Electromagnetic Theory-I (PHY F212)

Transcription

1 Electromagnetic Theory-I (PHY F212) Kaushar Vaidya Ph.D. (Astronomy) Office: 3242-N, Physics Department, FD III Building

2 Electromagnetic Theory-I (PHY F212) Textbook: Introduction to Electrodynamics by David J. Griffiths (3 rd Ed.), Pearson Education Inc Reference: Physics Vol. II by Halliday, Resnick & Crane (5 th Ed.) John Wiley & Sons, Inc Electricity and Magnetism (Berkeley Vol. II) by Edward Purcell (2 nd Ed.) The McGraw Hill Companies, 2008

3 Evaluation Criteria Component Weightage Comments Mid-Semester 35 % (100 Marks) on 4 th October 2012 (Closed Book) Comprehensive 40 % (120 Marks) on 13 th December 2012 (Open/Closed Book) Tutorial Tests 25 % (80 Marks) 4/5 Tests; Regularly Spaced & Announced in Advance Make-Up Policy: Only GENUINE cases of sickness leading to hospitalization would, after thorough investigations, be considered for make-up. Fake cases would be severely penalized and reported to the institute. Make-up to attend weddings of siblings are strongly discouraged. Absolutely No Make-up for Tutorial Tests.

4 Mechanics Goal of Mechanics Time evolution of a system under a given force Realms of Mechanics s p e e d Classical Mechanics (Newton) Special Relatively (Einstein) s i z e Quantum Mechanics (Bohr, Heisenberg, Schrodinger, et. al.) Quantum Field Theory (Dirac, Pauli, Feynman, Schwinger, et al.)

5 Fundamental Forces s t r e n g t h 1. Strong 2. Electromagnetic 3. Weak 4. Gravity Electromagnetic forces are the most dominant in every-day life. These forces are largely responsible for the physical and chemical properties of matter from atom to a living cell.

6 A bit of History Earlier than Coulomb 1767, Joseph Priestley: force between charges vary as the inverse square of the distance. 1769, John Robinson: force of repulsion between two spheres with charges of same sign varied as X , Henry Cavendish: discovers (but does not publish) dependence of the force between charged bodies on the distance

7 A bit of History Coulomb s Law 1785, Charles Augustin de Coulomb: 1 qq F r, where r r 2 r 4 r 0

8 A bit of History Unification of Electricity, Magnetism and Optics 1820, Oersted: electric current could deflect a magnetic compass needle. 1831, Faraday: a moving magnet generates an electric current 1861, 1862 Maxwell: four equations --- theory of classical electrodynamics 1888, Hertz: Experimental confirmation of Maxwell s theory

9 Validity of Coulomb s Law and Departure from Inverse Square Law Coulomb s law is valid for 24 orders of magnitude of length! (10-13 cm to cm) F r 1 2 Recent limit on ε = 2.7 x (Williams et al. 1971)

10 Overview of the Course 1. Survey of Scalar and Vector Fields and their Calculus (Ch. 1) Electrostatics (Ch. 2, 3, 4) 1. Electrostatics in Free Space 2. Electrostatics in the Presence of Conductors 3. Electrostatics in the Presence of Insulators Instructor: Dr. Kaushar Vaidya; first 20 lectures

11 Overview of the Course Magnetostatics (Ch. 5, 6) 1. Magnetostatics in Free Space 2. Magnetostatics in the Presence of Materials Electrodynamics (Ch. 7) 1. Faraday s Law of Electromagnetic Induction 2. Maxwell s Equations and Their Simple Solutions Instructor: Dr. Rakesh Choubisa; last 20 lectures

12 Chapter 1 Vector Analysis

13 Homework Section 1.1: Vector Algebra

14 What is a Vector? A quantity with a magnitude and direction?

15 Vector Transformation If you z rotate your coordinate-system by an angle φ about x- axis, how would the components of vectors in the rotated frame relate to the components in the un-rotated frame? θ A θ φ y A Acosθ, A Asinθ y A A cosθ, A Asin θ y z z Something is a vector if it transforms A like a vector. cos sin A y y A sin cosaz z

16 Vector Transformation For rotation about an arbitrary axis in three dimensions, A x Rxx Rxy R xz A x A y R yx R yy Ryz Ay Rzx Rzy R A zz A z z

17 Derivatives Function of single variable f (x) df df dx df dx dx how slow/fast a function varies

18 Gradient For a function of three variables, T(x, little distance, how much T varies? y, z), if we move a Displacement Vector dl dxx dyy dzz For a given magnitude of displacement, the answer depends on the direction! We need directional derivatives.

19 Gradient T T T dt dx dy dz x y z T T T dt x y z. dxx dyy dzz x y z T.dl T T x T y T z Gradient of T x y z

20 Gradient dt T.dl T.n dl dl dln dt dl n T.n Directional derivative The rate of change of a function in any direction at a point, is the component of the gradient of the function in the given direction at that point.

21 T Gradient points in the direction of maximum increase of the function T. T dt T.dl T dl cos θ gives the slope along this direction.. T = 0 stationary point of the function T. Maxima, minima, or a saddle point.

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23 Gradient Problem 1.12 The height of a certain hill (in feet) is given by, 2 2 h(x, y) 10(2xy 3x 4y 18x 28y 12) where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill? (c) How steep is the slope at a point 1 mile north and 1 mile east of South Hadley? In what direction is the slope steepest at that point?

24 Problem 1.13 (Tutorial) Gradient Let r be the separation vector from a fixed point (x, y, z ) to the point (x, y, z), and let r be its length. Find out (a) (r 2 ) (b) (1/r) (c) (r n )

25 Problem 1.14 Gradient Suppose that f is a function of two variables y, and z. Show that gradient f transforms as a vector under rotations.

26 The Operator Del x y z x y z Three ways Operator Del can act 1. On a scalar function T : T (Gradient) 2. On a vector function v :.v (Divergence) 3. On a vector function v : xv (Curl)

27 Divergence.v x y z. v x x vy y vzz x y z v v x y vz.v x y z Divergence is flux-density, or flux per unit volume. It tells you how much the vector (function) spreads out from the point in question. A point of positive divergence is a source, e.g. faucet, a point of negative divergence is a sink or drain.

28 Divergence Positive divergence Negative divergence

29 Divergence Zero divergence Positive divergence

30 Problem 1.16 Sketch the vector function Divergence r r The answer may surprise you. Can you explain it? v 2 and compute its divergence.

31 The Curl v x y z v x x vy y vz z x y z v v z y vx vz vy v x v x y z y z z x x y Curl is circulation-density, or circulation per unit area. It tells you how much the vector (function) curls about the point in question. A paddle wheel would start rotating when placed on a point of positive curl. A whirlpool is a region of large curl.

32 The Curl Curl pointing in the z-direction

33 The Curl Is there a curl in the n direction? or n 2? 1 Figure Source:

34 Maxwell s Equations 1.E ρ ε0.b 0 B E t B μ J+μ ε E t

35 Second Derivatives Gradient 1. T T T. T x y z. x y z x y z x y z T T T x y z 2 T (Laplacian of T) 2. T 0

36 Second Derivatives Divergence 3..v seldom occurs in physics problems 4. Curl. v 0 (Prob. 1.26) 2 v.v v 5.

37 Second Derivatives Problem 1.27 Prove that the curl of a gradient is always zero.

38 Line Integral b v.dl where, v is a vector function, and dl is the a infinitesimal displacement vector. The integral is to be carried out along a prescribed path P from point a to b.

39 Line Integral If the path of integral is a closed loop, i.e. a=b, then it is denoted as, v.dl Example: Work done by a force For some special class of vector functions (e.g. gradient of any scalar function), the line integral is independent of the path, i.e. it is determined entirely by its end points. A force that has this property is called conservative.

40 Problem 1.28 Line Integral Calculate the line integral of the function v x 2 x 2yzy y 2 z from the origin to the point (1,1,1) by three different routes: (a) (0,0,0) (1,0,0) (1,1,0) (1,1,1) (b) (0,0,0) (0,0,1) (0,1,1) (1,1,1) (c) The direct straight line (d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)?

41 Surface Integral v.da where, v is a vector function, and da is an s infinitesimal patch of area, with direction perpendicular to surface.

42 Surface Integral For closed surface, we put a circle on the integral sign, i.e. v.da, and outword is considered positive. For some special class of vector functions (e.g. curl of any vector function), the surface integral is independent of the particular surface chosen, i.e. it is determined entirely by the boundary line.

43 Problem 1.29 Surface Integral Calculate the surface integral of the function 2 v 2xzx x 2 y y z 3 z over the bottom of the box shown in the figure. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box including the bottom? z y x

44 Volume Integral V Td where T is a scalar function, and dτ is an infinitesimal volume element. In Cartesian coordinates, d dxdydz. For a vector function, v, vd x v d y v d z v d x y z

45 Volume Integral Problem 1.30 Calculate the volume integral of the function T = z 2 over the tetrahedron with corners at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). z 0,0,1 z 0,0,1 x 1,0,0 x 0,1,0 1,0,0 y 0,1,0 y

46 The Fundamental Theorem of Calculus a b df dx f (b) f (a) dx In words: Integral of a derivative over an interval is equal to value of the function at the boundary.

47 The Fundamental Theorem for b a P Gradients T.dl T(b) T(a) b Corollary 1: T.dl is a independent of path taken from a to b. In words: To know the height of Eiffell tower: Corollary 2: T.dl 0 Either you climb each step while measuring the height of it and summing it up until you reach the (End top points (LHS) are identical). Or you can use altimeter and subtract the reading you get at the bottom from the reading you get at the top (RHS).

48 The Fundamental Theorem for Divergences v.v d v.da Other names: Gauss s theorem, Green s theorem. s In words: If you have lots of faucets in some region, you can measure how much each outputs and add the contribution from all faucets (LHS). Alternatively you can deploy your men at the boundary of this region (a closed surface bounding this volume) and measure how much of the fluid leaves (RHS).

49 The Fundamental Theorem for Divergences Problem 1.32 Test the divergence theorem for the function, v xy x 2yz y 3zx z Take as your volume the cube shown in the figure with sides of length 2. z y x

50 The Fundamental Theorem for Curls s v.da v.dl P Other names: Stokes theorem. Direction Convention Right-hand Rule: If your fingers point in the direction of line integral, your thumb fixes the direction of da.

51 The Fundamental Theorem for Curls In words: You can measure the total amount of swirl by summing up how much your tiny paddle wheel rotates at various points over some surface or you can go around the edge and find out how much the flow is following the boundary.

52 The Fundamental Theorem for Curls Problem 1.32 Test Stokes theorem for the function, v xy x 2yz y 3zx z using the triangular shaded area as shown in the figure. 2 2

53 Spherical Polar Coordinates r: distance from the origin θ (polar angle): angle down from the z-axis φ (azimuthal angle): angle around from the x-axis x r sin cos, y r sin sin, z r cos A A r A A r

54 Spherical Polar Coordinates Unit Vectors r sin cosx sin sin y cos z cos cos x cos sin y sin z sin x cos y r r The directions of r,, and, change from point to point!

55 Spherical Polar Coordinates Infinitesimal elements of length in the r,, and, direction dl r θ dr dl =rdθ dl r sin θd

56 Spherical Polar Coordinates Infinitesimal volume element dτ=dl dl dl r θ 2 r sin θdrdθd Examples of Infinitesimal area elements 2 da dlθdl r r sin θdθdr da dl dl θ rdrdθ r

57 Spherical Polar Coordinates Gradient T ˆ 1 T 1 T T rˆ ˆ r r r sin Divergence 1 v v r v 2 r sin v r r r sin r sin

58 Spherical Polar Coordinates Curl 1 v v sin v r r sin 1 1 v 1 vr rv rv r sin r r r r Laplacian T 1 T 1 T T r sin r r r r sin r sin

59 Spherical Polar Coordinates Problem 1.39 Compute the divergence of the function, v r cos r r sin r sin cos Check the divergence theorem for this function, using as your volume the inverted hemispherical bowl of radius R, resting on the xy-plane and centered on the origin.

60 Cylindrical Polar Coordinates x z z s s cos x the sin z-axis y s z y s: perpendicular distance from φ (azimuthal angle): angle around from the x-axis sin x cos y z z z: z xthe sdirections cos, y of s sin s, ˆ ˆ,, and, z zzˆ change from point to point!

61 Cylindrical Polar Coordinates Infinitesimal elements of length in the s,,and, z direction dl dl dl s z ds sd dz Infinitesimal volume element d sdsddz

62 Cylindrical Polar Coordinates Gradient T 1 T T t sˆ zˆ s s z Divergence 1 1 v v v s s s s z z sv

63 Cylindrical Polar Coordinates Curl 1 v v v v 1 v ˆ ˆ s z z s s s z s z s v s sv z Laplacian T 1 T T T s s s s s z

64 Cylindrical Polar Coordinates Problem 1.42 (a) Compute the divergence of the function 2 v s 2 sin sˆ ssin cos ˆ 3z zˆ (b) Check the divergence theorem for this function, using the quarter cylinder shown in the figure. (c) Find the Curl of this vector function..

65 The One-Dimensional Dirac Delta Function The 1-D Dirac delta function: infinitely high, infinitesimally narrow spike with area 1 0, if x 0 δ x and, if x = 0 δ x dx 1 Example: linear mass density of a point mass

66 The One-Dimensional Dirac Delta Function Technically, it is generalized function, or distribution. Limit of a sequence of functions as shown in the figures R n x of height n and width 1 T n n x of height n and base 2 n

67 If The One-Dimensional Dirac Delta f x Function is any continuous function, then f x δ x f 0 δ x f x δ x dx f 0 δ x dx f 0 Under an integral, the delta function picks out the value of function at x = 0 (if x = 0 is included in the limits of integration).

68 Shifted 1-D Dirac Delta Function 0, if x a δ x a and, if x = a δ x a dx 1 f x δ x a f a δ x a f x δ x a dx f a If f x D x dx f x D x dx for all f(x), 1 2 Then D (x) 1 2 D (x)

69 Homework: Example 1.15 Show that constant. 1 k δ x δ kx =, where k is any non-zero

70 Prob (a) 2 1-D Dirac Delta Function 2x+3 δ 3x dx (b) (c) (d) x +3x+2 δ 1 - x dx 2 2 9x δ3x+1dx 1 a δ x - b dx

71 The Three-Dimensional Delta Function all space 3 δ (r)=δ x δ y δ z 3 δ r dτ δ x δ y δ z dxdydz 1 all space f r δ r a dτ f a 3

72 The Three-Dimensional Delta Function ˆr r 3. 4πδ r 2 1 rˆ 2 1 Since 4 δ 3 r r r 2 r

73 The Three-Dimensional Delta Function Prob (a) Write an expression for the volume charge density of a point charge q at r. (b) What is the volume charge density of an electric dipole, consisting of a point charge q at the origin and a point charge +q at a? (c) What is the volume charge density of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centered at the origin?