Magnetostatic Fields Electrostatic field stuck charge distribution E

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 Display Device Lab Dong-A University

• Bio-Savart’s law dl R I H field Experimental eq. Independent on material property Display Device Lab Dong-A University

• The direction of d. H is determined by right-hand rule • Independent on material property • Current is defined by Idl (line current) Kds (surface current) Current element Jdv (volume current) I Display Device Lab K Dong-A University

• Ampere’s circuital law I H dl I enc : enclosed by path By applying the Stoke’s theorem Display Device Lab Dong-A University

• Magnetic flux density From this Magnetic flux line always has same start and end point Display Device Lab Dong-A University

• Electric flux line always start isolated (+) pole to isolated (-) pole : • Magnetic flux line always has same start and end point : no isolated poles Display Device Lab Dong-A University

• Maxwell’s eq. For static EM field Time varient system Display Device Lab Dong-A University

• Magnetic scalar and vector potentials Vm : magnetic scalar potential It is defined in the region that J=0 A : magnetic vector potential Display Device Lab Dong-A University

• Magnetic force and materials • Magnetic force Q E u B Q Fm : dependent on charge velocity does not work (Fm dl = 0) only rotation does not make kinetic energy of charges change Display Device Lab Dong-A University

• Lorentz force • Magnetic torque and moment Current loop in the magnetic field H D. C motor, generator Loop//H max rotating power Display Device Lab Dong-A University

• Slant loop F 0 B F 0 Display Device Lab an Dong-A University

• Magnetic dipole A bar magnet or small current loop m m N S A bar magnet Display Device Lab I A small current loop Dong-A University

• Magnetization in material Similar to polarization in dielectric material Atom model (electron+nucleus) B Micro viewpoint Ib : bound current in atomic model Ib Display Device Lab Dong-A University

• Material in B field B Display Device Lab Dong-A University

• Magnetic boundary materials - Two magnetic materials - Magnetic and free space boundary Display Device Lab Dong-A University

• Magnetic energy Display Device Lab Dong-A University

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 Display Device Lab Dong-A University

• Faraday’s law – Time-varying magnetic field could produce electric current Electric field can be shown by emf-produced field Display Device Lab Dong-A University

• Motional EMFs E and B are related I E B(t): time-varying Display Device Lab Dong-A University

• Stationary loop, time-varying B field Display Device Lab Dong-A University

• Time-varying loop and static B field Display Device Lab Dong-A University

• Time-varying loop and time-varyinjg B field Display Device Lab Dong-A University

• 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 Display Device Lab Dong-A University

• Therefore, Displacement current density Display Device Lab Dong-A University

• Maxwell’s Equations in final forms Point form Integral form Gaussian’s law Nonexistence of Isolated M charge Faraday’s law Ampere’s law Display Device Lab Dong-A University

• Time-varying potentials - stationary E field - In the tme-varying field ? Display Device Lab Dong-A University

- Poisson’s eqation in time-varying field - poisson’s eq. in stationary field - poisson’s eq. in time-varying field ? Coupled wave equation Display Device Lab Dong-A University

• Relationship btn. A and V ? Display Device Lab Dong-A University

→ From coupled wave eq. Uncoupled wave eq. Display Device Lab Dong-A University

• 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 Explanation of phasor Z Z=x+jy=r Re Display Device Lab Dong-A University

• 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 Display Device Lab Dong-A University

• Maxwell’s eq. for time-harmonic EM field Point form Display Device Lab Integral form Dong-A University

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 Display Device Lab Dong-A University

• Waves in general form Sourceless 0 u : Wave velocity Display Device Lab Dong-A University

• Solution of general Maxwell’s equation Special case : time-harmonic Display Device Lab Dong-A University

• Solution of general Maxwell’s equation A, B : Amplitude t - z : phase of the wave : angular frequency : phase constant or wave number Display Device Lab Dong-A University

• Plot of the wave E A 0 /2 3 /2 z T/2 T 3 T/2 t A 0 Display Device Lab Dong-A University

• EM wave in Lossy dielectric material Time-harmonic field 0 Display Device Lab Dong-A University

• Propagation constant and E field If z-propagation and only x component of Es Display Device Lab Dong-A University

• Propagation constant and H field Display Device Lab Dong-A University

• E field plot of example t=t 0 x t=t 0+ t z Display Device Lab Dong-A University

• EM wave in free space Display Device Lab Dong-A University

• E field plot in free space x a. E a. H ak z y TEM wave (Transverse EM) Uniform plane wave Polarization : the direction of E field Display Device Lab Dong-A University

Reference • Matthew N. O. Sadiku, “Elements of electromagnetic” Oxford University Press, 1993 • Magdy F. Iskander, “Electromagnetic Field & Waves”, prentice hall Display Device Lab Dong-A University