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**Electromagnetic Field and Waves**

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

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

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**Line, Surface and Volume Integral**

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

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**Del operator : Volume Integral : Gradient Divergence Curl**

Laplacian of scalar

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**dV = potential difference btw the scalar field V**

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

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**Divergence, Gaussian’s law**

It is a scalar field

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Curl, Stoke’s theorem ds Closed path L

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**Practical solution method**

Laplacian of a scalar Practical solution method

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**Classification of the vector field**

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Electrostatic Fields Time-invariant electric field in free space

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**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

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**E : Field intensity to the normalized charge (1)**

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

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**Electric Flux density D**

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

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**From the Gaussian’s law**

From this From the Gaussian’s law

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**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

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**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

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**Relationship btn. E and V**

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

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Energy density We

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**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

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**Convection current ( In the case of insulator)**

Current related to charge, not electron Does not satisfy Ohm’s law

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**Conduction current (current by electron : metal)**

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**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

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**Multiple dipole moments**

- + 0 : permittivity of free space : permittivity of dielectric r : dielectric constant

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**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)

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Continuity equation Qinternal time

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**Boundary condition Dielectric to dielectric boundary**

Conductor to dielectric boundary Conductor to free space boundary

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**Poisson eq. and Laplacian**

Practical solution for electrostatic field

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**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

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**Independent on material property**

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

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**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

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**By applying the Stoke’s theorem**

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

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**Magnetic flux density From this Magnetic flux line always has**

same start and end point

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**Electric flux line always start isolated (+) pole to isolated (-) pole :**

Magnetic flux line always has same start and end point : no isolated poles

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**Maxwell’s eq. For static EM field**

Time varient system

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**Magnetic scalar and vector potentials**

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

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**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

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**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

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Slant loop an B F0

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**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

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**Magnetization in material**

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

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Material in B field B

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**Magnetic boundary materials**

Two magnetic materials Magnetic and free space boundary

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Magnetic energy

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**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

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**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

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

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**Stationary loop, time-varying B field**

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**Time-varying loop and static B field**

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**Time-varying loop and time-varyinjg B field**

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**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

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**Displacement current density**

Therefore, Displacement current density

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**Maxwell’s Equations in final forms**

Point form Integral form Gaussian’s law Nonexistence of Isolated M charge Faraday’s law Ampere’s law

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**In the tme-varying field ?**

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

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** 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

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**Relationship btn. A and V ?**

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From coupled wave eq. Uncoupled wave eq.

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**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

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**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

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**Maxwell’s eq. for time-harmonic EM field**

Point form Integral form

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**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

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Waves in general form Sourceless u : Wave velocity

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**Special case : time-harmonic**

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

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**Solution of general Maxwell’s equation**

A, B : Amplitude t - z : phase of the wave : angular frequency : phase constant or wave number

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

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**EM wave in Lossy dielectric material**

Time-harmonic field

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**Propagation constant and E field**

If z-propagation and only x component of Es

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**Propagation constant and H field**

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

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EM wave in free space

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**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

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

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