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Final EXAM: Monday, May 5th, LWSN room B155 3:30-5:30 pm Magnetic Forces in Moving Reference Frames 1 e2 æ v 2 ö 20 ns F = 4pe r 2 çç1 - c 2 ÷÷ 0 è ø +e 1 2 1 e 15 ns F = 4pe 0 r 2 Who will see protons hit floor and ceiling first? r v F21,m 2 B1 +e v E1 F21,e Relativistic Field Transformations Our detailed derivations are not correct for relativistic speeds, but the ratio Fm/Fe is the same for any speed: Fm v 2 = 2 Fe c According to the theory of relativity: (E - vBz ) E = Ex E = Bx' = Bx v æ ö B + E ç y z÷ 2 c ø By' = è 1 - v 2 / c2 ' x ' y y 1- v / c 2 2 E = ' z (E z + vB y ) 1 - v 2 / c2 v æ ö B E ç z y÷ 2 c ø Bz' = è 1 - v2 / c2 Magnetic Field of a Moving Particle q E= B=0 2 4pe 0 r v æ ö v B E ç z ÷ - 2 Ey y 2 c ø= c Moving: Bz' = è 1 - v2 / c2 1 - v 2 / c2 Still: 1 v 1 q Slow case: v<<c B = - 2 c 4pe 0 r 2 1 = c2 m0 qv Field transformation is consistent ' m0e 0 Bz = with Biot-Savart law 4p r 2 ' z Electric and magnetic fields are interrelated Magnetic fields are relativistic consequence of electric fields Electric Field of a Rapidly Moving Particle E = Ex ' x E = ' y E = ' y (E y - vBz ) 1- v / c 2 2 Ey 1- v / c 2 2 E = ' z E = ' z (E z + vB y ) 1 - v 2 / c2 Ez 1 - v 2 / c2 The Principle of Relativity There may be different mechanisms for different observers in different reference frames, but all observers can correctly predict what will happen in their own frames, using the same relativistically correct physical laws. Wave Description – wavelength: distance between crests (meters) T – period: the time between crests passing fixed location (seconds) v – speed: the distance one crest moves in a second (m/s) f – frequency: the number of crests passing fixed location in one second (1/s or Hz) – angular frequency: 2f: (rad/s) v T 1 f T v f Wave: Variation in Time and Space 2 E E0 cos t T 2 2 E E0 cos t T 2 E E0 cos x x ‘-’ sign: the point on wave moves to the right Wave: Phase Shift 2 2 E E0 cos t x T But E @ t=0 and x =0, may not equal E0 phase shift, =0…2 2 2 E E0 cos t x T 2 E E0 cos t E0 cost T Two waves are ‘out of phase’ (Shown for x=0) Wave: Amplitude and Intensity E E0 cost E0 is a parameter called amplitude (positive). Time dependence is in cosine function Often we detect ‘intensity’, or energy flux ~ E2. Intensity I (W/m2): I E02 Works also for other waves, such as sound or water waves. Interference Superposition principle: The net electric field at any location is vector sum of the electric fields contributed by all sources. Laser: source of radiation which has the same frequency (monochromatic) and phase (coherent) across the beam. Two slits are sources of two waves with the same phase and frequency. What can we expect to see on the screen? Can particle model explain the pattern? Interference: Constructive E1 Two emitters: E2 Fields in crossing point E1 E0 cost E2 E0 cost Superposition: E E1 E2 2 E0 cost Amplitude increases twice: constructive interference Interference: Energy E1 Two emitters: E2 E E1 E2 2 E0 cost What about the intensity (energy flux)? Energy flux increases 4 times while two emitters produce only twice more energy There must be an area in space where intensity is smaller than that produced by one emitter Interference: Destructive E1 E0 cost E1 E2 E0 cost E2 E E1 E2 E0 cost cost 0 cost Two waves are 1800 out of phase: destructive interference Two-Slit Experiment with Waves •We measure the Intensity of the wave motion at the detector (related to the square of the wave height) Two-Slit Experiment with Bullets •Bullets arrive in lumps •We measure the probability of arrival of a lump (bullet) •P1 = probability bullet went through slit 1 in arriving at x •P12 = P1 + P2 Differential Form of Gauss' Law (Sec. 22.8) GAUSS' LAW Think about a region of space, enclosed by a box. Divide Gauss' law by the volume of the box: E || x Take the limit of a small box Work on the left hand side of the equation: For a general case where E can point in any direction: GAUSS' LAW Differential Form "Parallel Derivative" where Differential Form of Ampere's Law (Sec. 22.9) Ampere's Law 3 Write I in terms of current density J: 2 4 1 Divide Ampere's Law by a very small ΔA: Current I out of the board In our geometry, n = z Differential Form of Ampere's Law (Sec. 22.9) Ampere's Law 3 2 4 We divided Ampere's Law by a very small ΔA, and got this: 1 Current I out of the board Now work on the left hand side: Definition of derivative! "Crossed derivative" Differential Form of Ampere's Law (Sec. 22.9) Ampere's Law 3 2 4 We divided Ampere's Law by a very small ΔA, and got this: 1 Current I out of the board For a loop in any direction, this can be re-expressed as: AMPERE'S LAW Differential Form Curl: Here's the Math copy 1st two colums +( - ) +( - ) +( - ) set up the answer Curl: Here's the Math +( - ) +( - ) +( - ) Maxwell's Equations – The Full Story Divergence GAUSS' LAW Flux GAUSS' LAW (Magnetism) Curl FARADAY'S LAW AMPERE'S LAW Circulation Maxwell's Equations – No Charges In the ABSENCE of "sources" = charges, currents: GAUSS' LAW GAUSS' LAW (Magnetism) FARADAY'S LAW AMPERE'S LAW This says once a wave starts, it keeps going! Maxwell's Equations – No Charges What happens if we feed one equation into the other? Use This Maxwell's Equations – No Charges What happens if we feed one equation into the other? ("Vector identity" -- see Wolfram alpha) Maxwell's Equations – No Charges How do you solve a Differential Equation? Know the answer! (Ask Wolfram Alpha) This is a WAVE EQUATION, with speed c Using similar ideas, you can show that E obeys the same equation: