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UNIVERSITI MALAYSIA PERLIS EKT 242/3: ELECTROMAGNETIC THEORY CHAPTER 1 - INTRODUCTION What is Electromagnetism? • Electromagnetism - Magnetic forces produced by electricity. Oscillating electrical and magnetic fields. • Electromagnetism - Magnetism arising from electric charge in motion. Electrostatic vs. Magnetostatic UNIVERSITI MALAYSIA PERLIS Electrostatic Magnetostatic Fields arise from a potential difference or voltage gradient Fields arise from the movement of charge carriers, i.e flow of current Field strength: Volts per meter (V/m) Field strength: Amperes per meter (A/m) Fields exist anywhere as long as there was a potential difference Fields exist as soon as current flows We will see how charged dielectric produces an electrostatic fields We will see how current flows through conductor and produces magnetostatic fields Example of electrostatics: vigorously rubbing a rubber rod with a piece of fur and bring to a piece of foil – foil will be attracted to the charged rod Example of magnetostatics: Current passes through a coil produces magnetic field about each turn of coil – combined will produce 3 two-pole field, south & north pole Timeline for Electromagnetics in the Classical Era UNIVERSITI MALAYSIA PERLIS 1785 Charles-Augustin de Coulomb (French) demonstrates that the electrical force between charges is proportional to the inverse of the square of the distance between them. 4 Timeline for Electromagnetics in the Classical Era UNIVERSITI MALAYSIA PERLIS 1835 Carl Friedrich Gauss (German) formulates Gauss’s law relating the electric flux flowing through an enclosed surface to the enclosed electric charge. 5 Timeline for Electromagnetics in the Classical Era UNIVERSITI MALAYSIA PERLIS 1873 James Clerk Maxwell (Scottish) publishes his “Treatise on Electricity and Magnetism” in which he unites the discoveries of Coulomb, Oersted, Ampere, Faraday and others into four elegantly constructed mathematical equations, now known as Maxwell’s Equations. 6 Units and Dimensions UNIVERSITI MALAYSIA PERLIS SI Units French name ‘Systeme Internationale’ Based on six fundamental dimensions 7 Multiple & Sub-Multiple Prefixes UNIVERSITI MALAYSIA PERLIS Example: 4 x 10-12 F becomes 4 pF 8 The Nature of Electromagnetism stop here UNIVERSITI MALAYSIA PERLIS Physical universe is governed by 4 forces: 1. nuclear force – strongest of the four but its range is limited to submicroscopic systems, such as nuclei 2. weak-interaction force – strength is only 10-14 that of the nuclear force. Interactions involving certain radioactive particles. 3. electromagnetic force – exists between all charged particles. The dominant force in microscopic systems such as atoms and molecules. Strength is of the order 10-2 of the nuclear force 4. gravitational force – weakest of all four forces. Strength is of the order 10-41 that of the nuclear force. Dominant force in macroscopic systems, e.g solar system 9 The Electromagnetic Force UNIVERSITI MALAYSIA PERLIS Gravitational force – between two masses Gm1m2 Fg 21 R̂ 12 R122 (N) Where; m2, m1 = masses R12 = distance G = gravitational constant R̂ 12 = unit vector from 1 to 2 10 Electric fields • Electric fields exist whenever a positive or negative electrical charge is present. • The strength of the electric field is measured in volts per meter (V/m). • The field exists even when there is no current flowing. • E.g rubbing a rubber sphere with a piece of fur. 11 Electric Fields UNIVERSITI MALAYSIA PERLIS Electric field intensity, E due to q ~ ER q (V/m) (in free space) 2 40 R ~ where R = radial unit vector pointing away from charge 12 Electric Fields UNIVERSITI MALAYSIA PERLIS Electric flux density, D D E (C/m ) 2 where E = electric field intensity ε = electric permittivity of the material 13 Magnetic Fields UNIVERSITI MALAYSIA PERLIS • Magnetic field arise from the motion of electric charges. • Magnetic field strength (or intensity) is measured in amperes per meter (A/m). • Magnetic field only exist when a device is switched on and current flows. • The higher the current, the greater the strength of the magnetic field. 14 Magnetic Fields UNIVERSITI MALAYSIA PERLIS • Magnetic field lines are induced by current flow through coil. north pole south pole • Magnetic field strength or magnetic field intensity is denoted as H, the unit is A/m. 15 Magnetic Fields UNIVERSITI MALAYSIA PERLIS Velocity of light in free space, c c 1 0 0 3 108 (m/s) where µ0 = magnetic permeability of free space = 4π x 10-7 H/m Magnetic flux density, B (unit: Tesla) B H where H = magnetic field intensity 16 Permittivity UNIVERSITI MALAYSIA PERLIS Describes how an electric field affects and is affected by a dielectric medium Relates to the ability of a material to transmit (or “permit”) an electric field. Each material has a unique value of permittivity. Permittivity of free space; Relative permittivity; r 0 17 Permeability UNIVERSITI MALAYSIA PERLIS The degree of magnetization of a material that responds linearly to an applied magnetic field. The constant value μ0 is known as the magnetic constant, i.e permeability of free space; Most materials have permeability of 0 except ferromagnetic materials such as iron, where is larger than 0 . Relative permeability; r 0 18 The Electromagnetic Spectrum UNIVERSITI MALAYSIA PERLIS Atmosphere Opaque Ionosphere Opaque 100% 0 X-rays V Medical diagnosis i Gamma rays Ultraviolet s Infrared i Heating, Sterilization Cancer therapy b Night vision l e 1 fm 1 pm 10-15 10-12 1 nm 10-10 10-9 Radio Spectrum Communication, radar, radio and TV broadcasting, radio astronomy 1 μm 1 mm 1m 10-6 10-3 1 1 km 1 Mm 103 106 Wavelength (m) 108 Frequency (Hz) 1 EHz 1023 1021 1018 1 PHz 1 THz 1015 1012 1 GHz 109 1 MHz 1 kHz 1 Hz 106 103 1 19 Electromagnetic Applications UNIVERSITI MALAYSIA PERLIS 20 Review of Complex Numbers UNIVERSITI MALAYSIA PERLIS • You can use calculator . j 1 • A complex number z is written in the rectangular form Z = x ± jy • x is the real ( Re ) part of Z • y is the imaginary ( Im ) part of Z • Value of • Hence, x =Re (z) , y =Im (z) 21 Forms of Complex Numbers UNIVERSITI MALAYSIA PERLIS • Using Trigonometry, convert from rectangular to polar form, z x jy z cos j z sin z (cos j sin ) • Alternative polar form, z ze j z 22 Forms of complex numbers UNIVERSITI MALAYSIA PERLIS • Relations between rectangular and polar representations of complex numbers. 23 Forms of complex numbers UNIVERSITI MALAYSIA PERLIS Applying Euler’s identity e cos θ j sin θ jθ thus z z e j z cos j z sin which leads to the relations x z cos , NB: θ in degrees z x 2 y2 , y z sin , tan 1 ( y / x ) , 24 Complex conjugate UNIVERSITI MALAYSIA PERLIS • Complex conjugate, z* • Opposite sign (+ or -) & with * superscript (asterisk) z* ( x jy)* x jy • Product of a complex number z with its complex conjugate is always a real number. • Important in division of complex number. z z* real number 25 Equality UNIVERSITI MALAYSIA PERLIS omplex numbers z1 and z 2 are given by 1 = z1 x1 jy1 z1 e j1 z 2 x 2 jy 2 z 2 e j2 x 1 x 2 and y1 y 2 or, equivalent • z1 = z2 if and only if x1=x2 AND y1=y2 z 2 if and only if 2 . • Or equivalently, z1 z 2 AND 1 2 26 Addition & Subtraction UNIVERSITI MALAYSIA PERLIS If two complex numbers z1 and z 2 are given by z1 a jb z 2 x jy Hence z 1 z 2 (a x ) j ( b y) and z 1 z 2 (a x ) j ( b y) 27 Multiplication in Rectangular Form UNIVERSITI MALAYSIA PERLIS plex •numbers and are given by z z bers and are given by z z 1 2 1 2 Given two complex numbers z1 and z2; j1 j1 z1 xz11 xjy jy1z1 e z1 e 1 1 j2 z 2 xz2 xjy2 jy z2 ze e j2 2 2 2 2 • Multiplication gives; z1z 2 (x1 jy1 )( x 2 jy 2 ) z1z 2 (x1 jy1 )( x 2 jy 2 ) (x1x 2 y1y2 ) j(x1y2 x 2 y1 ) (x1x2 y1y2 ) j(x1y2 x2 y1) 28 Multiplication in Polar Form UNIVERSITI MALAYSIA PERLIS • In polar form, form z1 z 2 z1 e j1 z2 e z1 z 2 e j2 j( 1 2 ) z1 z2 cos (1 2) j sin (1 2) 29 Division in Polar Form UNIVERSITI MALAYSIA PERLIS z2 0 • For z 0 z1 z2 x1 jy1 x2 jy2 x1 jy1 x 2 jy2 x 2 jy2 x 2 jy2 x(1xxx2x2 yy1 1yy22) jj(xx22yy11 xx11yy2 2) x2 2 y 2 2 x 2 y22 2 30 Division in Polar Form UNIVERSITI MALAYSIA PERLIS z1 z2 z1 e j1 z2 e z1 z2 z1 z2 j2 e j(θ1 θ2) cos (θ1 θ2 ) j sin (θ1 θ2 ) 31 Powers UNIVERSITI MALAYSIA PERLIS sitive • integer For anyn,positive integer n, z (z e ) n j n z e n jn z (cos n j sin n ) n • And, 1 2 1 2 z z e j 2 1 2 z (cos 2 j sin 2 ) 32 Useful Relations: Powers UNIVERSITI MALAYSIA PERLIS • Useful relations 1 e j je e j 2 j e j (e j 1 180 , 1 90 , j 2 j 2 j e ) e 1 2 j 4 j 2 e 1 90 j 4 ( 1 j) 2 ( 1 j) 2 , , 33