Maxwell s Equations & Hertz Waves

Transcription

1 Maxwell s Equations & Hertz Waves XI. Maxwell & Electromagnetic Waves A. Maxwell s Equations Dr. Bill Pezzaglia B. Hertz Waves & Poynting C. Polarization Updated: 3Aug5 A. Maxwell s Equations 3. Hints that Light is Electromagnetic 4 ) Light is Electromagnetic A number of experiments suggested that there was a connection between electricity, magnetism and the phenomena of light. ) Displacement Current 834, 857 Speed of electricity in wire measured to be very fast (close to speed of light) 3) Maxwell s Equations 85 Speed of light is measured accurately 85 by Foucault. 844 Faraday rotates the polarization of light with a magnetic field (implies light has magnetic properties). Charles Wheatstone (8-875) 5 Michael Faraday discovery by English physicist Charles Wheatstone that current traveled through long lengths of wire with great velocity almost 88, miles/second 844 Faraday rotates the polarization of light using a magnetic field. Suggests light is a transverse magnetic disturbance A bit off, it can t travel faster than speed of light (86, 8 miles/second) 837 Developed an early telegraph (5 needles) 857 Wilhem Weber shows Amp of current is a Coulomb per second, gets characteristic speed of electrical signals to be speed of light.

2 Gustav Kirchhoff (84-887) 7. Field Induction Telegraphy Equations Derives (based on earlier work by Faraday & Thomson 854) that speed of electrical signal in cable should be close to speed of light. Recall Faraday s law is that voltage (emf=electric firld times circumference) in a wire loop was generated by changing magnetic flux through the loop. Maxwell shows that the law is more general. A changing magnetic field generates a circular electric field even if there is no wire! V t James Maxwell (83-879) b. Ampere s circulation Law 9 c. Maxwell s Displacement Current Recall : Ampere s law says that a circular magnetic field is generated by a current. Or: B field multiplied by circumference of a circle is proportional to current flowing through circle B ( r ) I 86: Maxwell makes Ampere s law look like the complement of Faradays Law. A changing electric flux will generate circular B field. B( r) I c Note c is the speed of light, and Electric Flux is defined: E A t [details, you can ignore equation] 3. Maxwell s Equations b. Differential Form of Maxwell s Equations (a) The General Laws of Maxwell Gauss s Law shows that charge is the source of electric fields (electric flux through a closed surface is proportional to net enclosed charge) Gauss s Law for magnetism states that there are no magnetic charges (magnetic flux through a closed surface is zero). Faraday s Law: changing magnetic fields create electric fields Ampere s Circulation Law: current is the source of magnetic fields. Maxwell adds the displacement current to this equation such that changing electric fields create magnetic fields. 884 (with Gibbs) Heaviside reorganizes Maxwell s equations compactly into 4 vector equations For completeness, here they are, but don t worry about them. B E B E t E B J t Oliver Heaviside (85-95)

3 c. Relativity and Maxwell s Equations 3 B. Hertz Waves 4 95 Einstein s Relativity shows that time is the 4 th dimension. In our ordinary 3D view of the world, electric fields are different than magnetic fields, however we see they are complementary In 4D we see that they are both the same thing, i.e. we unify electricity with magnetism. We can write Maxwell s 4 equations in just : F j [ F ] For completeness, here they are, but don t worry about them. ) Equations predict waves ) Hertz Experiment 3) Energy in Waves. EM Wave Equation 5 b. Prediction of Electromagnetic Waves 6 (a) 865 Maxwell shows his equations predict that electromagnetic waves can exist in vacuum (note E & B are perpendicular to each other and direction of wave) The Theoretical speed: comes out very close to known speed of light c v Magnitude of electric and magnetic fields are simply related by wavespeed: E cb. Making EM Waves 7 b. Nikola Tesla ( ) 8 (a) 89 (888?) Hertz demonstration that electromagnetic waves can be transmitted and then received. Proves existence of waves with frequencies of million cycles per second. Heinrich Hertz ( ) 89 (893?) Chicago World s fair, demonstrates wireless telegraphy (3 feet) 894 Lodge transmits 5 yards 3

4 c Guglielmo Marconi ( Wavespeed Phenomena 899 Marconi steals Tesla s design and broadcasts across the English Channel 9 Across the Atlantic (a) Index of refraction: Speed of light v in media is slower where n is index of refraction (about.5 for glass). Index can be calculated from the electrical permittivity () and magnetic permeability () properties of the media. Index usually depends upon wavelength of the light (e.g. in glass red might have n=.5 while for blue n=.53) c v n n (b) Reflection & Transmission (c) Absorption As a wave (such as light) in media, with index n, enters a denser media (index n ) where the speed changes, part of the wave will be reflected. Proportion given by formula: The rest is transmitted. For glass (n=.5) we calculate that 4% is reflected, 96% transmitted n R n n n Good conductors: reflect nearly % Poor conductors: wave penetrates into media to skin depth and is absorbed (energy turned into heat). Wave exponentially decays with distance. For AC Signals traveling through a wire, at higher frequencies the skin depth is very small, and so electricity will travel only on the outside of a conductor (hence increasing its resistance). At 6 cycles the skin depth is 8.5 mm for copper, so making a bigger diameter wire is a waste of metal. Instead, use Litz wire, made of many small wires. C. Polarization 3 Linear Polarization 4 ) Linear Polarization ) Birefringence 3) Circular Polarization [8 Fresnel develops wave theory of transverse polarized light, well before the electromagnetic nature was known] light has two perpendicular linear polarizations (electric field) can be horizontal or vertical 888 Hertz shows electromagnetic waves have transverse polarization (equivalent to light ) 4

5 Polarization by Reflection 5 Detecting Polarized Light 6 88 Malus s Law: Reflected light is often polarization Linear polarizer can be used to detect polarized light, only lets one polarization through! 8 Fresnel develops wave theory of transverse polarized light 85 Brewster s angle: at this angle of incidence the reflected light is entirely s polarized such that electric field is parallel to the interface surface tan b n 88 Malus s Law: Linear polarized light passing through a second polarizer tilted at angle to first will be attenuated: I I [cos ] Hence no light gets through crossed polarizers (=±9 ) Note: Plane of incidence is the plane defined by the three beams above. The normal also lies in this plane. The plane is perpendicular to the surface. Optical Activity 7 Birefringence 8 Optically active materials can rotate the polarization If such a substance is put between crossed polarizers (9º angle) you will often see interesting colors. 669 Erasmus Bartolinus (Denmark) discovers the birefringence (double refraction) of calcite crystals. When polarization was understood better, it was realized the two different polarizations took different paths (they are refracted differently, or the index of refraction is dependent upon polarization) Index of refraction: n=c/v, so the different polarizations travel at slightly different speeds. Quarter Wave Plates 9 Circular Polarized Light 3 A quarter wave plate retards horizontal polarization by 9º to vertical. It can be used to make circular polarized light from linear polarized light. Another type of polarized light can be left or right handed circular polarized 5

6 Detecting Circular Polarized Light A quarter wave plate will turn circular back into linear, which can be detected by a linear polarizer 3 References Things to Do 33 Find tesla museum stuff Who first predicted circular polarized light? Can we make a 3D image for students using polarized light? Need two projectors? Ideally we d use circular polarized light, but one test so far shows either the transparency projector or the screen does not preserve the circular polarization. 6