Reference W.H. Hayt and J.A. Buck , Engineering Electromagnetics, McGraw-Hill, 8th Ed., 2011. J. Edminister, Schaum's Outline of Electromagnetics, McGraw-Hill, 2nd Ed., 1994. S.A. Nasar, 2000 Solved Problems in Electromagnetics, McGraw-Hill, 1992. Vector fundamental (1)

Vector algebra Coordinate systems Surface and volume integration Vector Fundamental Vector algebra Coordinate systems Surface and volume integration Vector fundamental (1)

Objectives To be able to convert the coordinate between Cartesian, cylindrical and spherical coordinate systems. To be able to formulate the line, surface and volume integration. For the given problem, to be able to select the proper coordinate system. Vector fundamental (1)

Common Shapes Box Can = Cylinder Sphere http://www.pepsico.com/Media/Multimedia-Downloads Box http://www.bevreview.com/wp-content/image_pepsithrowback_canada1.jpg Can = Cylinder http://www.ideasonemktg.com/assets/images/p7208.jpg Sphere Vector fundamental (1)

Coordinate Systems Cartesian Cylindrical Spherical Box Can Sphere Axes: x, y, z Axes: r, , z Axes: r ,  Polar coordinate: (x,y) =  r = radius  latitude   longitude x y   r r Counterclockwise: positive angle (x,y) = r+ Clockwise: negative angle (x,y) = r Vector fundamental (1)

Cartesian Coordinate: (x, y, z) Subscript shows the direction of vector (P) P: (Px, Py, Pz) Py Pz Px Position vector: Definition Unit vector: Magnitude of vector = 1 Vector from origin (0,0,0) to P. Subscript shows the point (P) Magnitude of Vector fundamental (1)

Unit Vector Used to indicate the direction In x,y,z axes Calculus: z In x,y,z axes P: (Px, Py, Pz) Py Pz Px In this subject, we use this format, because we have 3 coordinate systems and we don’t want to remember lots of acronym. Calculus: Vector fundamental (1)

No vector division/multiplication!! Cartesian Coordinate Constant Vector addition/subtraction:  Vector pointing from P to Q (direct calculation) Dot and cross product:  No vector division/multiplication!! Multiplication of scalar and vector is OK. Vector fundamental (1)

Integration The laws in physics is often the interaction between points (volume = 0). In the real world, everything has volume so the interaction is approximated as between two very small volumes(dv)/surfaces(dS)/lines(dL). The integration is used to sum the interaction to the large volume/surface/line. Vector fundamental (1)

Integration: Some Rules of Thumb Use coordinate that has the least free variables. Idea: use the coordinate system that has similar shapes to its edges/surfaces/volume. Vector fundamental (1)

Integration Along Line : Cartesian Basic shape = box Line integration: 1 axis changed, 2 axes fixed Assume f(x,y,z) as the interaction from (x,y,z). Find total interaction along given line. z 2 y changed, x, z fixed. z changed, x, y fixed. 3 y x changed, y, z fixed. 1 Vector fundamental (1) x

Surface Integration: Cartesian 2 axes changed, 1 axis fixed Find total interaction on the given surface z x, y changed, z fixed. 2 x, z changed, y fixed. 3 y y, z changed, x fixed. 1 x Vector fundamental (1)

Volume Integration: Cartesian Find total interaction from the given box. z 2 x, y, z changed. 3 y 1 x Vector fundamental (1)

Directional Surface Integration: Cartesian Direction: pointing outwards Parallel to the vector of the fixed axis. z y x Vector fundamental (1)

Integration: Summary () dL dS dv Cartesian Cylindrical Spherical x y z x y z x y z x y z x y z x y z x y z Vector fundamental (1)

Exercise (1): Surface Integration Find the total charge of the given surface (z = 0). y Solution: 2 methods 2 (1) Separating to 4 small rectangles. (2) Remove the rectangular hole from rectangle. = Subtract the hole from the rectangle. 1 1 2 3 4 x -2 -1 1 2 -1 -2  Vector fundamental (1)

Exercise (1): Solution Rectangle with fixed z coordinate.  dS = dxdy 2 3 4 2 1 x No dz. -2 -1 1 2 -1 Rectangle with fixed z coordinate.  dS = dxdy -2 Upper limit = maximum y co-ordinate Upper limit = maximum x co-ordinate 1 Lower limit = minimum y co-ordinate Lower limit = minimum x co-ordinate Vector fundamental (1)

Exercise (1): Solution (2) y 2 1 2 3 4 1 x -2 -1 1 2 -1 -2 Vector fundamental (1)

Exercise (1): Solution (3) y 2  1 x -2 -1 1 2 -1 -2 Vector fundamental (1)

Limitation of Cartesian Coordinate Surface integration for the circle, part of the circle, … Volume integration for cylinder, sphere, … y 1 y x -1 1 -1 Vector fundamental (1)

Cylindrical Coordinate (,,z) x y z P: (Px, Py, Pz) P is described as the point on the surface of a cylinder whose axis is z axis. x y z P: (Px, Py, Pz) Rotate for better viewing. Pz z = the height of a cylinder  = the radius of a cylinder P P Pxy: (Px, Py, 0)  = angle to x axis Vector fundamental (1)

Coordinate Conversion: Cylindrical x y z P: (Px, Py, Pz) Cartesian to cylindrical Pz P P Pxy: (Px, Py, 0) Projection to xy plane x y Pxy P P Vector fundamental (1)

Coordinate Conversion: Cylindrical (2) x y z P: (Px, Py, Pz) Cylindrical to Cartesian Pz P P Pxy: (Px, Py, 0) x y Pxy P P Conversion of xy plane to polar coordinate and then add z axis. Vector fundamental (1)

Directional Vector: Cylindrical x y z P: (Px, Py, Pz) Direction to increase the value in each axis. x y   Pz Function of  constant P P Pxy: (Px, Py, 0) x y Pxy P P Vector fundamental (1)

Position Vector: Cylindrical x y z P: (Px, Py, Pz) Q: (Qx, Qy, Qz) Pz Qz 2 components for 3D position. P P Q x y Pxy P P Possible because the direction of  changes with (x,y). Qxy Vector fundamental (1)

Cylindrical Coordinate Constant & non-constant Vector addition/subtraction: Dot and cross product:  Convert to the same coordinate system first. Not necessary Vector fundamental (1)

Integration: Cylindrical Small distance in ,  or z axes &z are the distance  d, dz  is an angle  use d Integration by path Vector fundamental (1)

Line Integration: Cylindrical Assume f(,,z) as the interaction from point (,,z). Find total interaction along given line. z 2  changed, , z fixed.  is multiplied to convert  from angle to length. z changed, ,  fixed. 0.25 y 4 r changed, , z fixed. x Vector fundamental (1)

Surface Integration: Cylindrical 2 axes changed, 1 axis fixed. Find the total interaction of the given surface. z 2 ,  changed, z fixed. , z changed,  fixed. y 4 x Vector fundamental (1)

Surface Integration: Cylindrical (2) r, z changed,  fixed. z 2 y 0.25 4 x Vector fundamental (1)

Directional Surface Integration: Cylindrical z y x Vector fundamental (1)

Volume Integration: Cylindrical z y x Vector fundamental (1)

Integration: Summary () dL dS dv Cartesian Cylindrical Spherical x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z x y z Vector fundamental (1)

Exercise (2): Line Integration Find the total charge of the given line (z = 0). y y 2 2 2 x x 2 -2 Vector fundamental (1)

Exercise (2): Solution y No d &dz. 2 Upper limit = maximum  co-ordinate Arc of a circle has fixed radius () & z coordinates.  dL = d x Lower limit = minimum  co-ordinate 2 Upper limit = maximum  co-ordinate y 2 1. Clockwise direction = negative value Lower limit = minimum  co-ordinate x 2 2. This point make 0.5 radian angle to x axis in clockwise direction.  = 0.5. -2 Vector fundamental (1)

Exercise (3): Surface Integration Find the total charge of the given surface. y y 1 1 x x 1 1 Vector fundamental (1)

Exercise (3): Solution y No dz. 1 Area inside a circle has fixed z coordinates.  dS = dd Upper limit = maximum radius x Lower limit = minimum radius Upper limit = maximum  Lower limit = minimum  1 Move from the beginning point in the counterclockwise direction for 2 radian. =2 Vector fundamental (1)

Exercise (3): Solution (2) y No dz. 1 Area inside a circle has fixed z coordinates.  dS = dd Lower limit = minimum  Upper limit = maximum radius x Lower limit = minimum radius Upper limit = maximum  1 Vector fundamental (1)

Ex. (4): Surface Integration of Vector Function Find the surface integration of ar when the surface is the unit circle. Repeat the first question but use ay instead of ar. When  changes, ar changes the direction, so ar is not a constant!! y 1 x 1 ANS ay does not changes the direction, so ay is a constant. ANS Vector fundamental (1)

Spherical Coordinate (r,, ) P is the point on the sphere described by the radius, latitude and longitude. x y z P: (Px, Py, Pz) Pr Pz P P P Pxy: (Px, Py, 0) r = radius of a sphere or the distance between P and the origin (0,0,0)  = angle between r and z axes  Latitude  = angle between Pxy and x axes  Longitude Vector fundamental (1)

Coordinate Conversion: Spherical x y z P: (Px, Py, Pz) Cartesian to spherical Pr Pz P P Pr Pxy: (Px, Py, 0) x y Pxy Pr P Vector fundamental (1)

Coordinate Conversion: Spherical (2) x y z P: (Px, Py, Pz) Spherical to Cartesian Pr Pz P P P Pxy: (Px, Py, 0) x y Pxy P P Projection of r onto x-y plane is  in cylindrical coordinate. Vector fundamental (1)

Directional Vector: Spherical x y z P: (Px, Py, Pz) Pr Pz  z r  P P P P Pxy: (Px, Py, 0) Function of, constant x y Pxy P P Vector fundamental (1)

Directional Vector: Spherical (2)  z r  P Use the relation between cylindrical and cartesian coordinate systems. Vector fundamental (1)

Position Vector: Spherical x y z P: (Px, Py, Pz) 1 component for 3D position. Possible because the direction of r changes with (x,y,z). Vector fundamental (1)

Spherical Coordinate Non-constant Vector addition/subtraction: Dot and cross product:  not necessary Vector fundamental (1)

Integration: Spherical Small distance in r,  or  axes r is the distance  dr  is an angle  use rd  is an angle  use rsin d Integration by path z Prsin P Pr P y Vector fundamental (1) x

Integration Along Line : Sphere Assume f(r,,) as the interaction from point f(r,,). Find the total interaction along the line. r changed, ,  fixed. z  changed, r,  fixed. r is multiplied to convert  from angle to length. /4 y 1  changed, r,  fixed. rsin  is multiplied to convert  from angle to length. x Vector fundamental (1)

Surface Integration: Sphere Find the total interaction from the given surface. z ,  changed, r fixed. r,  changed,  fixed. /4 y 1 x Vector fundamental (1)

Surface Integration: Sphere (2) z r , changed,  fixed. y 1 0.4 x Vector fundamental (1)

Directional Surface Integration: Spherical z y x Vector fundamental (1)

Volume Integration: Spherical z y x Vector fundamental (1)

Integration: Summary dL dS dv Cartesian Cylindrical Spherical x y z x Vector fundamental (1)

Exercise (4): Surface Integration Find the total charge of the given surface. y z 2 1 x y 1 1 Vector fundamental (1)

Exercise (4): Solution No d. xy plane has  fixed at 0.5 and a cone becomes a circle.  dS = rsin drd = rsin 0.5drd y 1 Upper limit = maximum spherical radius x Lower limit = minimum distance to origin (spherical radius) Upper limit = maximum  Lower limit = minimum  1 Vector fundamental (1)

Exercise (4): Solution Upper limit = maximum spherical radius No d. 2 Upper limit = maximum  Lower limit = minimum  The surface of a cone has fixed   dS = rsin drd  Lower limit = minimum distance to origin (spherical radius) y 1 Vector fundamental (1)

Ex. (5): Surface Integration of Vector Function Find the surface integration of ar when the surface is the surface of the unit sphere. Repeat the first question but use az instead of ar. When ,  changes, ar changes the direction, so ar is not a constant!! y 1 x 1 ANS az does not changes the direction, so az is a constant. ANS Vector fundamental (1)

Exercise: Given that f (x, y, z) = y is the charge at point (x,y,z). Find for the given surface. y y 2 2 1 45 x x -2 45 2 -2 Don’t forget to convert y to the coordinate you use for integration. Vector fundamental (1)

Summary: Coordinate Conversion Cartesian  Cylindrical Cylindrical  Cartesian

Summary: Coordinate Conversion (2) Cartesian  Spherical Spherical  Cartesian

Summary: Coordinate Conversion (3) Cylindrical  Spherical Spherical  Cylindrical

Summary: Dot Product Cylindrical coordinate Spherical coordinate

Summary: Dot Product (2) Spherical coordinate Cylindrical coordinate