Physics 2102 Jonathan Dowling James Clerk Maxwell Lecture 22 TUE 13 APR 2010 1831 1879 Ch 32 1 5 Maxwell s equations Ch 33 1 3 Electromagnetic Waves EXAM 03 6PM THU 15 APR LOCKETT 6 The exam will cover Ch 28 second half through Ch 32 1 3 displacement current and Maxwell s equations The exam will be based on HW07 HW10 The formula sheet for the exam can be found here http www phys lsu edu classes spring2010 phys2102 formulasheet3 pdf You can see examples of old exam IIIs here http www phys lsu edu classes spring2009 phys2102 Test3 oldtests pdf Maxwell I Gauss Law for E Fields charges produce electric fields field lines start and end in charges E dA q 0 S S S S S Maxwell II Gauss law for B Fields field lines are closed or there are no magnetic monopoles S S S B dA 0 S S S S Maxwell III Ampere s law electric currents produce magnetic fields B ds i 0 C C Maxwell IV Faraday s law changing magnetic fields produce induce electric fields d E ds B dA C dt S Maxwell Equations I IV E dA q 0 S B ds i 0 C B dA 0 S d E ds B dA C dt S In Empty Space with No Charge or Current E dA 0 S B dA 0 q 0 i 0 S B ds 0 C very suspicious NO SYMMETRY d E ds B dA C dt S Maxwell s Displacement Current B E B If we are charging a capacitor there is a current left and right of the capacitor Thus there is the same magnetic field right and left of the capacitor with circular lines around the wires But no magnetic field inside the capacitor The missing Maxwell Equation With a compass we can verify there is indeed a magnetic field equal to the field elsewhere But there is no current producing it E id 0d dt Maxwell s Fix We calculate the magnetic field produced by the currents at left and at right using Ampere s law B ds i 0 C We can write the current as dq d CV dV 0 A d Ed d EA d E C 0 0 i dt dt dt d dt dt dt q CV C 0A d V Ed E E dA EA Displacement Current B ds 0 C Maxwell proposed it based on symmetry and math no experiment d C B ds 0 0 dt S E dA B B B i i E Maxwell s Equations I V I E dA q 0 S II B dA 0 S V d C B ds 0 0 dt S E dA 0i III d IV E ds B dA dt S C Maxwell Equations in Empty Space E dA 0 Fields without sources S B dA 0 S d C B ds 0 0 dt S E dA d E ds B dA Changing E gives B C dt S Changing B gives E Maxwell Waves and Light A solution to the Maxwell equations in empty space is a traveling wave d C B ds 0 0 dt S E dA d C E ds dt S B dA electric and magnetic fields can travel in EMPTY SPACE d2E d2E 0 0 2 E E0 sin k x ct 2 dx dt c 1 3 10 8 m s 0 0 The electric magnetic waves travel at the speed of light Light itself is a wave of electricity and magnetism Electromagnetic waves First person to prove that electromagnetic waves existed Heinrich Hertz 1875 1894 First person to use electromagnetic waves for communications Guglielmo Marconi 1874 1937 1909 Nobel Prize first transatlantic commercial wireless service Nova Scotia 1909 Electromagnetic Waves One Velocity Many Wavelengths with frequencies measured in Hertz cycles per second and wavelength in meters http imagers gsfc nasa gov ems http www astro uiuc edu kaler sow spectra html How do E M Waves Travel Is there an ether they ride on Michelson and Morley looked and looked and decided it wasn t there How do waves travel Electricity and magnetism are relative Whether charges move or not depends on which frame we use This was how Einstein began thinking about his theory of special relativity We ll leave that theory for later Electromagnetic Waves A solution to Maxwell s equations in free space E Em sin k x t B Bm sin k x t c speed of propagation k Em c Bm 1 0 0 m 299 462 954 187 163 miles sec s Visible light infrared ultraviolet radio waves X rays Gamma rays are all electromagnetic waves Radio waves are reflected by the layer of the Earth s atmosphere called the ionosphere This allows for transmission between two points which are far from each other on the globe despite the curvature of the earth Marconi s experiment discovered the ionosphere Experts thought he was crazy and this would never work Maxwell s Rainbow The wavelength frequency range in which electromagnetic EM waves light are visible is only a tiny fraction of the entire electromagnetic spectrum Fig 33 2 Fig 33 1 33 2 The Traveling Electromagnetic EM Wave Qualitatively An LC oscillator causes currents to flow sinusoidally which in turn produces oscillating electric and magnetic fields which then propagate through space as EM waves E c m Bm Next slide 1 0 0 Fig 33 3 Oscillation Frequency 1 LC Em c Bm 1 0 0 33 3 Mathematical Description of Traveling EM Waves Electric Field E Em sin kx t Magnetic Field B Bm sin kx t Wave Speed c 1 0 0 All EM waves travel a c in vacuum 2 Wavenumber k EM Wave Simulation 2 Angular frequency Vacuum Permittivity 0 Vacuum Permeability 0 Fig 33 5 Amplitude Ratio Em c Bm E t Magnitude Ratio c B t 33 5 The Poynting Vector Points in Direction of Power Flow Electromagnetic waves are able to transport energy from transmitter to receiver example from the Sun to our skin The power transported by the wave and its direction is quantified by the Poynting vector 1 S E B 0 John Henry Poynting 1852 1914 For a wave since E is perpendicular to B In a wave the fields change with time Therefore the Poynting vector changes too Units Watt m2 E S B 1 1 2 S EB E 0 c 0 The direction is constant but the magnitude changes from 0 to a maximum value EM Wave Intensity Energy Density A better measure of the amount of energy in an EM wave is obtained by averaging the Poynting vector over one wave cycle The resulting quantity is called intensity Units are also Watts m2 I S 2 2 2 1 1 Em sin kx t E c 0 c 0 I 1 Em 2 or 2c 0 Both fields have the same energy density I The average of sin2 over one cycle is 1 Erms 2 c 0 1 1 1 B2 1 B2 2 2 uE 0 E 0 cB 0 uB …
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