Coordinate Systems

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

Visualization (Animation) Cylindrical Symmetry Spherical Symmetry Visualization (Animation)

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

Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates z 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

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

Cartesian Coordinates Differential quantities: Length: Area: Volume:

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

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

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

Volume of a Cylinder of radius ‘R’ and Height ‘H’ Try yourself: Surface Area of Cylinder = 2πRH . Base Area of Cylinder (Disc)=πR2.

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

Spherically Symmetric problem (r,θ,φ) Z θ r Y φ X

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)

Volume of a sphere of radius ‘R’ Try Yourself: 1)Surface area of the sphere= 4πR2 .

Spherical Coordinates: Volume element in space

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)

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

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

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

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

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.

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

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

Operator In Spherical Coordinate System Gradient : Divergence: Curl:

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

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