1 Electromagnetics I EELE 3331 Dr Assad AbuJasser
1 Electromagnetics I (EELE 3331) Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
2 Assad Abu-Jasser, Ph. D Electric Power Engineering The Islamic University of Gaza ajasser@iugaza. edu. ps site. iugaza. edu. ps/ajasser Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
3 Chapter 3 Vector Calculus Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
4 Introduction Some fundamental concepts in Electromagnetics or mathematics are presented in this chapter Applications of these concepts will be presented in next chapter Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
5 Differential Length Area & Volume Cartesian Coordinate Systems Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
6 Differential Length Area & Volume Cylindrical Coordinate Systems Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
7 Differential Length Area & Volume Spherical Coordinate Systems Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
8 Example 3. 1 Consider the object shown. Calculate a) the length BC b) the length CD d) the surface area ABO e) the surface area AOFD Dr. Assad Abu-Jasser - EE Department - IUGaza c) the surface area ABCD f) the volume ABDCFO Electromagnetics I
9 Line, Surface, and Volume Integral Cartesian Coordinate Systems Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
10 Example 3. 2 Given that F=x 2 ax-xzay-y 2 az, calculate the circulation of F around the (closed) path shown. Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
11 Del Operator (Gradient Operator) Cartesian, Cylindrical & Spherical Coordinates http: //copaseticflow. blogspot. com/2009/03/cool-math-tricks-deriving-divergence-or. html Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
12 Gradient of A Scalar Cartesian, Cylindrical & Spherical Coordinate Magnitude Gradient of a scalar fieldof. V ∇V is a equals vector maximum that of change and in V per distance represents both therate magnitude theunit direction of points in the direction of the maximum ∇V space rate of increase of V maximum rate of change in V ∇V at any point is perpendicular to the constant V surface The projection of ∇V in the direction of a is ∇V·a and called directional derivative of V along a If A=∇V, V is said to be the scalar potential of A Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
13 Example 3. 3 (a) V=e-zsin 2 xcoshy (b) U=ρ2 zcos 2φ (c) W=10 rsin 2ϴcosφ Find the gradient of the following scalar fields: (a) V=e-zsin 2 xcoshy (b) U=ρ2 zcos 2φ (c) W=10 rsin 2ϴcosφ Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
14 Example 3. 4 Given W=x 2 y 2+xyz, compute ∇W and the directional derivative d. W/dl in the direction 3 ax+4 ay+12 az, at (2, -1, 0) Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
15 Example 3. 5 x=y=2 z intersect the ellipsoid x 2+y 2+2 z 2=10 Find the angle at which line x=y=2 z intersect the ellipsoid x 2+y 2+2 z 2=10 Let line and the ellipsoid meet at angle ψ On line x=y=2 z, two unit increments along z correspond to one unit increment along x and y Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
16 Divergence of A Vector Divergence of vector A at a given point P is the outward flux per unit volume as the volume shrinks about P Produces a scalar field ∇. (A+B)=∇. A+∇. B ∇. (VA)=V∇. A+A. ∇V Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
17 Divergence Theorem Gauss-Ostrogradsky Theorem The total outward flux of a vector field A through the closed surface S is the same as the volume integral of the divergence of A Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
18 Example 3. 6 (a) P=x 2 yzax+xzaz (b) Q=ρsinφaρ+ρ2 zaφ+zcosφaz (c) T=(1/r 2)cosϴar+rsinϴcosφaϴ+cosϴaφ Determine the divergence of these vector fields: (a) P=x 2 yzax+xzaz (b) Q=ρsinφaρ+ρ2 zaφ+zcosφaz (c) T=(1/r 2)cosϴar+rsinϴcosφaϴ+cosϴaφ Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
19 Example 3. 7 If G(r)=10 e-2 z(ρaρ+az), determine the flux of G out of the entire surface of the cylinder ρ=1, 0≤z≤ 1. confirm the result by using the divergence theorem. Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
20 Curl of A Vector The curl of A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
21 Stokes’s Theorem The circulation of a vector field A around (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L provided that A and ∇×A are continuous Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
22 Example 3. 8 (a) P=x 2 yzax+xzaz (b) Q=ρsinφaρ+ρ2 zaφ+zcosφaz (c) T=(1/r 2)cosϴar+rsinϴcosφaϴ+cosϴaφ Determine the curl of these vector fields: (a) P=x 2 yzax+xzaz (b) Q=ρsinφaρ+ρ2 zaφ+zcosφaz (c) T=(1/r 2)cosϴar+rsinϴcosφaϴ+cosϴaφ Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
23 Example 3. 9 A=ρcosφ aρ+sin. Ф aφ If A=ρcosφaρ+sin. Фaφ, evaluate ∮A. dl around the path shown in the figure. Confirm this by using Stokes’s theorem. Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
24 Example 3. 10 For a vector field A, show explicitly that ∇. ∇×A =0; that is, the divergence of the curl of any vector field is zero Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
25 Laplacian of A Scalar Harmonic Scalar Field The Laplacian of a scalar vector V, written as ∇2 V, is the divergence of the gradient of V Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
26 Example 3. 11 (a) V=e-zsin 2 xcoshy (b) U=ρ2 zcos 2φ (c) W=10 rsin 2ϴcosφ Find the Laplacian of the following scalar fields: (a) V=e-zsin 2 xcoshy (b) U=ρ2 zcos 2φ (c) W=10 rsin 2ϴcosφ Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
27 Classification of Vector Fields AA vector (or divergeceless) vectorfieldisissolenoidal irrotational potential) ifif ∇. A=0 ∇х. A=0 A vector field is uniquely characterized by its divergence and curl Neither the divergence nor the curl of a vector is sufficient to completely describe the field. All fields can be classified in terms of their vanishing or nonvanishing divergence or curl Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
28 Example 3. 12 Show that the vector field A is conservative if A possesses one of these two properties: (a) The line integral of the tangential component of A along a path extending from a point P to a point Q is independent of the path (b) The line integral of the tangential component of A around any closed path is zero Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
29 End Of Chapter Three Dr. Assad Abu-Jasser - EE Department - IUGaza Electromagnetics I
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