Vector Analysis Electromagnetic Theory PHYS 401 Fall 2017 1
Vector Analysis Vector analysis is a mathematical formalism with which EM concepts are most conveniently expressed and best comprehended. Many physical quantities are completely describes by their value e.g. temperature, pressure, mass, frequency Such quantities are called scalars, and their values can be given in numbers. But many physical quantities have a direction in addition to magnitude e.g.: velocity, force, displacement To describe such quantities their direction as well as their magnitudes have to be specified. So just a regular number is not inadequate. Vector: a mathematical object that has a magnitude and a direction. Physical quantities which possess a direction as well as a magnitude are represented by vectors Vector Notation: A vector is usually written as a bold face letter (like A)or with an little arrow or line above it, The magnitude of a vector Ais written as A or A 2
Often a vector is graphically represented by a directed line segment (an arrow): Whose length represents the magnitude and its orientation in the direction of the vector. direction: direction of vector A length = A, A magnitude of vector NB.: This is just a geometrical representation, vectors are not arrows, they are abstract mathematical objects. In representing a vector graphically, it is defined only by its magnitude and direction, not its position. All these represent the same vector 3
When a vector is multiplied by a positive number, it multiplies its magnitude, its direction stays the same. Multiplying by a negative number flips the direction of a vector A 2A 2.5A -2A Sum of two vectors is the single equivalent vector which has same effect as application of the two vectors. e.g. consider adding two displacements. Graphically, vectors are added using the triangular rule A+B B (also called head-to-tail rule; Move Bso that the head of Atouches the tail of B) A 4
This can be repeated to add any number of vectors A+B+C A+B C A B given any three vectors A, B, and C vector addition obey following properties: Commutative Addition Multiplication by a scaler Associative Distributive where k and lare scalars 5
Vector Decomposition Just like vectors can be added to form a single vector, any given vector can be written as a combination of other vectors. This is called vector decomposition. One particularly useful decomposition is, decomposing a vector as a sum of vectors parallel to coordinate axes. Y Z A X Y A=X+Y X X X Y A Z A=X+Y+Z Y Those are called components of the vector along coordinate axes 6
Unit vectors: A vector whose magnitude is 1. Usually unit vectors are written with a hat (like, =1 ) Any vector can be written as a product of its magnitude times the unit vector in that direction. a a= Suppose,, are unit vectors along X,Y,Z coordinate directions, Then X= X, let X = x then X=x Similarly Y=y, Z=z So A=X+Y+Z = x+y+z Often it is just written as a coordinate triplet A= (x,y,z) leaving the sum over unit vectors to be understood. = : the unit vector in the direction of the vector a X X Y A Z Z A=X+Y+Z Y Magnitude of the vector A= 7
Z z Position Vector (Radius Vector) P P 1 r r 1 r 2 r 12 P2 X x O y The position vector rof point P is defined as the directed distance from the origin O to P; that is, r = x+y+z The unit vector in the direction of r is Y = = O If there are two points with position vectors r 1 and r 2, the distance vector between them r 12 = r 2 r 1 8
Components of sum of vectors is sum of components e.g. Let A = + + and B = + + Then A+B = + + ) +( + + ) = + + + + + = + + + 9
Vector Products Vector quantities are often combined to form a new quantities. the result could be a scale or a vector. O r " p = cos" # = $%sin" e.g. Work done by a force is the product of displacement times the force in the direction of displacement. Both force and displacement are vector quantities, and work is a scalar. Angular momentum is the product of momentum and distance perpendicular to momentum to origin (axis of rotation). Momentum is a vector, since distance is taken in a specified direction it also becomes a vector. Resulting angular momentum is also a vector. So two types of vector products are defined, scalar and vector. 10
The Scalar Product (Dot Product) B " -. A The scalar product (also called of the two vectors Aand Bis defined as the product of the magnitude of Aand the projection of Bonto A(or vice versa): ( * = ABcos" -. where " -. is the angle between Aand B. dot product is commutative: ( * = * ( and distributive : ( *+/ = ( *+( / If two vectors are perpendicular to each other their scaler product is zero (cos90 = 0 ). Therefore for unit vectors,, along coordinate axes = 4; = 4; = 4 and = 5; = 5; = 5 11
if A = + + and B = + + using above properties ( * = + + = + + ( ( = + + = = ( Since ( * = ABcos" -. cos" -. = ( * between two vectors. ( * useful in finding angle In general the component of a vector (in the direction of vector Cis given by ( /7 12
The Vector Product (Cross Product) (8* = 9ABsin" -. B : " -. (8* The vector product of two vectors, A and Bis a vector, A its magnitude is equal to the product of the magnitudes of A and Band the sine of theangle between them its direction is perpendicular to the plane containing A and B, in the direction of advance of a right handedscrewwhen it is turned from A to B. (8* = 9ABsin" -. (8* =area of the parallelogram determined by A and B 13
The vector product has following properties: It is not commutative: (8*= ;*8( It is not associative: (8 *8/ < (8* 8/ It is distributive: (8 */ = (8*(8/ (8( = 4 sinθ = 0 For unit vectors,, along coordinate axes In tems of components 8 = 4; 8 = 4; 8 = 4 8 = ; 8 = ; 8 = 8 = ;... and so on A = + + and B = + + (8* = + + 8 = ; ; ; 14
Triple vector products Since result of the vector product between two vectors gives a vector it can it canbe multipied with another vector. The scalar triple product between A,B and C: C B A It is equal to the volume of the parallelepiped spanned by A, B, and C In terms of components The vector triple product between A,B and C: (8 *8/ = * ( / ;/( * ( bac-cab rule) It is not associative: (8 *8/ < (8* 8/ 15
Vector Calculus Fields: A field can be defined as a function that specifies a value for a particular quantity everywhere in a region. It could be a scalar, vector or other type of field. Air temperature in a room: every location has a specific temperature so temperature can be considered as scalar field. Wind speed: Speed of air in the atmosphere is another example, since wind speed has a direction (velocity), it is a vector field. Wind speed at sea level during hurricane Katrina 16
Gradient of a scalar Field Z P 1 (r) dz dy dx P 2 (r+dr) Suppose a certain scalar field(i.e. temperature) given by T(x,y,z) temperatureatthepointp 1 ist(r)=t(x,y,z) temperatureatthepointp 2 ist(r+dr)=t(x+dx,y+dy,z+dz) The displacement from P1 to P2 is the displacement vector dr with components(dx,dy,dz). =dx+dy+dz Temperature difference between P1,P2: dt = T(r+dr)-T(r) X dt=t(x+dx,y+dy,z+dz)- T(x,y,z) Y 17
So the change in temperature dtis given as the projection of a change of temperature vector (inside square brackets) corresponds to the displacement dr. This vector is called the Gradient of the scalar T, written as Grad T or >?. It is a generalization of one the dimensional differential operator to 3 (or higher) dimensions: 18
Gradient operator >@ rate of change in this direction = >? T constant curves T(x,y) 9 P T low >?, maximum rate of change in this direction T high A surface of constant @ (a level surface), >@ is normal to it. Gradient is the slope of a scalar field at a point. It gives the direction and magnitude of the greatest rate of change of the field. Rate of change in any other direction is given by the projection of gradient in that direction Rate of change in a given direction = >?, unit vector in that direction 19
Example: Find the directional derivative of f(x,y,z)= x 2 +y 2 +z 2 along the direction 3x2y ;z and evaluate it at the point (2,1, 2). Lets denote the given direction as a The unit vector in the direction of a= F = F F = GHIJKL G M I M IKN M = GH IJKL NO directional derivative in the direction 3x2y ;z= P >Q = GHIJKL NO 2xx2yy 2zz = RHIOKL NO at the point (2,1,2) = R IO NK NO = N NO 20
Del Operator can be considered as the operator acting on the scalar field T. It is called the del operator when grad T is written as >?it represent this process. 21
Divergence of a Vector Field The divergence of a vector field at a given point is the net outward flux per unit volume. It is a scalar quantity. flow of A integrated over the surface It represents the amount of field sources at each point. Net outward flow, positive divergence Net inward flow, positive divergence No net outward flow zero divergence 22
Divergence A X Z Y (x- x,y- y,z- z) Suppose a vector filed A =(A x,a y,a z ) at the the point (x,y,z). Lets calculate the net outward flus of this field over a small rectangular box of size (2 x,2 y, z) centered at (x,y,z) 2 x 2 z On the left face of the box outward flux is P(x,y,z) 2 y (x+ x,y+ y,z+ z) x,yx- y,z) A z 2 z A y field normal to face at the center of face area A x 2 x On the right face of the box outward flux 23
Z A X Y 2 z 2 x P(x,y,z) 2 y Total outward flux from left and right faces Similarly Total outward flux from front and back faces Total outward flux from top and bottom faces Total outward flux Divergence 24
Divergence of the vector field A Divergence Theorem: The total outward flux of a vector field Aat the closed surface Sis the same as volume integral of divergence of A. Here is the surface integral of A over the surface flux through the surface element of area S S A " : W: directed area element (direction of area is the normal direction) 25
The Curl of a Vector Field Since the del operator > = X Y +Z Y + Y is a vector, it is possible to Y Y Y take the vector product of it with a vector field. The vector field produced by this operation is called the curl of the vector field. 26
The Curl of a Vector Field The curl of a vector field describes the infinitesimal rotation (circulation) of the vector field. A good measure of the circulation is the line integral of the field around a closed curve at a given point. If the vector field is rotational it contributes to the integral. is the line integral of the field Aalong the closed A " curve C, ( \ is the integral of component ] (projection) of the vector field along the curve. For a small segment of the curve l : \ ( cos" = ( \ 27
no rotation rotation exits away from the center bulk rotation around center, no local rotation elsewhere local rotation everywhere Watch the video: www.youtube.com/watch?v=vvztebp9lrc 28
Z d (x 0,y 0 - Z,z 0 + Z) 2 Y c (x 0,y 0 + Z,z 0 + Z) 2 Z a (x 0,y 0 - Y,z 0 - Z) P(x 0,y 0,z 0 ) b (x 0,y 0 + Y,z 0 - Z) X To see this lets calculate the circulation of field Aaround a closed rectangular contour abcdof size 2 Y82 Z around point P(x,y,z), perpendicular to X axis. Y evaluated at (x 0,y 0,z 0 ) 29
The Curl of a Vector Field area of loop curl of the vector field Acan be defined as: : S l " A C S is the area enclosed by an enclosed curve C oriented such that the integral has maximum value. : : unit vector normal to area S, in the direction of motion of a right handed screw when it is turned in the direction of integral is taken. 30
Stokes Theorem Stokes theorem converts the surface integral of the curl of a vector field over an open surface into a line integral of the vector field along the curve bounding the surface. ds Validity of this can be intuitively seen directly from the definition of curl C dl Divide the surface in to a large number of small areas and apply above to each and take the sum the line integrals along the common sides of adjacent areas mutually cancel. only those sides in the periphery of the surface contribute to the sum. In the limit areas are infinitesimal this becomes the Stokes theorem 31
Laplacian Operator The Laplacian operator is the scalar product of the del operator with itself. The result is a scalar operator. It can be applied to a scalar or vector field. 32
Cartesian coordinates : Coordinate Systems Three mutually orthogonal axes X,Y,Z, unit vectors,, arein the direction of increasing coordinate value. Apoint P in space is given by the Z z P projections x, y, z on coordinate axes. x O y Y P(x, y, z) = x+y+z < <, < <, < < X Infinitesimal volume element = dxdydz dz dr Infinitesimal volume element dr=dx+dy+dz P(x,y,z) dx dy 33
Cylindrical Coordinates Z In cylindrical coordinates position of a point P is given by: S : the radial distance from Y axis z s P e f the azimuthal angle, measured from the X-axis in the XY plane z : the distance from the XY plane (same as in the Cartesian system) X x O s y Y 0 d h d, 0 d g d 2k, ; d d Unit vectors e, f, are in the direction of increasing coordinate values. Unlike in Cartesian system not all unit vectors are fixed. Directions of e and f are depend on the position (azimuthal angle g). Relation between Cartesian and cylindrical coordinates. = hcosg, y = s sing, = e = cosfsinf, f = ;sinfcosf 34
ds s s+ds sdφ Sides of the infinitesimal volume element: ; h ; hg infinitesimal volume element= hhg Del operator: 35
Spherical Coordinates Z z In cylindrical coordinates position of a point P is given by: P f r: radial distance the origin azimuthal angle, measured from the X-axis in the XY plane " : angle between the Z axis and the line from origin to point P x O " r s m y Y 0 d $ d, 0 d g d 2k, 0 d " d k X Unit vectors, f, m are in the direction of increasing coordinate values. Their directions depend on the position. Relation between Cartesian and spherical coordinates. = $sin"cosg, y = $ sin"sing, = $cos" = sin"cosgsin"sing cos" = sin"cosgmcos"cosg;f sing m = cos"cosgcos"sing; sin" = sin"singmcos"singf cosg f = ;sing cosg = cos";m sing 36
Z z " r P m f dr rdθ rsinθdφ X x O s y Y Line element: =drrdθm+rsinθdφf Volume element: drrdθrsinθdφ = r 2 sinθdθdφ Del operator : Example: A sphere of radius 2 cm contains a volume charge density ρ given by Find the total charge Q contained in the sphere 37
Summary 38
Show that: curl of gradient of a scalar field is zero >8>@ = 0 Show that : divergence of curl of a vector field is zero > >8n = 0 Calculate grad of 5 o Calculate the divergence of : o M 39
(1)Gradient : (2)Divergence : (3)Curl : the Fundamental Theorems combine (1) and (3) combine (3) and (2) curl of grad zero when the boundary shrinks to a point div of curl zero
Few basic identities for grad, div, curl 41
> (8p =n. >8p p >8n : 42
Delta function: Naive calculation gives: but it leads to contradiction with the divergence theorem, say applied over a sphere: x >yzv = }~o 0 Problem is the field is at r=0 and not correctly expressed
Delta Function To work with such situations the Dirac delta function is used: In 1dimention it is defined as: It is an even fiction with unit area K x = 1 1 1 1 Q = lim kˆk M 44
Other properties of delta function 45
3D Delta function Which can be generalized to 3 D as and Now consider Since according to divergence theorem and > = 0 for $ < 0 46
Since for any sphere arbitrary small, it shows that the entire contribution comes from the point at the origin r=0 Which implies: So that As required by the divergence theorem 47
Helmholtz Theorem (The fundamental theorem of vector analysis) The Helmholtz theorem states that any continuous vector field can be written as a sum of a gradient of a scalar field and a curl of a vector field. Œ = > $> Ž(r) U is called the scalar potential and W is called the vector potential of the field Proof: dv r r P 0 (Both U(r), W(r) have to go to zero faster than 1/r 2 as r ) 48
The divergence and curl of a vector field uniquely define a vector field. So any vector can be written as a sum of a Divergence less field and a Curl less field. 49