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