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Basic Theoretical Physics: A Concise Overview P16

Basic Theoretical Physics: A Concise Overview P16. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 150 18 Magnetic Field of Steady Electric Currents b In the dipole case one would obtain outside the fictitious dipole film the equivalent result B f H with HM - - d cP A n r r - r 18 10 h r grad4 jy d A r - . i8.io Proof of the equivalence of the two results proceeds analogously to Stokes s theorem but since it is somewhat difficult in detail we only give an outline in a footnote3. An example is given in Fig. 18.1. In this context we additionally keep two useful identities in mind r A m x - t 4nr3 Fig. 18.1. The diagram illustrates a typical section of the magnetic field lines produced by a current loop of two infinitely long straight wires. The wires intersect the diagram at the points 1 0 . The plane of the loop of area A to and carrying a current I i.e. of opposite signs in the two long wires is perpendicular to the plane of the diagram. Exactly the same induction B J0H is also produced by a layer of magnetic dipoles inserted into the current loop with the quantitative relation dm p 0Ind2 A given in the text 3 In the following we use the antisymmetric unit-tensor eijk and Einstein s summing convention i.e. all indices which appear twofold are summed over. With these conventions Stokes s theorem becomes eF Ejdxj e Fejim9iEmnjd2A. Now the following chain of equations is true eijk dx Xk-3k fap dx j e.jkid k- RF ejimdiemki d kl rn d2A dF ijk r r 3 dF j jki k r r F jim l mki k r r j - Feikmejimd k r-njd2A . With the basic identity eikmejim SijSki - SiiSkj and the simple relations dp dn and dkk- 0 for r r i rr 4 rr kk rr our statement of equivalence is obtained. 18.5 Gyromagnetic Ratio and Spin Magnetism 151 for the vector potential of a magnetic dipole and m x curl mdiv -3 grad m r cf. problem 5 of the exercises summer 2002 2 . Because of the above-mentioned equivalence it would be natural to suggest that all magnetic dipole moments are generated in this way by Amperian current loops. However this suggestion would be wrong There are magnetic moments which cannot be .