Fields Lec 2
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This document discusses coordinate systems and vector calculus concepts needed for electromagnetic field theory. It introduces Cartesian, cylindrical, and spherical coordinate systems. It explains that vector integration requires defining appropriate differential elements (length, area, volume) that vary based on the coordinate system. It also introduces concepts of gradient, divergence, and curl - vector operators used to take derivatives of vector fields. The gradient represents the maximum rate of change, divergence measures flux, and curl represents rotational nature. Expressions for these operators are given in the three coordinate systems.
Fields Lec 2
- 1. 18/06/1440 1 1 Frequency Spectrum 2 Coordinate systems • EM fields vary in space and time – need to be able to uniquely describe all points in space • This may be done with curvilinear coordinate systems – orthogonal coordinate systems (i.e. coordinates are perpendicular) are easiest to work with, and may simplify problem solving!!
- 2. 18/06/1440 2 3 Scalar- Magnitude only (e.g. charge, energy, temperature). Vector – Both magnitude and direction (e.g. velocity, electric field intensity). Scalar and Vector 4 Cartesian coordinates Independent variables: x y z Ranges of independent variables? Base unit vectors: Represent vector as A=Axax+ Ayay + Azaz (Ax,Ay,Az) Point is intersection of 3 planes x a y a z a
- 3. 18/06/1440 3 5 Circular cylindrical coordinates z a a az Independent variables: Ranges of independent variables? Base unit vectors: The base vectors change their orientation at different points in the system. Point is intersections of cylindrical surface, plane and half-plane 6 Circular cylindrical coordinates Base vectors are mutually orthogonal is in radians, not degrees Represent vector A as: A=Aa+ Aa + Azaz (A,A,Az)
- 4. 18/06/1440 4 7 Spherical coordinates Independent variables: Ranges of independent variables? Base unit vectors: aaa r r Point is intersection of a spherical surface, cone and half-plane 8 Spherical coordinates Base vectors are mutually orthogonal , are in radians, not degrees Represent vector A as: A=Arar+ Aa + Aa (Ar, A, A)
- 5. 18/06/1440 5 9 Why we need vector integration Maxwell’s equations emf = E dl d dt B ds c s D ds dvvvs B ds s 0 + ssc sdE dt d sdJldH We need to define the lines, surfaces and volumes for integration, and to express the associated dl, ds, dv appropriately. 10 Differential length – used in line integrals sc sdJldH We want to move along contour c (closed path) in small steps (dl). In Cartesian coordinates, our contour could have components in x, y and z directions the most general form of dl should have small changes in each base vector direction For example,
- 6. 18/06/1440 6 11 Cartesian coordinates To define differential length, area and volume, we use a basic cube: -small changes in x, y and z (dx, dy, dz) • Differential volume: -Volume of the cube: dv = dx dy dz 12 dl
- 7. 18/06/1440 7 For example, 13 Differential normal area - used in surface integrals The surface is enclosed by the contour, c (assume that c has changes in 2 coordinates – e.g. x and y) It has both magnitude (small) and direction (normal to surface). sc sdJldH 14 dS
- 8. 18/06/1440 8 15 16 Cylindrical coordinates In Cartesian coordinates, we formed the cube by moving the three defining planes incrementally In cylindrical coordinates, we consider the shape formed by expanding or moving the three defining planes incrementally
- 9. 18/06/1440 9 17 18 Spherical coordinates Again, consider “moving” the three defining planes incrementally ddrdrdv sin2 aorasinorasin=s 2 rdrddrdrddrd r asinaa=l drrddrd r ++
- 10. 18/06/1440 10 19 Summary To perform vector integration (line, surface and volume), we need to define appropriate differential elements (length, surface and volume) dl ds dv Cart. Cylind. Spher. aaa=l zyx dzdydxd ++ aaa=l zdzddd ++ asin aa=l dr rddrd r ++ a a a z y x dxdy dxdz dydz a a a zdd dzd dzd a asin asin2 rdrd drdr ddr r dxdydzdv dzdddv ddrdrdv sin2
- 11. 18/06/1440 11 21 Differential length – used in line integrals sc sdJldH We want to move along contour c (closed path) in small steps (dl). In Cartesian coordinates, our contour could have components in x, y and z directions the most general form of dl should have small changes in each base vector direction For example, c b a d dlcos|A|lA 22 Line integrals Line integral: Integral of tangential component of A along curve C From points a to b: Closed path, L: (also called circulation of A around L) (slide 88) c dlA L dlA
- 12. 18/06/1440 12 23 Surface integral Surface integral: flux of F through surface s or, sum of components of F normal to the surface (direction of s is perpendicular to surface) Closed surface (calculate net outward flux of F) s dsF s dsF 24 n F S F S . n n F Flux parallel to area, total flux crossing area is zero Flux perpendicular to area Total flux = F cos s
- 13. 18/06/1440 13 25 Volume integral Volume integral: Integral of scalar over volume In all cases, meaning of integral depends on the quantity that we are integrating. dvv 26 Why we need vector differentiation Maxwell’s equations (differential form) D v B 0 + + H J E E t E B t We need to take the divergence and curl (two kinds of derivatives) of vector fields!
- 14. 18/06/1440 14 27 Del operator Vector differential operator () Definition: Operates on a function zyx a z a y a x ++ 28 Del operator – cylindrical and spherical coordinates za z a 1 a ++ a sin 1 a 1 a r r rr ++
- 15. 18/06/1440 15 29 Gradient Definition: the gradient of a scalar field, V, is a vector that represents both the magnitude and direction of the maximum space rate of increase of V aaa zyx z V y V x V V ++ aa 1 a z z VVV V ++ a sin 1 a 1 a V r V rr V V r ++ Example: A scalar, time independent temperature field given in a region of space by T= 3x + 2xyz - z2 – 2 1. Determine a unit vector normal to the isotherms (constant temp surface) at the point (0,1,2) 2. Find the maximum rate of change of temperature at the same point Solution: The grad T is a vector perpendicular to the isotherms. A unit vector in this direction is given by 65 4 65 7 65 4 )22(2)23( z x zx zyx zyx zxyxzyz z T y T x T a a aa n aaa aaa n +++ + + 7 TT T 2.The max. rate of temp change equals the magnitude of gradT. Maximum rate of temp change at (0,1,2)is =sqrt (65)
- 16. 18/06/1440 16 31 Important points about gradient Gradient vector is perpendicular to a surface of constant value of the scalar field. This property can be used to find a unit vector normal to a surface. Magnitude of the gradient vector describes how strongly the field varies with position, while the direction of the gradient vector is in the direction in which the scalar field increases most rapidly. If A= V then V is the scalar potential of A. 32 Divergence of a vector Divergence of A at given point P is a measure of the outflow of flux from a small closed surface per unit volume as the volume shrinks about P to zero (an operator that measures a vector field’s tendency to originate from or converge upon a given point P) vv s 0 sdA limA
- 17. 18/06/1440 17 33 The divergnece of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point. A vector field denoting the velocity of air expanding as it is heated positive divergence. At a given point, the divergence of A is just a singular number representing how much the flow is expanding at that point. 34 Flux in =Flux out No source or sink Flux out Flux out Flux out >Flux in Source case Flux out < Flux in Sink case Flux out s s s If C = x ax C = a r find .C
- 18. 18/06/1440 18 35 Divergence of vector E in the three coordinate systems + + zy zyx EE x E E + + z zEE1E1 E + + E sin 1Esin sin 1E1 E 2 2 rrr r r r 36 Curl Definition: curl of A is a rotational vector magnitude - maximum net circulation of A per unit area as the area tends to zero direction – normal to direction of area (area oriented such that circulation is maximum) curl orA A max0 )lim( n L S a S ldA A
- 19. 18/06/1440 19 37 Curl describes the properties of a field that cause rotation, and has both a magnitude and a direction. To get a sense of curl, hold your arms straight out, palms facing forward and pretend that you are in a vector field a) uniform vector field b) non-uniform vector field causes rotation 1. The field is uniform (part a of figure): your palms are experiencing the same amount of force. The field may “push” you forward or backwards but ignore this part for now. 2. The field is non-uniform (part b of figure): the left palm experiences greater force than the right, causing you to rotate. Point the fingers of your right hand in the direction of rotation. Your thumb points in the direction of the curl. The magnitude of the curl is determined by the difference between forces on your left and right palms (i.e. how quickly you rotate) 3. The field is non-uniform: the right palm experiences greater force than the left, causing a different rotation. Point the fingers of your right hand in the direction of rotation. In which direction is the curl?
- 20. 18/06/1440 20 40 Laplacian Definition: Laplacian of a scalar field V is the divergence of the gradient of V
- 21. 18/06/1440 21 41 Classification of vector fields A vector field is uniquely characterized by divergence AND curl Solenoidal (divergenceless)field: divergence is zero No sources or sinks of flux e.g. magnetic fields divergence of the curl of a vector field = 0 a solenoidal field may be expressed as the curl of another vector field 0 A 42 Classification of vector fields Irrotational field: curl is zero Line integral of A is independent of chosen path conservative field E.g. electrostatic field Gradient is irrotational can express irrotational field as gradient of scalar field A A dl dsc S ( ) 0 A
- 22. 18/06/1440 22 43 Example: 0,0 AA 0,0 AA 0,0 AA 0,0 AA 44
- 23. 18/06/1440 23 45 Summary GRAD computes the gradient of a scalar function It finds the gradient, the slope, how fast you change in any given direction. DIV computes the divergence of a vector. It finds how much “stuff” is leaving a point in space. CURL computes the rotational aspects of a vector function. Assume the vector field: B = r cos ar + sin a Find: over the surface of the semicircular contourarsemicircultheoverdB c . s dsB . dl = dr ar + d a + dz az
- 24. 18/06/1440 24 yx ayxayxA 2332 3 contourtriangulartheoverdA c . s dsA .