http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics 6. Magnetic fields in matter 7. Electrodynamics 8. Conservation laws 9. Electromagnetic waves 10. Potentials and fields 11. Radiation 12. Electrodynamics and relativity
Introduction to Electrodynamics, David J. Griffiths
Electromagnetics? History of Electromagnetics
BASIC EQUATIONS OF ELECTRODYNAMICS in SI units Maxwell's Equations Lorentz force law Auxiliary Fields Potentials
Review: Chapter 1. Vector Analysis
1.1.4 Position, Displacement, and Separation Vectors Position vector: Infinitesimal displacement vector: Separation vector from source point to field point:
1.2.3 The Del Operator: : a vector operator, not a vector. (gradient) Gradient represents both the magnitude and the direction of the maximum rate of increase of a scalar function. (divergence) (curl)
1.2.2 Gradient What s the physical meaning of the Gradient: Gradient is a vector that points in the direction of maximum increase of a function. Its magnitude gives the slope (rate of increase) along this maximal direction. Gradient represents both the magnitude and the direction of the maximum rate of increase of a scalar function.
Gradient of Separation distance Magnitude of separation vector: ( x, yz, ) r r ( x x) ( y y) ( zz) 2 2 2 1 2 2 2 1/2 ( xx) ( y y) ( zz) r 1 1 1 1 ax ay az r x r y r z r ax( xx) ay( y y) az( zz) r a = 2 2 2 ( xx) ( y y) ( zz) r r r 3/2 3 2 r r' r ( x, y, z) 1 r a r r 2 : in field coordinates 1 r a r r 2 : in source coordinates
1.2.4 The Divergence div A A A A x y A A x y z z : scalar, a measure of how much the vector A spread out (diverges) from the point in question : positive (negative if the arrows pointed in) divergence : zero divergence : positive divergence
1.2.5 The Curl curl A rot A A : vector, a measure of how much the vector A curl (rotate) around the point in question. Zero curl : Non-zero curl :
1.2.7 Second Derivatives The curl of the gradient of any scalar field is identically zero! If a vector is curl-free, then it can be expressed as the gradient of a scalar field E 0 E V The divergence of the curl of any vector field is identically zero! If a vector is divergence-free, then it can be expressed as the curl of a vector field B 0 B A
Laplacian and Laplace equation Laplacian = the divergence of the gradient of 2 2 V V x y z x y z 2 2 2 2 V 2 2 2 V V V a a a a a a x y z x y z V V V x y z Laplace equation: Poisson equation: 2 2 V V 0 0
Useful product rules Triple Products (BAC-CAB rule) Product Rules Second Derivatives
[Appendix A] Vector Calculus in Curvilinear Coordinates (Orthogonal) Curvilinear Coordinates: ( uvw,, )
Gradient Theorem Gradient in arbitrary curvilinear coordinates. f g h x, y, z 1 1 1 s,, z 1 r 1 r,, 1 r r sin Fundamental theorem for gradients
Divergence in Curvilinear Coordinates: The divergence of A in curvilinear coordinates is defined by f g h x, y, z 1 1 1 s,, z 1 r 1 r,, 1 r r sin Divergence theorem It converts a volume integral to a closed surface integral, and vice versa.
Divergence Theorem For a very small differential volume element bounded by a surface s A A ds A lim 0 0 1 1 lim 0 lim 0 N N A lim Ads s N 1 s A Ad V N A ds A ds s S 1 s A ds Ad Ads V S
Curl in Curvilinear Coordinates: The curl of A in curvilinear coordinates is defined by f g h x, y, z 1 1 1 s,, z 1 r 1 r,, 1 r r sin Stokes theorem It converts a volume integral to a closed surface integral, and vice versa.
Stokes s Theorem S dl A ds A C lim A ds Adl s 0 1 lim N N s 0 1 c C A s A ds Adl Adl C S
Laplacian in Curvilinear Coordinates: 2 Laplacian = the divergence of the gradient of Gradient of t Divergence of A f g h x, y, z 1 1 1 (Ex) Laplace equation: Poisson equation: 2 2 V V 0 0 s,, z 1 r 1 r,, 1 r r sin
1.5 The Dirac Delta Function Consider the divergence of E: (divergence in terms of r) Since the r-dependence is contained in r = r - r', r How do we solve the divergence? 2 r
The Divergence of Consider the vector function directed radially: Let s apply the divergence theorem to this function: Does this mean that the divergence theorem is false? What's going on here? The divergence theorem MUST BE right since it s a fundamental theorem. The source of the problem is the point r = 0, where v blows up! ( ) vanishes everywhere except r = 0, its integral must be 4. The entire contribution of must be coming from the point r = 0! No ordinary function behaves like that. It's zero except at the source location, yet its integral is finite! It s called the Dirac delta function. It is, in fact, central to the whole theory of electrodynamics.
1.5.3 The Three-Dimensional Dirac Delta Function Generalize the delta function to three dimensions: with its volume integral is 1: As in the one-dimensional case, integration with picks out the value f at r = 0. The divergence of is zero everywhere except at the origin. The integral of over any volume containing the origin is a constant (= 4) More generally, or Since
Dirac Delta Function and Divergence of E E () r 0 This is Gauss's law in differential form
1.6 The Theory of Vector Fields 1.6.1 The Helmholtz Theorem Maxwell reduced the entire theory of electrodynamics to four differential equations, specifying respectively the divergence and the curl of E and B. The Helmholtz theorem guarantees that the field, E or B is uniquely determined by its divergence and curl. For example, in electrostatics (V: Scalar potential) In magnetostatics, (A: Vector potential)
1.6.2 Potentials Note the two null identities the curl of the gradient of any scalar field is identically zero: The divergence of the curl of any vector field is identically zero: V 0 A 0 F 0 If the curl of a vector field (F) vanishes (everywhere), F can be written as the gradient of a scalar potential (V) F V (The minas sign is purely conventional.) F 0 If the divergence of a vector field (F) vanishes (everywhere), F can be written as the curl of a vector potential (A) F A For all cases, any vector field can be written as F V A