Physics 2102 Jonathan Dowling James Clerk Maxwell 1831 1879 Lecture 31 MON 30 MAR Ch 32 1 3 Maxwell s equations 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 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 With a compass we can verify there is The missing Maxwell Equation 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 S B dA 0 S d C B ds 0 0 dt S E dA d E ds B dA C dt S Fields without sources 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 forces can travel 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 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 Summary Changing electric fields produce induce magnetic fields displacement currents Maxwell s laws allow us to calculate electric and magnetic fields everywhere in space if we are given the sources electric charges and currents If there are no sources we can still have electric and magnetic fields as electromagnetic waves which travel at the speed of light
View Full Document
Unlocking...