Waves can be represented by simple harmonic motion Standing wave y = Asin(kx − ωt) + Asin(kx + ωt) y (t ) A sin( kx t ) The amplitude of a wave is a measure of the maximum disturbance in the medium during one wave cycle. (the maximum distance from the highest point of the crest to the equilibrium). The wavelength (denoted as λ) is the distance between two sequential crests (or troughs). This generally has the unit of meters. A wavenumber k 2 2 Period T Phase velocity: v k T f Electromagnetic waves f c Light as a Wave (1) c = 300,000 km/s = 3*108 m/s • Light waves are characterized by a wavelength and a frequency f. • f and are related through f = c/ The Electromagnetic Spectrum Wavelength Frequency Need satellites to observe High flying air planes or satellites Dual, wave-particle nature of light E hf h c h 6.6 1034 Joule sec 1 eV = 1.6x10-19 J c = 3x108 m/s 1 Angstrom = 10-10 m Speed of light in matter: cm = c/n, where n is refractive index Note: n is a function of Light as a Wave (2) • Wavelengths of light are measured in units of nanometers (nm) or Ångström (Å): 1 nm = 10-9 m 1 Å = 10-10 m = 0.1 nm Visible light has wavelengths between 4000 Å and 7000 Å (= 400 – 700 nm). Light as Particles • Light can also appear as particles, called photons (explains, e.g., photoelectric effect). • A photon has a specific energy E, proportional to the frequency f: E = h*f h = 6.626x10-34 J*s is the Planck constant. The energy of a photon does not depend on the intensity of the light!!! Maxwell’s Equations Q E dS 1 E B dS 0 B 0 d E dr dt B dS E B t 0 d B dr 0i 0 0 dt E dS 0 B 0 j 0 0 E t Information Age The cost of the transmission, storage and processing of data has been decreasing extremely fast Information is available anytime, any place, and for everyone Information and knowledge became a capital asset All of this became possible because of several revolutionary ideas Telecommunications How it all started … Samuel Morse's telegraph key, 1844. Today's information age began with the telegraph. It was the first instrument to transform information into electrical signal and transmit it reliably over long distances. Alexander Graham Bell’s commercial telephone from 1877. Speaking into the handset's transmitter or microphone makes its diaphragm vibrate. This varies the electric current, causing the receiver's diaphragm to vibrate. This duplicates the original sound. • Telephone connection requires a dedicated wire line; • Only one communication is possible at a time Radio: communication through radio waves 1895 www.nrao.edu Frequency measured in Hertz 1 Hz = 1 cycle/second 1 kHz = 1000 cycles/second Guglielmo Marconi How many channels are possible? How many signals can be transmitted at the same time?? Alexander Popov Radio stations have to broadcast at different carrier frequencies to avoid cross-talk Human ear: 10 Hz-20 kHz Range of frequencies (Bandwidth) needs to be at least 20 kHz for each station Frequencies of different stations should be at least 20 kHz apart Need more channels? Need higher speed? Use higher frequencies for transmission! Higher carrier frequencies Wider bandwidth Higher data rate, more channels Using light? Optical frequencies ~ 1014 Hz ! How can we send light over long distances Air? Only within line of sight; High absorption and scattering, especially when it rains Are there any “light wires” (optical waveguides)? Copper wire? High absorption, narrow bandwidth 300 MHz Glass? Window glass absorbs 90% of light after 1 m. Only 1% transmission after 2 meters. Sand?! Ultra-low absorption in silica glasses Transmisson 95.5% of power after 1 km 1% of power after 100 km: need amplifiers and repeaters Total bandwidth ~ 100,000 GHz!! Predicted 1965, in first low-loss fiber in 1970 Silica (Silicon dioxide) is sand – the most abundant mineral on Earth How to trap light with transparent material?? Total internal reflection! n1 > n2 Light coming from more refractive to less refractive medium experiences total reflection – get trapped there! No charges, no real currents E dS 0 B dS 0 d E dr dt B dS d B dr 0 0 dt E dS E 0ix E y ( x, t )iy 0iz B 0ix 0iy Bz ( x, t )iz Wave equation 2Ey x 2 0 0 2Ey t 2 2 Bz 2 Bz 0 0 2 2 x t 2Ey x 2 0 0 2Ey t 2 E y A sin( kx t ) k 2 2 T k is a wave number, is a wave length, T is the period Velocity of propagation v k 1 0 0 c Coulomb’s Law 1 q1q2 FE 40 x 2 1 40 9 109 Newton meter 2 / coulomb 2 9 109 N m 2 / C 2 0 is the permittivity of free space Charge Charge q1 q2 Conservation of electric charge Charge is conserved: in any isolated system, the total charge cannot change. If it does change, then the system is not isolated: charge either went somewhere or came in from somewhere r̂12 r̂21 Charge Charge q1 q2 rˆ12 rˆ21 F2 Let’s denote the force that q1 exerts on q2 as and force exerted by q2 on q1 as F1 ; r is the distance between charges. 1 q1q2 1 q1q2 F1 rˆ12 rˆ21 F2 2 2 40 r 40 r (Newton’s third law works!) Like charges repel; opposites attract 8 Exercise: If two electrons are placed 10 meters apart, what is the magnitude of the Coulomb force between them? Compare this to the gravitational force between them. r Solution: The magnitude of electric force 19 2 q2 ( 1 . 6 10 ) 9 12 FE 9 10 N 2 . 3 10 N 2 8 2 40 x (10 ) 1 The magnitude of gravitational force 31 2 m1m2 ( 9 10 ) 11 55 FG G 2 6.67 10 N 5 . 4 10 N 8 2 x (10 ) FE q2 2.3 43 10 FG 40Gm1m2 5.4 (no matter what the separation is) Gauss’s Law Q E dS 0 A conducting sphere, conducting shell, insulating sphere, shell ….. Two parallel conducting plates - + + + l - + a + - + d E dS EdS Ea cap a Ea 0 E 0 (the total field at any point between the plates) Capacitors Consider two large metal plates which are parallel to each other and separated by a distance small compared with their width. y Area A L The field between plates is V E 0 [V (top) V (bottom)] E y dy dy L 0 0 0 0 L L A QL [V (top) V (bottom)] L L 0 0 A 0 A QL V A 0 The capacitance is: A 0 Q Q C QL V L A 0 A 0 Q Q C V QL L A 0 Capacitors in series: 1 1 1 1 ... Ctot C1 C2 C3 Capacitors in parallel: Ctot C1 C2 C3 ... 1 1 2 2 W CV Q 2 2C [C ] farad Current, Ohm’s Law, Etc. dQ i dt V Ohm ' s Law : R ; R Const (independent of V ) i l R A j E E j Current Density i j dS S Consider current flowing in a homogeneous wire with cross sectional area A. i j dS jdS j dS jA A A i j A A For steady state situation j d S 0 E dr 0 Circuits will be included! Joule’s Law 2 V P Vi i R R 2 The force F on a charge q moving with a velocity F q( E v B) If E 0 v the magnitude of the force F qvB sin [ B] Newtons /(Coulomb meter / sec) 1T (tesla) 1w / m 1Newton / C m / s 10 gauss 2 4 4 BEarth 1Gauss 10 T v || B F 0 v B F qvB F ma ir Fr mar F 0 2 v qvB mr 2 m r mv r qB The angular velocity v v qB r mv m qB Using Crossed E and B Fields Velocity selector qvB qE 0 E vB E v B independent of the mass of the particle! Ampere’s Law B d r i 0 The field produced by an infinite wire 0 i B 2 a Biot-Savart Law Infinitesimally small element of a current carrying wire produces an infinitesimally small magnetic field dS i ( ds r ) dB 3 r i r 0 i (ds r ) dB 4 r3 0 is called permeability of free space 0 4 10 7 webers /( amp meter) 4 10 7 N /( amp) 2 (Also called Ampere’s principle) Force exerted on a current carrying wire dF ids B For a straight, finite wire of length F il B l and uniform magnetic field Faraday’s Law of Induction The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop d B EMF dt d B d E dr dt dt B dS There can be EMF produced in a number of ways: • • • • A time varying magnetic field An area whose size is varying A time varying angle between B and dS Any combination of the above From Faraday’s law: a time varying flux through a circuit will induce an EMF in the circuit. If the circuit consists only of a loop of wire with one resistor, with resistance R, a current R EMF i R Which way? Lenz’s Law: if a current is induced by some change, the direction of the current is such that it opposes the change. d B E dr dt A Simple Generator Faraday’s Law is used to obtain differential equations for some simple circuits. E dr B dS Self-inductance L B B dS Li Displacement current d iD 0 E dS dt d B dr 0i 0 0 dt E dS Thank you for a great semester!