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## Electromagnetic Field and Waves

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**Electromagnetic Field and Waves**Outline: Electrostatic Field Magnetostatic Field Maxwell Equation Electromagnetic Wave Propagation Gi-Dong Lee**Vector Calculus**• Basic mathematical tool for electromagnetic field solution and understanding.**Path L**• Line, Surface and Volume Integral • Line Integral : Circulation of A around L ( ) Perfect circulation : • Surface Integral : Net outward flux of A**Volume Integral :**• Del operator : Gradient Divergence Curl Laplacian of scalar**V1**V2 dV = potential difference btw the scalar field V • Gradient of a scalar →**Divergence, Gaussian’s law**• It is a scalar field**ds**Closed path L • Curl, Stoke’s theorem**Practical solution method**• Laplacian of a scalar**Electrostatic Fields**• Time-invariant electric field in free space**Q1**Q2 • Coulomb’s law and field intensity • Experimental law • Coulomb’s law in a point charge • Vector Force F12 or F21 Q1 Q2 F21 F12 r1 r2**1**R Q r r’ • Electric FieldE E : Field intensity to the normalized charge (1)****• Electric Flux densityD Flux density D is independent on the material property (0) • Maxwell first equation from the Gaussian’s law**From this**From the Gaussian’s law**Q**B A E • Electric potential Electric Field can be obtained by charge distribution and electric potential In case of a normalized charge Q + : work from the outside - : work by itself**Q=1**r E O : origin point Absolute potential • Second Maxwell’s Equ. From E and V**3**4 5 E • Second Maxwell’s Equ • Relationship btn. E and V 3,4,5 : EQUI-POTENTIAL LINE**Conductor**Material Insulator Non conductor Dielctric material • E field in material space ( not free space) 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**After field is induced**- - - - - - - - - - - + + - - - - - - - - - Displacement can be occurred - - - - - • Equi-model - + Dipole moment - -Q +Q + • Polarization in dielectric 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 distancer :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**dl** I H field • Bio-Savart’s law R Experimental eq. Independent on material property**I**K • 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**H dl • Ampere’s circuital law 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**u**B Q Q E • Magnetic force and materials • Magnetic force Fm : dependent on charge velocity does not work (Fm dl = 0) only rotation does not make kinetic energy of charges change**Lorentz force**• Magnetic torque and moment Current loop in the magnetic field H D.C motor, generator Loop//H max rotating power**F0** B an F0 • Slant loop**m**m N I S A bar magnet or small current loop • Magnetic dipole A bar magnet A small current loop**B**Ib • Magnetization in material Similar to polarization in dielectric material Atom model (electron+nucleus) Micro viewpoint Ib : bound current in atomic model**B**• Material in B field**Magnetic boundary materials**• Two magnetic materials • Magnetic and free space boundary**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**Faraday’s law**• Time-varying magnetic field could produce electric current Electric field can be shown by emf-produced field**I**E B(t):time-varying E and B are related • Motional EMFs**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