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Electromagnetic Field Theory: A Problem Solving Approach Part 40

Electromagnetic Field Theory: A Problem Solving Approach Part 40. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Magnetic Field Boundary Value Problems 365 Because the curl of H is zero we can define a scalar magnetic potential H V 6 where we avoid the use of a negative sign as is used with the electric field since the potential x is only introduced as a mathematical convenience and has no physical significance. With B proportional to H or for uniform magnetization the divergence of H is also zero so that the scalar magnetic potential obeys Laplace s equation in each region V2 0 7 We can then use the same techniques developed for the electric field in Section 4-4 by trying a scalar potential in each region as Ar cos 0 l Dr C r2 cos 0 r R r R 8 366 The Magnetic Field The associated magnetic field is then TT dX. 1 . 1 . A ir cos 0 ie sin d liI r R t D 2C r3 cos dir D 4-C7r3 sin di r R For the three cases the magnetic field far from the sphere must approach the uniform applied field H r oo Hoiz H0 ir cos 0 ie sin d Ho 10 The other constants A and C are found from the boundary conditions at r R. The field within the sphere is uniform in the same direction as the applied field. The solution outside the sphere is the imposed field plus a contribution as if there were a magnetic dipole at the center of the sphere with moment mz 4trC. i If the sphere has a different permeability from the surrounding region both the tangential components of H and the normal components of B are continuous across the spherical surface He r R He r R_ A D C R3 Br r R Br r _ n Hr r R K_ l which yields solutions a 1h2 c _f r3Ho 12 The magnetic field distribution is then ir cos d -i sm d ------------ A2 2 41 r R SfiiHo 13 cos dir The magnetic field lines are plotted in Figure 5-25a when - oo. In this limit H within the sphere is zero so that the field lines incident on the sphere are purely radial. The field lines plotted are just lines of constant stream function X found in the same way as for the analogous electric field problem in Section 4-4-36. Magnetic Field Boundary Value Problems 367 . ii If the .