Electromagnetic Field Theory: A Problem Solving Approach Part 5. 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. | Vector Algebra 15 cylindrical coordinates r z or the spherical coordinates EXAMPLE 1-3 CROSS PRODUCT Find the unit vector in perpendicular in the right-hand sense to the vectors shown in Figure 1-10. A -L ij L B ix - i ix What is the angle between A and B SOLUTION The cross product A X B is perpendicular to both A and B AxB det 1 2 ix i 1 -1 1 The unit vector in is in this direction but it must have a magnitude of unity AxB - 1 r x- ï tn - I . f-r t AXB V2 A ix i i2 Figure 1-10 The cross product between the two vectors in Example 1-3. 16 Review of Vector Analysis The angle between A and B is found using 12 as sin 9 AxB 2 2 AB V3V3 3 2 0 or The ambiguity in solutions can be resolved by using the dot product of 11 cos0 - - - 0 AB 1-3 THE GRADIENT AND THE DEL OPERATOR 1-3-1 The Gradient Often we are concerned with the properties of a scalar field f x y z around a particular point. The chain rule of differentiation then gives us the incremental change df in f for a small change in position from x y z to x dx y dy z dz df df df df dx dy dz 1 dx dy dz If the general differential distance vector dl is defined as dl dx ix dy i dz i 2 1 can be written as the dot product dx dy dz grad f dl 3 where the spatial derivative terms in brackets are defined as the gradient of grad V ix-F i ix 4 dx dy dz The symbol V with the gradient term is introduced as a general vector operator termed the del operator . d . d . d V ix Fl F ix 5 dx y dy dz By itself the del operator is meaningless but when it premultiplies a scalar function the gradient operation is defined. We will soon see that the dot and cross products between the del operator and a vector also define useful operations. The Gradient and the Del Operator 17 With these definitions the change in f of 3 can be written as d V dl V dZ cos fl 6 where 0 is the angle between V and the position vector dl. The direction that maximizes the change in the function f is when dl is colinear with V fl 0 . The gradient thus has
