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Electromagnetic Field Theory: A Problem Solving Approach Part 8. 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. | Problems 45 . z sin . b A rcos piIH----------ir r 2 . . cos 0 sin . c A r sin 0 cos pir ------ ------ r 19. Using Stokes theorem prove that b dl - V XdS L Js Hint Let A if where i is any constant unit vector. 20. Verify Stokes theorem for the rectangular bounding contour in the xy plane with a vector field A x a y b z c ix Check the result for a a flat rectangular surface in the xy plane and b for the rectangular cylinder. 21. Show that the order of differentiation for the mixed second derivative dx dy dy dx does not matter for the function x2lny y 22. Some of the unit vectors in cylindrical and spherical coordinates change direction in space and thus unlike Cartesian unit vectors are not constant vectors. This means that spatial derivatives of these unit vectors are generally nonzero. Find the divergence and curl of all the unit vectors. 46 Review of Vector Analysis 23. A general right-handed orthogonal curvilinear coordinate system is described by variables u v w where iu x iu iw Since the incremental coordinate quantities du dv and dw do not necessarily have units of length the differential length elements must be multiplied by coefficients that generally are a function of u v and w dLu hu du dLv hv dv dLw hw dw a What are the h coefficients for the Cartesian cylindrical and spherical coordinate systems b What is the gradient of any function f u v w c What is the area of each surface and the volume of a differential size volume element in the u v w space d What are the curl and divergence of the vector A Auitt Awiw e What is the scalar Laplacian V2 V V f Check your results of b - e for the three basic coordinate systems. 24. Prove the following vector identities a V g Vg gV b V A-B A-V B B V A Ax VxB Bx VxA c V fA V A A V d V AxB B VxA A VxB e V x A x B A V B - B V A B V A - A V B Problems 47 f Vx A V xA VxA g VxA xA A- V A- V A- A h Vx VxA V V- A -V2A 25. Two points have Cartesian coordinates 1 2 1 and 2 -3 1 . a What is the distance between these two points b
