Top Banner

Click here to load reader

Coor

Dec 06, 2015

ReportDownload

Documents

tooshank

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

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.