1. Coordinate Systems

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

3. Visualization (Animation)
Cylindrical Symmetry
Spherical Symmetry
Visualization (Animation)

4. Orthogonal Coordinate Systems:
1. Cartesian Coordinates z
P(x,y,z) Or y
Rectangular Coordinates
P (x, y, z) x z z
P(r, Φ, z)
2. Cylindrical Coordinates
P (r, Φ, z) r y 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. Cartesian Coordinates Cylindrical Coordinates Spherical Coordinatesz z
Cartesian Coordinates
P(x, y, z)
P(x,y,z)
P(r, θ, Φ) θ r y y x x Φ
Cylindrical Coordinates
P(r, Φ, z)
Spherical Coordinates
P(r, θ, Φ) z z
P(r, Φ, z) r y 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. Cartesian Coordinates
Differential quantities:
Length:
Area:
Volume:

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

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

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 dz
r dφ 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 φ direction.
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
r cos θ P
r dθ θ r Y φ
r sinθ
r sinθ dφ 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 Cartesian x,y,z dx dy dz
System Coordinates dl1 dl2 dl3
Cartesian x,y,z dx dy dz
Cylindrical r, φ,z dr rdφ dz
Spherical r,θ, φ dr 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. b) Volume covered by these surfaces.
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=ABSinθ n Vector
(Result of cross product is always
perpendicular(normal) to the plane
of A and B n B A

20. Scalar and Vector Fields
A scalar field is a function that gives us a single value of some variable for every point in space.
voltage, current, energy, temperature
A vector is a quantity which has both a magnitude and a direction in space.
velocity, momentum, acceleration and force

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

22. 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.

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

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

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

26. Divergence or Gauss’ Theorem
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

27. 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