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Magnetic Forces in Moving Reference Frames Two protons Electric force: F21,e +e 1 r v F21,m B1 +e v E1 2 F21,e 1 e2 = q2 E1 = rˆ 2 4pe 0 r Magnetic field: m0 q1v1 ´ rˆ B1 = 4p r 2 Magnetic force: F21,m = q2v2 ´ B1 F21,m m0 e 2 v 2 = q2vB1 = 4p r 2 Magnetic Forces in Moving Reference Frames 1 e2 æ v 2 ö 20 ns F = 4pe r 2 çç1 - c 2 ÷÷ 0 è ø 2 1 e 15 ns F = 4pe 0 r 2 Who will see protons hit floor and ceiling first? +e 1 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. Chapter 22 Patterns of Fields in Space • Electric flux • Gauss’s law • Ampere’s law • Maxwell equations Patterns of Fields in Space What is in the box? no charges? vertical charged plate? Gauss’s law: If we know the field distribution on closed surface we can tell what is inside. Electric Flux: Surface Area flux through small area: flux ~ E nˆA Definition of electric flux on a surface: E nˆA surface Adding up the Flux E nˆA surface E nˆdA E dA dA electric flux on a closed surface E dA Gauss’s Law E nˆA surface E nˆdA q inside e0 q inside e0 Features: 1. Proportionality constant 2. Size and shape independence 3. Independence on number of charges inside 4. Charges outside contribute zero 1. Gauss’s Law: Proportionality Constant E nˆA q surface inside e0 1 Q rˆ nˆA 2 surface 4e0 r 1 Q 4e0 r 2 A surface 1 Q Q 2 4r 2 4e0 r e0 What if charge is negative? Works at least for one charge and spherical surface 1 Q E 4e0 r 2 2. Gauss’s Law: The Size of the Surface E nˆA surface q inside e0 1 E~ 2 r A ~ r2 1 E~ 2 r universe would be much different if exponent was not exactly 2! 1 Q E 4e0 r 2 3. Gauss’s Law: The Shape of the Surface E nˆA surface q inside e0 E n̂A EA surface surface A2 R2 (r2 tan )2 r22 All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux A2 / A1 r22 / r12 E2 A2 / E1A1 1 4. Gauss’s Law: Outside Charges E nˆA surface E nˆA surface A ~ r 2 1 E~ 2 r q inside e0 EA surface A1 E1 – A2 E2 Outside charges contribute 0 to total flux 5. Gauss’s Law: Superposition Q1 ˆ E1 nA surface e0 Q2 E2 nˆA surface e0 E3 nˆA 0 surface E nˆA surface q inside e0 Gauss’s Law E nˆA q inside e0 surface E nˆdA q inside e0 Is it a law or a theorem? Can derive one from another 1 Q E 4e0 r 2 Gauss’s law is more universal: works at relativistic speeds Clicker Question What is the net electric flux on the box? A. 0 V*m B. 0.36 V*m C. 0.84 V*m D. 8.04 V*m E. 8.52 V*m Applications of Gauss’s Law E nˆA surface q inside e0 1. Knowing E can conclude what is inside 2. Knowing charges inside can conclude what is E The Electric Field of a Large Plate Symmetry: Field must be perpendicular to surface Eleft=Eright E nˆA q inside e0 surface 2EAbox Q / AAbox e0 Q / A E 2e 0 The Electric Field of a Uniform Spherical Shell of Charge Symmetry: 1. Field should be radial 2. The same at every location on spherical surface E nˆA q inside e0 surface A. Outer sphere: Q 2 E 4r e0 B. Inner sphere: 0 2 E 4r e0 1 Q E 4e0 r 2 E 0 The Electric Field of a Uniform Cube E nˆA surface q inside e0 Is Gauss’s law still valid? Can we find E using Gauss’s law? Clicker Question What is the electric flux through the area A? cos(30o) 0.866 sin(30o) 0.5 E = 100 V/m q 30o A = 2 m2 A) B) C) D) E) 0 100 V*m 173 V*m 50 V*m 87 V*m Gauss’s Law: Properties of Metal Can we have excess charge inside a metal that is in static equilibrium? Proof by contradiction: E nˆA q surface =0 q inside e0 0 inside e0 Gauss’s Law: Hole in a Metal E nˆA surface =0 q inside e0 q inside e0 0 What is electric field inside the hole? VACB = 0 VADB E dl 0 1. No charges on the surface of an empty hole 2. E is zero inside a hole Gauss’s Law: Screening E nˆA surface q inside e0 Gauss’s Law: Charges Inside a Hole E nˆA q surface =0 inside e0 q inside e0 +5nC q + qinside 0 q 5 nC surface surface 0 Gauss’s Law: Circuits Can we have excess charge inside in steady state? E nˆA surface E nˆA left _ surface E nˆA right_ surface q inside e0 q inside e0 0 Gauss’s Law: Junction Between two Wires n2<n1 u2<u1 i1=i2 n1Au1E1 = n2Au2E2 n1u1 E2 E1 E1 n2u2 E nˆA surface q inside q inside e0 e 0 (E1 A E2 A) 0 There is negative charge along the interface! Patterns of Magnetic Field in Space Is there current passing through these regions? There must be a relationship between the measurements of the magnetic field along a closed path and current flowing through the enclosed area. Ampere’s law Quantifying the Magnetic Field Pattern m0 2 I Bwire 4 r Curly character – introduce: B dl m0 2 I m0 2 I B dl 4 r dl 4 r 2r B dl m0 I Similar to Gauss’s law (Q/e0) Will it work for any circular path of radius r ? A Noncircular Path (home study) B dl m0 I Need to compare B1 dl1 and B2 dl2 m0 2 I B1 B dl Bdl|| 4 r1 dl1 dl2|| r1 r2 m0 2 I r1 B2 B1 4 r2 r2 r1 r2 B2 dl2 B2dl2|| B1 dl1 r2 r1 B2 dl2 B1 dl1 Currents Outside the Path (home study) B dl m0 I Need to compare B1 dl1 and B2 dl2 dl1 dl2|| r1 r2 r1 B2 B1 r2 B2 dl2 B1dl1 B dl 0 for currents outside the path Three Current-Carrying Wires (home study) B1 dl m0 I1 B2 dl m0 I 2 B3 dl 0 r r r — B1 + B2 + B3 dl m0 I1 I2 Ampere’s law B dl m0 Iinside_ path Ampère’s Law B dl m0 Iinside_ path All the currents in the universe contribute to B but only ones inside the path result in nonzero path integral Ampere’s law is almost equivalent to the Biot-Savart law: but Ampere’s law is relativistically correct Ampere, 1826: Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience Maxwell: We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it.