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Coordinate Systems
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Coordinate Systems

Feb 24, 2016

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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:. - PowerPoint PPT Presentation
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Page 1: Coordinate Systems

Coordinate Systems

Page 2: 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.

Page 3: Coordinate Systems

Cylindrical Symmetry Spherical Symmetry

Visualization (Animation)

Page 4: Coordinate Systems

Orthogonal Coordinate Systems:

3. Spherical Coordinates

2. Cylindrical Coordinates

1. Cartesian Coordinates

P (x, y, z)

P (r, θ, Φ)

P (r, Φ, z)

x

y

zP(x,y,z)

Φ

z

rx y

z

P(r, Φ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Rectangular CoordinatesOr

X=r cos Φ,Y=r sin Φ,Z=z

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

Page 5: Coordinate Systems

Cartesian CoordinatesP(x, y, z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, Φ, z)

x

y

zP(x,y,z)

Φ

z

rx y

z

P(r, Φ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 6: Coordinate Systems

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 dzEx: Show that volume of a cube

of edge a is a3.

P(x,y,z)

X

Y

Z

3

000adzdydxdvV

aa

v

a

dxdy

dz

Page 7: Coordinate Systems

Cartesian Coordinates

Differential quantities:

Length:

Area:

Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysddydzxsd

z

y

x

ˆ

ˆˆ

dxdydzdv

Page 8: Coordinate Systems

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

Page 9: Coordinate Systems

Cylindrical coordinate system (r,φ,z)

X

Y

Z

Z

Page 10: Coordinate Systems

Spherical polar coordinate system

• 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 .

φ is azimuth angle

Cylindrical coordinate system (r,φ,z)

X

Y

Z

r dφ

dz

dr

r dφ

dr

Page 11: Coordinate Systems

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 = 2πRH . 2) Base Area of Cylinder (Disc)=πR2.

Page 12: Coordinate Systems

Differential quantities:

Length element:

Area element:

Volume element:

dzardadrald zr ˆˆˆ

rdrdasd

drdzasddzrdasd

zz

rr

ˆ

ˆˆ

dzddrrdv

Limits of integration of r, θ, φ are 0<r<∞ , 0<z <∞ , o<φ <2π

Cylindrical Coordinates: Visualization of Volume element

Page 13: Coordinate Systems

Spherically Symmetric problem (r,θ,φ)

X

Y

Z

r

φ

θ

Page 14: Coordinate Systems

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 <2πEx: Show that Volume of a

sphere of radius R is 4/3 π R3 .

P(r, θ, φ)

X

Y

Z

r

φ

θ

drP

r dθ

r sinθ dφ

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)

r cos θ

r sinθ

Page 15: Coordinate Systems

Volume of a sphere of radius ‘R’

33

0 0

2

0

2

2

342.2.

3

sin

sin

RR

dddrr

dddrrdvV

R

v

Try Yourself:1)Surface area of the sphere= 4πR2 .

Page 16: Coordinate Systems

Spherical Coordinates: Volume element in space

Page 17: Coordinate Systems

Points to rememberSystem Coordinates dl1 dl2 dl3

Cartesian x,y,z dx dy dzCylindrical r, φ,z dr rdφ dzSpherical 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: NOTEArea element da=r dr dφ in both the coordinate systems (because θ=900)

drrdvQv l 24

Page 18: Coordinate Systems

Quiz: Determine a) Areas S1, S2 and S3.b) Volume covered by these surfaces.

Radius is r,Height is h,

X

Y

Z

r

S1S2

S3

21

hrdzrddrVb

rrddrSiii

rhdzdrSii

rhdzrdSia

Solution

h r

r

r h

h

)(2

..)

)(2

.3)

2)

)(1))

:

12

2

0 0

12

2

0

0 0

120

2

1

2

1

2

1

Page 19: Coordinate Systems

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 planeof A and B

A

B

n

Page 20: Coordinate Systems

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

xAA

.

The Del Operator

Page 21: Coordinate Systems

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.

Page 22: Coordinate Systems

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

Divergence:

Curl:

Operator in Cartesian Coordinate System

kzTj

yTi

xTT ˆˆˆ

y zxV VV

Vx y z

kyV

xV

jxV

zVi

zV

yVV xyzxyz ˆˆˆ

kVjViVV zyxˆˆˆ

where

as

Page 23: Coordinate Systems

Operator in Cylindrical Coordinate System

Volume Element:

Gradient:

Divergence:

Curl:

dzrdrddv

zzTˆT

rr

rTT

1

1 1 zr

V VV rV

r r r z

zVrVrr

ˆrV

zVr

zVV

rV rzrz

11

zVVrVV zr ˆˆˆ

Page 24: Coordinate Systems

Operator In Spherical Coordinate System

Gradient :

Divergence:

Curl:

ˆTsinr

ˆTr

rrTT

11

2

2

sin1 1 1sin sin

rr V VVV

r r r r

ˆVrV

rr

ˆrVr

Vsinr

rV

Vsinsinr

V

r

r

1

111

ˆˆˆ VVrVV r

Page 25: Coordinate Systems

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

Page 26: Coordinate Systems

dSnSd

Oriented boundary L

n

Stokes’ Theorem

S L

ldFSdF

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