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Dec 06, 2015

coordinates

Coordinate Systems

COORDINATE SYSTEMS

RECTANGULAR or Cartesian

CYLINDRICAL

SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

To understand the Electromagnetics, we must know basic vector algebra and

coordinate systems. So let us start the coordinate systems.

Cylindrical Symmetry Spherical Symmetry

Visualization (Animation)

Orthogonal Coordinate Systems:

3. Spherical Coordinates

2. Cylindrical Coordinates

1. Cartesian Coordinates

P (x, y, z)

P (r, , )

P (r, , z)

x

y

zP(x,y,z)

z

rx

y

z

P(r, , z)

r

z

yx

P(r, , )

Rectangular Coordinates

Or

X=r cos ,Y=r sin ,Z=z

X=r sin cos ,Y=r sin sin ,Z=z cos

Cartesian Coordinates

P(x, y, z)

Spherical Coordinates

P(r, , )

Cylindrical Coordinates

P(r, , z)

x

y

zP(x,y,z)

z

rx

y

z

P(r, , z)

r

z

yx

P(r, , )

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.

P(x,y,z)

X

Y

Z

3

000

adzdydxdvVaa

v

a

dx

dy

dz

Cartesian Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld

dxdyzsd

dxdzysd

dydzxsd

z

y

x

dxdydzdv

AREA INTEGRALS

integration over 2 delta distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant

Cylindrical coordinate system

(r,,z)

X

Y

Z

r

Z

Spherical polar coordinate system

dr is infinitesimal displacement along r, r d is along and dz is along z direction.

Volume element is given by

dv = dr r d dz

Limits of integration of r, , are

0

Volume of a Cylinder of radius R and Height H

HR

dzdrdr

dzddrrdvV

R H

v

2

0

2

0 0

Try yourself:

1) Surface Area of Cylinder = 2RH . 2) Base Area of Cylinder (Disc)=R2.

Differential quantities:

Length element:

Area element:

Volume element:

dzardadrald zr

rdrdasd

drdzasd

dzrdasd

zz

rr

dzddrrdv

Limits of integration of r, , are 0

Spherically Symmetric problem

(,,)

X

Y

Z

r

d

d

Spherical Coordinates: Volume element in space

Spherical polar coordinate system (,,)

d is infinitesimal displacement along , d is along and r sin d is along direction.

Volume element is given by

dv = d .d. sin d

Limits of integration of , , are

0<

Volume of a sphere of radius R

33

0 0

2

0

2

2

3

42.2.

3

sin

sin

RR

dddr

dddrdvV

R

v

Try Yourself:

1)Surface area of the sphere= 4R2 .

Points to remember

System Coordinates dl1 dl2 dl3

Cartesian x,y,z dx dy dz

Cylindrical r, ,z dr rd dz

Spherical r,, dr rd r sind

Volume element : dv = dl1 dl2 dl3 If Volume charge density depends only on r:

Ex: For Circular plate: NOTE

Area element da=r dr d in both the

coordinate systems (because =900)

drrdvQv l

24

Quiz: Determine

a) Areas S1, S2 and S3.

b) Volume covered by these surfaces.

Radius is r,

Height is h,

X

Y

Z

r

d

S1S2

S3

21

hr

dzrddrVb

rrddrSiii

rhdzdrSii

rhdzrdSia

Solution

h r

r

r h

h

)(2

..)

)(2

.3)

2)

)(1))

:

12

2

0 0

12

2

0

0 0

12

0

2

1

2

1

2

1

Vector Analysis

What about A.B=?, AxB=? and AB=?

Scalar and Vector product:

A.B=ABcos Scalar or

(Axi+Ayj+Azk).(Bxi+Byj+Bzk)=AxBx+AyBy+AzBz

AxB=ABSin n Vector(Result of cross product is always

perpendicular(normal) to the plane

of A and BA

B

n

Gradient, Divergence and Curl

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.

f Vector

xA

A

.

The Del Operator

Fundamental theorem for

divergence and curl

Gauss divergence theorem:

Stokes curl theorem

v s

daVdvV .).(

s l

dlVdaVx .).(

Conversion of volume integral to surface integral and vice verse.

Conversion of surface integral to line integral and vice verse.

Gradient:

gradT: points the direction of maximum increase of the

function T.

Divergence:

Curl:

Operator in Cartesian Coordinate System

kz

Tj

y

Ti

x

TT

y zxV VV

Vx y z

ky

V

x

Vj

x

V

z

Vi

z

V

y

VV x

yzxyz

kVjViVV zyx

where

as

Operator in Cylindrical Coordinate System

Volume Element:

Gradient:

Divergence:

Curl:

dzrdrddv

zz

TT

rr

r

TT

1

1 1 z

r

V VV rV

r r r z

zVrVrr

r

V

z

Vr

z

VV

rV rzrz

11

zVVrVV zr

Operator In Spherical Coordinate System

Gradient :

Divergence:

Curl:

T

sinrT

rr

r

TT

11

22

sin1 1 1

sin sin

rr V VVV

r r r r

VrVrr

rVr

V

sinrr

VVsin

sinrV

r

r

1

111

VVrVV r

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

Basic Vector Calculus

2

( )

0, 0

( ) ( )

F G G F F G

F

F F F

Divergence or Gauss Theorem

SV

SdFdVF

dSnSd

Oriented boundary L

n

Stokes Theorem

S L

ldFSdF

Stokess 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

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