* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
Physics 2102 Jonathan Dowling Lecture 22 James Clerk Maxwell (1831-1879) Maxwell’s equations the dawn of the 20th century revolution in physics Gauss’ Law: charges produce electric fields, field lines start and end in charges E dA q / 0 S S S S Gauss’ law for magnetism: field lines are closed or, there are no magnetic monopoles B dA 0 S S Ampere’s law: electric currents produce magnetic fields B ds i 0 C C Faraday’s law: changing magnetic fields produce (“induce”) electric fields d E ds B dA C dt S All together: E dA q / 0 S B ds i 0 B dA 0 S C d E ds B dA C dt S No charges or currents: E dA 0 q=0 S B dA 0 ? i=0 S B ds 0 C …very suspicious…! d E ds B dA C dt S Something is not right… 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 indeed a magnetic field, equal to the field elsewhere. But there is no current producing it! ? E id=0d/dt Maybe we can make it right… 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 i C 0 0 dt dt dt d dt dt dt q=CV C=0A/d V=Ed E=E•A=EA Displacement current B ds 0 C Maxwell proposed it, and it was confirmed. d C B ds 0 0 dt S E dA B B! B i i E “Maxwell” equations: E dA q / 0 S B dA 0 S d B ds E dA i 0 0 0 C dt S d E ds B dA C dt S Maxwell equations in free space: E dA 0 S B dA 0 S d B ds E dA 0 0 C dt S d E ds B dA C dt S Fields without sources? Maxwell, waves and light A solution to the Maxwell equations in free 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! d 2E d 2E 0 0 2 E sin k ( x vt) 2 dx dt v 1 0 0 =3 108 m/s 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 frequencies! 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 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…maybe. 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.