opposite directions. E Emax sin( kx t ) Emax sin( kx t ) 2 Emax sin kx cos t B Bmax sin( kx t ) Bmax sin( kx t ) 2 Bmax cos kx sin t nodal planes (for E) 3 x , , , n 2 2 2 Standing wave mode when L n 2L c c n ; f n n n n 2L 2 p202c33: 18 Example: EM standing waves are set up in a cavity used for electron spin resonance studies. The cavity has two parallel conducting plates seperated by 1.50 cm. a) Calculate the longest weavelength and lowest frequency of EM standing waves between the walls. b) Where in the cavity is the maximum magnitude electric field and magnetic field? p202c33: 19 Electromagnetic Spectrum (see graphic) in vacuum, v = c = 2.99792458x108 m/s f = v increasing frequency <=> decreasing wavelength visible spectrum: 400 nm (violet) to 700 nm (red) p202c33: 20 Radiation from a Dipole Q Q p0 k 2 sin E sin( kr t ) 4 r p0 k 2 sin B sin( kr t ) 4v r sin I r 2 p202c33: 21 p202c33: 22