Introduction to electromagnetics

Electromagnetic theory EM-theory Material Electric field (E) Magnetic field (H) Electro-magnetic field (E,H ) Sources (q, J) Material (ε, μ) Mathematics Coordinate systems Vector calculus (divergence, curl, gradient)

Contents Electric field Coulomb’s law Gauss’s law (divergence) Electric potential (gradient) Capacitance Ohm’s law Magnetic field Biot-Savart law Ampere’s law (curl) Inductance Sources Charge Current Material Conductor (semi-conductor, lossy material) Dielectric (insulator) Magnetic material

Electro-magnetic field Faraday’s law Displacement current Maxwell’s equations Plane wave Reflection/transmission Transmission lines Impedance matching Smith chart Waveguides Radiation

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

Example of a vector field Magnitudes and directions of vectors change with positions. +q The electric field is a field quantity because its magnitude and direction changes with positions. Electric field generated by a charge (+q1)

Electric field of a moving charge https://phet.colorado.edu/sims/radiating-charge/radiating-charge_en.html

Usefulness of the field concept 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. +q -q +q -q The electric field due to +q spread into the space. Then (–q) feels the attractive force by way of the electric field.

Analogy to the mechanical law Gravitational field Moon (m) Earth (M) Gravitational field mediates interactions between the earth and the moon.

Electric phenomena

Coulomb force +q2 fixed If q1, q2 have the same polarity, the force is repulsive. Otherwise, the force is attractive. +q1 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.

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.

Electric field due to multiple charges

Induced charges in electric fields + + Electric fields moves charges.

Electromagnetic problems

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.

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 - External E-field Tightly bound electron

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

Movement of electrons in a conductor External E-field The electrons can move freely in conductor atoms.

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.

Charges in an insulator 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. 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.

Generation of charges : friction charging

Friction charging Electrons “gained” Contact Electrons “lost” Separation

Balloon and static electricity https://phet.colorado.edu/sims/html/balloons-and-static-electricity/latest/balloons-and-static-electricity_en.html

(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.

Induction charging Metallic sphere

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. Electrons are generated via electro-chemical reaction.

Current flow Steady state current (simple DC circuit) The globe lights up due to the work done by electric current (moving charges).

Charge transport example : battery with open wire Charges in a wire are moved by diffusion and electromagnetic laws. Positive charges are plenty. Diffusion Charge movement by diffusion Negative charges are plenty.

Coulomb’s law 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. ε0 : permittivity of vacuum. If q1, q2 have the same polarity, the force is repulsive. +q2 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. +q1 O

Definition of electric field +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.

Magnetic phenomena

Magnetic field A charged particle in motion generates magnetic field nearby. In the same way, currents generate magnetic field nearby.

Biot-Savart law Current segment Direction of H-field The generated magnetic field can be predicted by Biot-Savart’s law

Magnetic material Magnetic flux density Permeability

Motion of a charge in a magnetic field Lorentz force (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.

Motion of a charge in a magnetic field Charged particles in motion are influenced by magnetic fields

Electro-magnetic phenomena

Electromagnetic law – Maxwell equations Electromagnetic phenomena are explained by the four Maxwell equations. Through the equations, electric field and magnetic field are coupled to each other. Quantities on the right hand side are the source terms. Quantities on the left side are the resulting phenomena. The independent variables are current density vector J and charge density . Maxwell equations E: electric field D: electric displacement flux density H: magnetic field B: magnetic flux density

Ampere’s law Current or increase of electric field strength E , J H

Faraday’s law H Increase of magnetic field E

Faraday’s law The time-varying magnetic field generates electric field nearby.

Gauss’ law E +Q -Q Electric field lines emanate from positive charges and sink into negative charges.

Magnetic field lines always form closed loops

Example – Signal propagation over a line trace H-field due to moving charges  E  H ZL     

Electromagnetic wave : signal propagation The electrical signal propagate along the line trace at the speed of light.

Example – Hertzian dipole antenna spheres for storing electric charges Heinrich Hertz (1857-1894) arc monitoring

Schematic diagram of Hertz experiment Transformer for high voltage generation

Propagation of electromagnetic wave Electric field : red Magnetic field : blue

Radio communication

Reception of EM wave current V Transmitting antenna Receiving antenna The charges on the receiving antenna move toward the antenna terminal, which causes voltage drop across them.

Radiation by oscillating charges

Generation of electromagnetic wave Oscillator circuit Output voltage Oscillating voltage source forces electrons to be accelerated, which generates electromagnetic wave

Importance of electromagnetic theory 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.

Electromagnetic wave : radio communication Moving charges on the antenna generate electromagnetic waves.

Electromagnetic wave generation : antennas Many kinds of antennas are built and utilized.