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Chapter 22 Patterns of Fields in Space • Electric flux • Gauss’s law • Ampere’s law • Maxwell equations Gauss’s Law E nˆA surface q inside 0 E nˆdA q inside 0 Can derive one from another Gauss’s law is more universal: works at relativistic speeds 1 Q E 40 r 2 Gauss’s law: If we know the field distribution on closed surface we can tell what is inside. 1. Knowing E can conclude what is inside 2. Knowing charges inside can conclude what is E Clicker Q2 A solid conducting sphere of radius rA has charge Q1 uniformly distributed over its surface. Q1 rA rB rC Concentric with the solid sphere is a conducting shell of with charge Q2 on its outer surface. What is the charge Q on the inner surface of the outer shell? A. B. C. D. E. 0 +Q2 -Q2 +Q1 -Q1 Gauss’s Law: Charges Inside a Hole E nˆA q surface 0 inside 0 q inside 0 +5nC q qinside 0 q 5 nC surface surface 0 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 0 2 I Bwire 4 r Curly character – introduce: B dl 0 2 I 0 2 I B dl 4 r dl 4 r 2r B dl 0 I Similar to Gauss’s law (Q/0) Will it work for any circular path of radius r ? A Noncircular Path (home study) B dl 0 I Need to compare B1 dl1 and B2 dl2 0 2 I B1 B dl Bdl|| 4 r1 dl1 dl2|| r1 r2 0 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 0 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 B1 dl 0 I1 B2 dl 0 I 2 B3 dl 0 Ampere’s law B dl 0 Iinside_ path Ampère’s Law B dl 0 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. Inside the Path Ampere’s law B dl 0 Iinside_ path 1. Choose the closed path 2. Imagine surface (‘soap film’) over the path 3. Walk counterclockwise around the path adding up B dl 4. Count upward currents as positive, inward going as negative (right hand rule: 4 fingers along the path then thumb is positive current) I inside_ path I1 I 2 I inside_ path I up I up I down I up Ampere’s Law: A Long Thick Wire B dl 0 Iinside_ path Can B have an out of plane component? Is it always parallel to the path? B dl B2r B2r 0 I for thick wire: 0 2 I B 4 r (the same as for thin wire) Would be hard to derive using Biot-Savart law Clicker What is the magnetic field inside a wire (wire radius is R) at distance r1 from its center? A. B=20r1I/(R2) B. B=r1I/(20R2) C. B=0r1I/2R2 D. B=0R2I/2r1R2 E. 0 R r1 Ampere’s Law: find B inside a solenoid B dl 0 Iinside_ path Number of wires: (N/L)d What is B dl on sides? B outside is very small B dl Bd Bd 0 I N / L d B 0 IN L Uniform: same B no matter where is the path (solenoid) Gauss’s Law for Magnetism Dipoles: Electric field: ‘+’ and ‘–’ charges can be separated Magnetic field: no monopoles Suppose magnetic dipole consists of two magnetic monopoles, each producing a magnetic field similar to the electric field. One cannot separate them total magnetic ‘charge’ is zero. E nˆA surface q inside 0 B nˆA 0 surface or B nˆA 0 Gauss’s law for magnetism Maxwell’s Equations Three equations: Gauss’s law for electricity E nˆdA q inside 0 Gauss’s law for magnetism Ampere’s law for magnetism B dl 0 I inside_ path Is anything missing? ‘Ampere’s law for electricity’ E dl 0 First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence (incomplete)