송석호 ( 물리학과 )

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1 송석호 ( 물리학과 ) Field and Wave Electromagnetics, David K. Cheng Reviews on (Week 1). Vector Analysis 3. tatic Electric Fields (Week ) 4. olution of Electrostatic Problems 5. teady Electric Currents (Week 3) 6. tatic Magnetic Fields 7. Time-varying Fields: Faraday s Law Introduction to Electromagnetics, 3 rd Edition, David J. Griffiths (Week 4-5) 7. Electrodynamics: Maxwell s Equations (Week 6) 8. Conservation Laws (Week 7-8) 9. Electromagnetic Waves (Week 9-1) 1. Potentials and Fields (Week 11-1) 11. Radiation (Week 13-14) 1. Electrodynamics and Relativity

2 Chapter. Vector Analysis 한양대학교, 전기공학과정진욱

=const In Cartesian coordinate system u 1 = x, u = y and u 3 = z a a =a, a a =a, a a =a

3 -4. Orthogonal coordinate systems Cartesian, cylindrical, spherical coordinates In 3D space, the three families of surface are described by u 1 =const, u =const and u 3 =const In Cartesian coordinate system u 1 = x, u = y and u 3 = z a a =a, a a =a, a a =a x y z y z x z x y a a a a a a x y y z x z OP a x a y a z x 1 y 1 z 1 dl a dxa dya dz d dxdydz x y z

4 Cylindrical coordinates u 1, u, u 3 = ( r,, z ) a a =a, a a =a, a a =a r z z r z r dl a dr a rd a dz r d rdrddz z r Differential volume element d rdrddz

5 pherical Coordinates u 1, u, u 3 = ( R,, ) a a =a, a a =a, a a =a R R R dl dr a Rd a Rsin d a R d R sin drd d x Rsin cos y Rsin sin z Rcos d R sindrdd

6 Generalized Orthogonal Coordinate Base vectors a, a, a u u u 1 3 Aa A a A a A Displacement vector Differential volume Differential area u u u u u u dl a h du a h du a h du u 1 1 u u dl h du h du h du dv h h h du du du ds ands, ds h h du du, ds h h du du Metric coefficients h 1 h h 3 x, y, z r,, z 1 r 1 R,, 1 R R sin

-6. Gradient of a calar Field Gradient : the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar.

7 -6. Gradient of a calar Field Gradient : the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar. d du h du dl dl i i i i i i ui i hiui i dl ( hdu, h du, h du ), a ui i hiui ( u, u, u ) ( x, y, z) 1 3 V ax ay az x y z V In general orthogonal coordinates ( u, u, u ) a a a u1 u u3 h1u1 hu h3u3 1 3

8 -7. Divergence of a Vector Field Flux lines : representation of field variations graphically by directed field lines. Magnitude of the field at a point : either depicted by the density or by the length of the directed lines in the vicinity of the point Divergence at a point: the net outward flux of A per unit volume as the volume about the point tends to zero div A d A lim A s s : Equal by definition Net outward flux indicates the presence of a source Flow source Div A is a measure of the strength of the flow source The flow of an incompressible fluid

9 Divergence of a Vector Field A lim A s A ds s front back right left top bottom A ds s d On the front face x Ads A front sfront A front ax y zax ( x, y, z) y z front x x Ax Ax( x, y, z) Ax( x, y, z) x ( x, y, z ( x, y, z ) ( x, y, z ) higher-order terms x Ads Aback sback Aback ax yz Ax ( x, y, z) yz back x x Ax Ax( x, y, z) Ax( x, y, z) higher-order terms x Ax Ads H.O.T. front back x A s A A A x y z ( x, y, z ) x y z ) Following the same procedure for 4 faces x y z d s x y z A A x y A A x y z z

Divergence of a Vector Field In general orthogonal curvilinear coordinates (u 1,u,u 3 ) 1 A hhh u u u hha hha hha 3 1

10 Divergence of a Vector Field In general orthogonal curvilinear coordinates (u 1,u,u 3 ) 1 A hhh u u u hha hha hha Divergence Theorem V Ad Ads It converts a volume integral to a closed surface integral, and vice versa.

11 -9. Curl of a Vector Field A C dl : Circulation of a vector field A around contour C caused by a vortex source 1 A lim an dl s s A C A A z y A A x Az y A x A ax ay z y z z x a x y In general orthogonal curvilinear coordinates a a a x y z A x y z A A A x y z A ha h a ha 1 u u 3 u hhh u u u ha ha ha

12 Laplace equation Laplacian = the divergence of the gradient of V V x y z x y z V V V V a a a a a a x y z x y z V V V x y z Laplacian in orthogonal curvilinear coordinates (u 1,u,u 3 ) 1 hh 3V hh 1 3V hh 1 V V V hh 1 h3 u1 h1 u1 u h u u3 h3 u3 1 A hhh u u u hha hha hha Laplace equation: Poisson equation: V V

-1. tokes s Theorem dl A ds A C The surface integral of the curl of a vector field over open surface Is equal to the closed line integral of the vector along the contour bounding the surface.

13 -1. tokes s Theorem dl A ds A C The surface integral of the curl of a vector field over open surface Is equal to the closed line integral of the vector along the contour bounding the surface. It converts a surface integral of the curl of a vector to a line integral, and vice versa. Note! Divergence Theorem V Ad Ads It converts a volume integral to a closed surface integral, and vice versa.

14 -11. Two Null Identities (I) The curl of the gradient of any scalar field is identically zero. V s V d V dl dv V dl C (ex) If a vector is curl-free, E then it can be expressed as the gradient of a scalar field. E V (II) The divergence of the curl of any vector field is identically zero. A V d A A ds A a ds A a ds Ad Ad n1 n C C 1 1 B (ex) If a vector is divergenceless, then it can be expressed as the curl of another vector field. B A

15 Field Classification and Helmholtz s Theorem A field is olenoidal and irrotational if F, F (static electric field in a charge-free region) olenoidal but not irrotational if F, F (A steady magnetic field in a current-carrying conductor) Irrotational but not solenoidal if F, F (A static electric field in a charged region) Neither solenoidal nor irrotational if F, F, (An electric field in a charged medium with a time-varying magnetic field) A vector field is determined if both its divergence and its curl are specified everywhere. Helmholtz s theorem A general vector function F can be written as the sum of the gradient of a scalar function and the curl of a vector function

16 ome useful vector formulas V V V A A A A A A A B B A A B A A A A B C B C A C A B A B C =B A C -C A B V A

17 Chapter.3 tatic Electric Fields 한양대학교전기공학과정진욱

18 Coulomb s law The experimental law of Coulomb (1785) F qq k 1 r 9 a R k 91 Nm /C

19 Electrostatics in Free pace Electric field density : the force per unit charge (very small) E lim q F q V/m The two fundamental postulates of electrostatics in free space. E 1 E d d E d s V V Q Gauss s law E dl E C Kirchhoff s voltage law tatic electric field is irrotational!

20 tatic E is Conservative!! calar line integral of E is independent of the path; it depends only on the end points. E E ds Edl C C P PC C P PC Edl Edl P PC P P C Edl Edl Edl Edl!!

21 Electrical potential E EV From the null identity, calar quantities are easy to handle than vector quantities. If we can determine V more easily, then E can be found by a gradient operation. Work done from point P 1 to point P W q P P 1 V V1 Ed P1 Edl P l J/C or V potential difference (Electric potential) V Relative direction of E and increasing V.

3-6. Conductors in static electric field Inside a conductor ( under static conditions), E hielding from outside electric fields Boundary conditions at a conductor-free space interface En d E E s s

22 3-6. Conductors in static electric field Inside a conductor ( under static conditions), E hielding from outside electric fields Boundary conditions at a conductor-free space interface En d E E s s s n E Edl Ew t abcda t Under static conditions, The E field on a conductor surface is everywhere normal to the surface. The surface of a conductor is an equipotential surface under static conditions.

3-7. Dielectrics in static E field Insulators ( or dielectrics) Bound charges The induced electric dipoles will modify the electric field both inside and outside the dielectric material

23 3-7. Dielectrics in static E field Insulators ( or dielectrics) Bound charges The induced electric dipoles will modify the electric field both inside and outside the dielectric material + E applied p induced + E applied Polarization vector P : Average volume density of electric dipole moment Polarization charge densities (bound-charge densities) P n p k k 1 lim C/m Pn P

24 Physical meaning of polarization charge Polarization charge P n P when P n 1 P n1 P n =-n 1 P n P n1 P

25 Physical meaning of polarization charge P n1 P n P n P n P n P n when P P P

26 3 C/m C/m p free E P E E P D E P D e R R e e e 1 1 E D E E E D E P Gauss's law inside dielectric with no surface charge P p on charge, Polarizati Relative permittivity (dielectric constant) Permittivity (dielectric constant) electric susceptibility 3-8. Electric Flux Density and Dielectric Constant : Generalized Gauss's law d Q D s

27 Common misunderstanding on E & D 오류 1 E 를인가했더니 e e P 가유도? induced polarization? applied field? : not at all! P E e E = E applied + E by dipoles 오류 E 는인가된전기장,D 는유도된전기장? : not at all! E 와 D 는서로다른물리량. D E D=D applied + D by dipoles

28 3-9. Boundary conditions Tangential component of E E E1 t Et E dl E 1d w E d w E1 tdw Etdw abcda Normal component of D D D1 n Dn s (C/m ) s lim h h D Dds D a D a a D D d h 1 n n1 n 1 V V d

29 ummary Electrostatic case Charge density V 1 4 V dv R V / E V E E /, E 1 4 V Rˆ dv R Potential Electric Field V E d l

송석호 ( 물리학과 )

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