# Chap5 - Static and Stationary Magnetic Fields Andr´ e-Marie Amp` ere(1775 1836 Contents 1 Introduction and Definitions 2 1.1 Magnetic Induction 2

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Unformatted text preview: Static and Stationary Magnetic Fields Andr´ e-Marie Amp` ere (1775 - 1836) December 23, 2000 Contents 1 Introduction and Definitions 2 1.1 Magnetic Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Current Density and Conservation . . . . . . . . . . . . . . . . . . . . 3 2 Amp` ere’s Law 5 2.1 Induction of an Arbitrary Current Density . . . . . . . . . . . . . . . 8 2.2 An Alternate Form of Amp` ere’s law . . . . . . . . . . . . . . . . . . . 9 2.3 Example: Force Between Parallel Wires . . . . . . . . . . . . . . . . . 10 3 Differential Equations of Magnetostatics 11 4 Vector and Scalar Potentials 15 4.1 Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Vector Potential and Gauge Invariance . . . . . . . . . . . . . . . . . 16 4.3 Example: A Circular Current Loop . . . . . . . . . . . . . . . . . . . 19 5 The Field of a Localized Current Distribution 22 1 6 Forces on a Localized Current Distribution 28 7 Macroscopic Magnetostatics 31 7.1 Magnetization Current Density . . . . . . . . . . . . . . . . . . . . . 33 7.2 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8 Examples of Boundary-Value Problems in Magnetostatics 41 8.1 Uniformly Magnetized Sphere . . . . . . . . . . . . . . . . . . . . . . 41 8.1.1 Scalar Potential for the Induction . . . . . . . . . . . . . . . . 42 8.1.2 Scalar Potential for the Field . . . . . . . . . . . . . . . . . . 44 8.1.3 Direct Calculation of B . . . . . . . . . . . . . . . . . . . . . . 46 8.2 Shielding by a Paramagnetic Cylinder . . . . . . . . . . . . . . . . . . 48 2 1 Introduction and Definitions As far as anyone knows, there is no such thing as a free magnetic charge or magnetic monopole , although there are people who look for them (and occasionally claim to have found one); certainly they may exist. Because no known phenomena require their existence, we shall develop magnetostatics and eventually electrodynamics assuming that they do not exist. In this case there is a fundamental difference between elec- trostatics and magnetostatics, explaining in part why the two subjects developed independently and were regarded as distinct rather than dif- ferent limits or aspects of one type of phenomenon (electromagnetic phenomena). 1.1 Magnetic Induction In the absence of monopole moments, the most fundamental source of magnetic effects is the magnetic dipole . In the presence of other magnetic materials, a point dipole will experience some force. One defines the magnetic flux density or magnetic induction B in terms of the torque N exerted on the dipole. Given that the dipole moment is μ , the defining relation is N ≡ μ × B . (1) 3 Thus, as for electrostatics, the basic field of magnetostatics is defined by the effect produced on an elementary source....
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## This note was uploaded on 01/08/2012 for the course PHYSICS 707 taught by Professor Electrodynamics during the Fall '11 term at LSU.

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Chap5 - Static and Stationary Magnetic Fields Andr´ e-Marie Amp` ere(1775 1836 Contents 1 Introduction and Definitions 2 1.1 Magnetic Induction 2

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