Coordinate Systems

1. Coordinate Systems

2. Choice is based on symmetry of problem To understand the Electromagnetics, we must know basic vector algebra and coordinate systems. So let us start the coordinate systems. COORDINATE SYSTEMS • RECTANGULAR or Cartesian • CYLINDRICAL • SPHERICAL Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

3. Cylindrical Symmetry Spherical Symmetry Visualization (Animation)

4. Orthogonal Coordinate Systems: 1. Cartesian Coordinates z P(x,y,z) Or y Rectangular Coordinates x P (x, y, z) z z P(r, Φ, z) 2. Cylindrical Coordinates P (r, Φ, z) y r x Φ X=r cos Φ, Y=r sin Φ, Z=z z 3. Spherical Coordinates P(r, θ, Φ) θ r P (r, θ, Φ) X=r sin θ cos Φ, Y=r sin θ sin Φ, Z=z cos θ y x Φ

5. z z Cartesian Coordinates P(x, y, z) P(x,y,z) P(r, θ, Φ) θ r y x y x Φ Cylindrical Coordinates P(r, Φ, z) Spherical Coordinates P(r, θ, Φ) z z P(r, Φ, z) y r x Φ

6. Cartesian coordinate system • dx, dy, dz are infinitesimal displacements along X,Y,Z. • Volume element is given by dv = dx dy dz • Area element is da = dx dy or dy dz or dxdz • Line element is dx or dy or dz Ex: Show that volume of a cube of edge a is a3. dz Z dy dx P(x,y,z) Y X

7. Differential quantities: Length: Area: Volume: Cartesian Coordinates

8. dy dx y 6 2 3 7 x AREA INTEGRALS • integration over 2 “delta” distances Example: AREA = = 16 Note that: z = constant

9. Z Z Y r φ X Cylindrical coordinate system (r,φ,z)

10. Spherical polar coordinate system Cylindrical coordinate system (r,φ,z) • dr is infinitesimal displacement along r, r dφ is along φ and dz is along z direction. • Volume element is given by dv = dr r dφ dz • Limits of integration of r, θ, φ are 0<r<∞ , 0<z <∞ , o<φ <2π Ex: Show that Volume of a Cylinder of radius ‘R’ and height ‘H’ is π R2H . Z r dφ dz dr Y dφ φ r r dφ dr X φ is azimuth angle

11. Volume of a Cylinder of radius ‘R’ and Height ‘H’ • Try yourself: • Surface Area of Cylinder = 2πRH . • Base Area of Cylinder (Disc)=πR2.

12. Cylindrical Coordinates: Visualization of Volume element Differential quantities: Length element: Area element: Volume element: Limits of integration of r, θ, φ are 0<r<∞ , 0<z <∞ , o<φ <2π

13. Spherically Symmetric problem (r,θ,φ) Z θ r Y φ X

14. Spherical polar coordinate system (r,θ,φ) • dr is infinitesimal displacement along r, r dθ is along θ and r sinθ dφ is along φdirection. • Volume element is given by dv = dr r dθ r sinθ dφ • Limits of integration of r, θ, φ are 0<r<∞ , 0<θ <π , o<φ <2π Ex: Show that Volume of a sphere of radius R is 4/3 π R3 . P(r, θ, φ) Z dr P r cos θ r dθ θ r Y φ r sinθ dφ r sinθ X θ is zenith angle( starts from +Z reaches up to –Z) , φ is azimuth angle (starts from +X direction and lies in x-y plane only)

15. Volume of a sphere of radius ‘R’ Try Yourself: 1)Surface area of the sphere= 4πR2 .

16. Spherical Coordinates: Volume element in space

17. Points to remember System Coordinates dl1 dl2 dl3 Cartesian x,y,z dx dy dz Cylindrical r, φ,z dr rdφ dz Spherical r,θ, φdr rdθ r sinθdφ • Volume element : dv = dl1 dl2 dl3 • If Volume charge density ‘ρ’ depends only on ‘r’: Ex: For Circular plate: NOTE Area element da=r dr dφ in both the coordinate systems (because θ=900)

18. Quiz: Determine a) Areas S1, S2 and S3. b) Volume covered by these surfaces. S3 Z Radius is r, Height is h, r S2 S1 Y dφ X

19. Vector Analysis • What about A.B=?, AxB=? and AB=? • Scalar and Vector product: A.B=ABcosθ Scalar or (Axi+Ayj+Azk).(Bxi+Byj+Bzk)=AxBx+AyBy+AzBz AxB=ABSinθn Vector (Result of cross product is always perpendicular(normal) to the plane of A and B n B A

20. Gradient, Divergence and Curl • Gradient of a scalar function is a vector quantity. • Divergenceof a vector is a scalar quantity. • Curl of a vector is a vector quantity. The Del Operator Vector

21. Fundamental theorem for divergence and curl • Gauss divergence theorem: • Stokes curl theorem Conversion of volume integral to surface integral and vice verse. Conversion of surface integral to line integral and vice verse.

22. Operator in Cartesian Coordinate System Gradient: gradT: points the direction of maximum increase of the function T. Divergence: Curl: as where

23. Operator in Cylindrical Coordinate System Volume Element: Gradient: Divergence: Curl:

24. Operator In Spherical Coordinate System Gradient : Divergence: Curl:

25. Basic Vector Calculus Divergence or Gauss’ Theorem The divergence theorem states that the total outward flux of a vector field F through the closed surface S is the same as the volume integral of the divergence of F. Closed surface S, volume V, outward pointing normal

26. Stokes’ Theorem Stokes’s theorem states that the circulation of a vector field F around a closed path L is equal to the surface integral of the curl of F over the open surface S bounded by L Oriented boundary L