PHY2049: Chapter 291Biot-Savart LawÎDeduced from many experiments on B field produced by currents, including B field around a very long wire Magnitude Direction: RHR #2 Vector notation Applications Reproduces formula for B around long, current-carrying wire B by current loop (on axis) In more complicated cases, numerically integrate to find B20sin 4 rθdsiπµdB =30 4 rrsdiπµBdrrr×=1/r2depedenceLike Coulomb’s lawdB=0 ahead of ds and behind it. Maximum on plane perp. to ds.PHY2049: Chapter 292Biot-Savart LawAmpere’s LawProof of equivalence not in the book (Require vector calculus and relies on the absence of magnetic monopoles) equivalentLaw of MagnetismÎUnlike the law of electrostatics, comes in two parts Part 1 Effect of B field on moving charge Part 2 Current produces BBvqFrr×=PHY2049: Chapter 293ÎRadius R and current i: find B field at center of loop Direction: RHR #3 (see picture)ÎIf N turns close togetherÎB field on axis, including center z=0: checks z>>R:B Field on Axis of Circular Current Loop()232220 2zRRiµB+=02iBRµ=02NiBRµ=From B-S law by integrationFrom B-S law by integration320 2 zRiµB =()232220 2zRRiµB+=Like E field around electric dipole!PHY2049: Chapter 294Current Loop ExampleÎi = 500 A, r = 5 cm, N=20()()7020 4 10 5001.26T2 2 0.05iBNrπµ−×== =×PHY2049: Chapter 295Field at Center of Partial LoopÎDirection of B?ÎSuppose partial loop covers angle φ Calculate B field from proportion of full circleÎUse example where φ = π (half circles) Define direction into page as positive022iBRµφπ=001201222 2 2114iiBRRiBRRµµππππµ =− =−PHY2049: Chapter 296Partial Loops (cont.)ÎNote on problems when you have to evaluate a B field at a point from several partial loops Only loop parts contribute, proportional to angle (previous slide) Straight sections aimed at point contribute exactly 0 Be careful about signs, e.g. in (b) fields partially cancel, whereas in (a) and (c) they addPHY2049: Chapter 297Solenoid and ToroidÎAnother application of Ampere’s law Read the bookPHY2049: Chapter 298FAQ on Magnetism (2)ÎAccording to the law of magnetism, a current produces a magnetic field, which exerts a force on a moving charge. In the phenomenon of two bar magnets attracting each other, I see no current in magnet 1 and no moving charge in magnet 2, and vise versa. Doesn’t this example show that the theory is incomplete? A: magnet 1 comprises magnetic ions, which produce magnetic field due to the orbital motion of electrons and the spins of electrons. Magnet 2 also comprises magnetic ions, in which electrons (negative charges) are orbiting around the nuclei and electrons are also magnetic dipole moments. The force between two magnets can be derived from the law, although the calculation is lengthy and you first need to derive the formula for the force exerted on a magnetic dipole by non-uniform
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