Electromagnetic Field and Waves Gi-Dong Lee Outline: Electrostatic Field Magnetostatic Field Maxwell Equation Electromagnetic Wave Propagation

Vector Calculus Basic mathematical tool for electromagnetic field solution and understanding.

Line, Surface and Volume Integral Line Integral : Circulation of A around L ( ) Perfect circulation : Surface Integral : Path L Net outward flux of A

Del operator : Volume Integral : Gradient Divergence Curl Laplacian of scalar

dV = potential difference btw the scalar field V Gradient of a scalar → V1 V2  dV = potential difference btw the scalar field V

Divergence, Gaussian’s law It is a scalar field

Curl, Stoke’s theorem ds Closed path L

Practical solution method Laplacian of a scalar Practical solution method

Classification of the vector field

Electrostatic Fields Time-invariant electric field in free space

Coulomb’s law and field intensity Experimental law Coulomb’s law in a point charge Q1 Q2 Vector Force F12 or F21 Q1 Q2 F21 F12 r1 r2

E : Field intensity to the normalized charge (1) Electric Field E r r’ 1 Q R E : Field intensity to the normalized charge (1)

Electric Flux density D  Flux density D is independent on the material property (0) Maxwell first equation from the Gaussian’s law

From the Gaussian’s law From this From the Gaussian’s law

In case of a normalized charge Q Electric potential Electric Field can be obtained by charge distribution and electric potential E A Q B In case of a normalized charge Q + : work from the outside - : work by itself

Second Maxwell’s Equ. From E and V Absolute potential E r O : origin point  Q=1 Second Maxwell’s Equ. From E and V

Relationship btn. E and V Second Maxwell’s Equ Relationship btn. E and V 3,4,5 : EQUI-POTENTIAL LINE E 3 4 5

Energy density We

E field in material space ( not free space) Conductor Non conductor Insulator Dielctric material Material can be classified by conductivity   << 1 : insulator  >> 1 : conductor (metal :  ) Middle range of  : dielectric

Convection current ( In the case of insulator) Current related to charge, not electron Does not satisfy Ohm’s law

Conduction current (current by electron : metal)

Displacement can be occurred Polarization in dielectric + - After field is induced Displacement can be occurred Equi-model -Q +Q Dipole moment Therefore, we can expect strong electric field in the dielectric material, not current

Multiple dipole moments - + 0 : permittivity of free space : permittivity of dielectric r : dielectric constant

Linear, Isotropic and Homogeneous dielectric D  E : linear or not linear When (r) is independent on its distance r : homogeneous When (r) is independent on its direction : isotropic  anisotropic (tensor form)

Continuity equation Qinternal time

Boundary condition Dielectric to dielectric boundary Conductor to dielectric boundary Conductor to free space boundary

Poisson eq. and Laplacian Practical solution for electrostatic field

Magnetostatic Fields Electrostatic field : stuck charge distribution E, D field to H, B field Moving charge (velocity = const) Bio sarvart’s law and Ampere’s circuital law

Independent on material property Bio-Savart’s law  I dl  H field R Experimental eq. Independent on material property

The direction of dH is determined by right-hand rule Independent on material property Current is defined by Idl (line current) Kds (surface current) Jdv (volume current) Current element I K

By applying the Stoke’s theorem Ampere’s circuital law I H dl I enc : enclosed by path By applying the Stoke’s theorem

Magnetic flux density From this Magnetic flux line always has same start and end point

Electric flux line always start isolated (+) pole to isolated (-) pole : Magnetic flux line always has same start and end point : no isolated poles

Maxwell’s eq. For static EM field Time varient system

Magnetic scalar and vector potentials Vm : magnetic scalar potential It is defined in the region that J=0 A : magnetic vector potential

Magnetic force and materials Magnetic force Q E B u Q Fm : dependent on charge velocity does not work (Fm dl = 0) only rotation does not make kinetic energy of charges change

Magnetic torque and moment Lorentz force Magnetic torque and moment Current loop in the magnetic field H D.C motor, generator Loop//H  max rotating power

Slant loop   an B F0 

A bar magnet or small current loop Magnetic dipole A bar magnet or small current loop N S m I m A bar magnet A small current loop

Magnetization in material Similar to polarization in dielectric material Atom model (electron+nucleus) Ib B Micro viewpoint Ib : bound current in atomic model

Material in B field B

Magnetic boundary materials Two magnetic materials Magnetic and free space boundary

Magnetic energy

Maxwell equations Maxwell equations In the static field, E and H are independent on each other, but interdependent in the dynamic field Time-varying EM field : E(x,y,z,t), H(x,y,z,t) Time-varying EM field or waves : due to accelated charge or time varying current

Electric field can be shown by emf-produced field Faraday’s law Time-varying magnetic field could produce electric current Electric field can be shown by emf-produced field

Motional EMFs E and B are related B(t):time-varying I E

Stationary loop, time-varying B field

Time-varying loop and static B field

Time-varying loop and time-varyinjg B field

Displacement current → Maxwell’s eq. based on Ampere’s circuital law for time-varying field In the static field In the time-varying field : density change is supposed to be changed

Displacement current density Therefore, Displacement current density

Maxwell’s Equations in final forms Point form Integral form Gaussian’s law Nonexistence of Isolated M charge Faraday’s law Ampere’s law

In the tme-varying field ? Time-varying potentials stationary E field In the tme-varying field ?

 Coupled wave equation Poisson’s eqation in time-varying field poisson’s eq. in stationary field poisson’s eq. in time-varying field ?  Coupled wave equation

Relationship btn. A and V ?

From coupled wave eq. Uncoupled wave eq.

Explanation of phasor Z Time-harmonic fields Fields are periodic or sinusoidal with time → Time-harmonic solution can be practical because most of waveform can be decomposed with sinusoidal ftn by fourier transform. Im Re   Explanation of phasor Z Z=x+jy=r 

Phasor form If A(x,y,z,t) is a time-harmonic field Phasor form of A is As(x,y,z) For example, if

Maxwell’s eq. for time-harmonic EM field Point form Integral form

EM wave propagation Most important application of Maxwell’s equation → Electromagnetic wave propagation First experiment → Henrich Hertz Solution of Maxwell’s equation, here is General case

Waves in general form Sourceless u : Wave velocity

Special case : time-harmonic Solution of general Maxwell’s equation Special case : time-harmonic

Solution of general Maxwell’s equation A, B : Amplitude t - z : phase of the wave : angular frequency  : phase constant or wave number

Plot of the wave E A /2  3/2 z A T/2 T 3T/2 t

EM wave in Lossy dielectric material Time-harmonic field

Propagation constant and E field If z-propagation and only x component of Es

Propagation constant and H field

E field plot of example x z t=t0 t=t0+t

EM wave in free space

E field plot in free space x z ak aE aH y TEM wave (Transverse EM) Uniform plane wave Polarization : the direction of E field

Reference Matthew N. O. Sadiku, “Elements of electromagnetic” Oxford University Press,1993 Magdy F. Iskander, “Electromagnetic Field & Waves”, prentice hall