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Coordinate Systems. 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:.

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slide2

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

slide3

Cylindrical Symmetry

Spherical Symmetry

Visualization (Animation)

slide4

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

slide5

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

cartesian coordinate system
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

slide7

Differential quantities:

Length:

Area:

Volume:

Cartesian Coordinates

slide8

dy

dx

y

6

2

3

7

x

AREA INTEGRALS

  • integration over 2 “delta” distances

Example:

AREA =

= 16

Note that: z = constant

spherical polar coordinate system
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

φ

r

r dφ

dr

X

φ is azimuth angle

volume of a cylinder of radius r and height h
Volume of a Cylinder of radius ‘R’ and Height ‘H’
  • Try yourself:
  • Surface Area of Cylinder = 2πRH .
  • Base Area of Cylinder (Disc)=πR2.
slide12

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

spherical polar coordinate system r
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)

volume of a sphere of radius r
Volume of a sphere of radius ‘R’

Try Yourself:

1)Surface area of the sphere= 4πR2 .

points to remember
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)

slide18

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

X

vector analysis
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

gradient divergence and curl
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

fundamental theorem for divergence and curl
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.

slide22

Operator in Cartesian Coordinate System

Gradient:

gradT: points the direction of maximum increase of the function T.

Divergence:

Curl:

as

where

slide23

Operator in Cylindrical Coordinate System

Volume Element:

Gradient:

Divergence:

Curl:

slide24

Operator In Spherical Coordinate System

Gradient :

Divergence:

Curl:

basic vector calculus
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

slide26

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