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1 Introduction to electromagnetics EMLAB 2 Electromagnetic theory EM-theory Material Electric field (E) Sources (q, J) Magnetic field (H) Electro-magnetic field (E,H ) Material (ε, μ) Mathematics Coordinate systems Vector calculus (divergence, curl, gradient) EMLAB 3 Contents 1. Electric field ① Coulomb’s law ② Gauss’s law (divergence) ③ Electric potential (gradient) 1. ④ Capacitance ⑤ Ohm’s law 2. 2. Magnetic field ① Biot-Savart law ② Ampere’s law (curl) Sources ① Charge ② Current Material ① Conductor (semi-conductor, lossy material) ② Dielectric (insulator) ③ Magnetic material ③ Inductance EMLAB 4 3. Electro-magnetic field ① Faraday’s law ② Displacement current ③ Maxwell’s equations ④ Plane wave ⑤ Reflection/transmission 4. Transmission lines ① Impedance matching ② Smith chart ③ Waveguides 5. Radiation EMLAB 5 Mathematics -Glossary • Scalar : a quantity defined by one number (eg. Temperature, mass, density, voltage, ... ) • Vector : a quantity defined by a set of numbers. It can be represented by a magnitude and a direction. (velocity, acceleration, …) • Field : a scalar or vector as a function of a position in the space. (scalar field, vector field, …) Air temperature Sea water velocity Scalar field Vector field EMLAB 6 Example of a vector field Ek +q q1 r 2 rˆ •Magnitudes and directions of vectors change with positions. •The electric field is a field quantity because its magnitude and direction changes with positions. Electric field generated by a charge (+q1) EMLAB Electric field of a moving charge 7 https://phet.colorado.edu/sims/radiating-charge/radiating-charge_en.html EMLAB 8 Usefulness of the field concept Fk +q -q E +q -q q 1q 2 r 2 rˆ This equation states only the forces between the two charges +q and –q. It does not state about the interactions that occur between them. It is misleading that this equation may imply that the interaction occurs instantaneously. q1 F E k 2 rˆ q2 r F q2 E The electric field due to +q spread into the space. Then (–q) feels the attractive force by way of the electric field. EMLAB 9 Analogy to the mechanical law Gravitational field Moon (m) GMm F 2 rˆ r Earth (M) Gravitational field mediates interactions between the earth and the moon. F GM G 2 rˆ m r EMLAB 10 Electric phenomena EMLAB 11 Coulomb force q 1q 2 ˆ Fk 2 R R +q2 fixed +q1 R If q1, q2 have the same polarity, the force is repulsive. Otherwise, the force is attractive. • This law is discovered by Coulomb experimentally. • In the free space, the force between two point charges is proportional to the charges of them, and is inversely proportional to the square of the distance between those charges. EMLAB Difficulties of electrostatic problems • The electrostatic forces between two isolated charges are simple enough to calculate. • In practical cases, however, numerous charges are clustered on objects, which complicates the calculation of forces. 12 EMLAB Electric field due to multiple charges 13 EMLAB Induced charges in electric fields F ma + 14 + Electric fields moves charges. EMLAB Electromagnetic problems 15 EMLAB 16 Electrons in an isolated atom Electron energy level 1 atom - + - - - - - - - Tightly bound electron - More freely moving electron Energy levels and the radii of the electron orbit are quantized and have discrete values. For each energy level, two electrons are accommodated at most. EMLAB 17 Electrons in a solid Atoms in a solid are arranged in a lattice structure. The electrons are attracted by the nuclei. The amount of attractions differs for various material. Freely moving electron + Electron energy level + - + Tightly bound electron + + - + - - - + + + - + - - - + + + - + - - - + + - Eext - External E-field - - EMLAB 18 Insulator and conductor Insulator atoms + + + - + - Conductor atoms + - + - + - + - + - + - + - + - - - External E-field + + - - + + - - + + External E-field + - + - + - + - - - + + - - Empty energy level - - Occupied energy level Energy level of conductor atom Energy level of insulator atoms EMLAB Movement of electrons in a conductor 19 External E-field The electrons can move freely in conductor atoms. EMLAB Difficulties due to conductor • The electrons in conducting objects move freely, which means the positions of electrons changes easily. • In a conductor, the density of electrons and positions of them are difficult to find, which complicates the prediction of electrostatic phenomena. 20 EMLAB 21 Charges in an insulator 1. If an electric field problem contains a physical media, it is difficult to predict electric field in the space due to the charges contained on it. 2. If the positions of the charges are unknown, Coulomb’s law cannot be applied. molecule Molecules in a solid are aligned in the direction of the external electric field. EMLAB Generation of charges : friction charging 22 EMLAB 23 Friction charging Contact Electrons “lost” Separation Electrons “gained” EMLAB Balloon and static electricity 24 https://phet.colorado.edu/sims/html/balloons-and-staticelectricity/latest/balloons-and-static-electricity_en.html EMLAB 25 (a) A negatively charged rubber rod suspended by a thread is attracted to a positively charged glass rod. (b) A negatively charged rubber rod is repelled by another negatively charged rubber rod. EMLAB 26 Induction charging Metallic sphere EMLAB 27 Generation of charges : battery An amount of positive charges are generated such that the terminal voltage is sustained. Electrons(-) are absorbed. (+) charges are generated Electrons(-) are generated. (+) charges are absorbed. 2NH 4 2e 2 NH3 H 2 Zn Zn 2 2e Electrons are generated via electro-chemical reaction. EMLAB Current flow 28 Steady state current (simple DC circuit) The globe lights up due to the work done by electric current (moving charges). EMLAB Charge transport example : battery with open wire 29 Charges in a wire are moved by diffusion and electromagnetic laws. Positive charges are plenty. Diffusion Charge movement by diffusion Negative charges are plenty. EMLAB Coulomb’s law 30 • This law is discovered by Coulomb experimentally. • In the free space, the force between two point charges is proportional to the charges of them, and is inversely proportional to the square of the distance between those charges. Fk q 1q 2 r 2 k 9 109 [ Nm2C2 ] rˆ 1 40 ε0 : permittivity of vacuum. If q1, q2 have the same polarity, the force is repulsive. +q1 +q2 R R 2 R1 R2 R1 Coulomb’s law only states that the force between two charge is related to the distance between them and their charges. It does not tells us how the interaction occurs. O EMLAB 31 Definition of electric field q1 ˆ F R 2 q2 0 q 40 r 2 E lim +q1 +q2 Electric field is measured by the force divided by charge quantity with the amount infinitesimally small. This limit process is necessary for not disturbing the original electric field by q1. EMLAB 32 Fk q 1q 2 r 2 rˆ EMLAB 33 Magnetic phenomena EMLAB 34 Magnetic field A charged particle in motion generates magnetic field nearby. In the same way, currents generate magnetic field nearby. EMLAB 35 EMLAB 36 Biot-Savart law ˆ Ids R dH 4R 2 Current segment Id s r' R r r' r Direction of H-field The generated magnetic field can be predicted by Biot-Savart’s law EMLAB 37 Magnetic material H ext r 1 B H r 0 H Magnetic flux density Permeability 0 4 107 [ H/m ] EMLAB Motion of a charge in a magnetic field 38 Lorentz force F qv B (a) A wire suspended vertically between the poles of a magnet. (b) The setup shown in part (a) as seen looking at the south pole of the magnet, so that the magnetic field (blue crosses) is directed into the page. When there is no current in the wire, it remains vertical. (c) When the current is upward, the wire deflects to the left. (d) When the current is downward, the wire deflects to the right. EMLAB Motion of a charge in a magnetic field 39 F qv B Charged particles in motion are influenced by magnetic fields EMLAB 40 Electro-magnetic phenomena EMLAB Electromagnetic law – Maxwell equations Maxwell equations B E t D H J t D B 0 41 1. Electromagnetic phenomena are explained by the four Maxwell equations. 2. Through the equations, electric field and magnetic field are coupled to each other. 3. Quantities on the right hand side are the source terms. 4. Quantities on the left side are the resulting phenomena. 5. The independent variables are current density vector J and charge density . E: electric field D: electric displacement flux density H: magnetic field B: magnetic flux density EMLAB Ampere’s law 42 E H J t Current or increase of electric field strength E,J H EMLAB Faraday’s law 43 E H H t Increase of magnetic field E EMLAB Faraday’s law 44 The time-varying magnetic field generates electric field nearby. EMLAB Gauss’ law 45 E / E +Q -Q Electric field lines emanate from positive charges and sink into negative charges. EMLAB 46 H 0 Magnetic field lines always form closed loops EMLAB Example – Signal propagation over a line trace 47 V H-field due to moving charges t E V H E H J t ZL H E t EMLAB Electromagnetic wave : signal propagation 48 The electrical signal propagate along the line trace at the speed of light. EMLAB Example – Hertzian dipole antenna 49 spheres for storing electric charges Heinrich Hertz (1857-1894) arc monitoring EMLAB Schematic diagram of Hertz experiment 50 Transformer for high voltage generation EMLAB Propagation of electromagnetic wave 51 Electric field : red Magnetic field : blue EMLAB Radio communication 52 EMLAB Reception of EM wave 53 current E Transmitting antenna V Receiving antenna The charges on the receiving antenna move toward the antenna terminal, which causes voltage drop across them. EMLAB Radiation by oscillating charges 54 EMLAB 55 Generation of electromagnetic wave Oscillating voltage source forces electrons to be accelerated, which generates electromagnetic wave Oscillator circuit Output voltage EMLAB Importance of electromagnetic theory • • 56 EM theory helps understand how electrical signals propagate along conductors as well as free space. Predicts voltages and currents using the concept of electric and magnetic field. EMLAB Electromagnetic wave : radio communication 57 Moving charges on the antenna generate electromagnetic waves. EMLAB Electromagnetic wave generation : antennas 58 Many kinds of antennas are built and utilized. EMLAB
