Electromagnetism INEL 4152

1. ElectromagnetismINEL 4152 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR

2. Electricity => Magnetism • In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. Cruz-Pol, Electromagnetics UPRM

3. Would magnetism would produce electricity? • Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current! Cruz-Pol, Electromagnetics UPRM

4. Electromagnetics was born! • This is the principle of motors, hydro-electric generators and transformers operation. This is what Oersted discovered accidentally: *Mention some examples of em waves Cruz-Pol, Electromagnetics UPRM

5. Cruz-Pol, Electromagnetics UPRM

6. Electromagnetic Spectrum Cruz-Pol, Electromagnetics UPRM

7. Some terms • E = electric field intensity [V/m] • D = electric field density • H = magnetic field intensity, [A/m] • B = magnetic field density, [Teslas] Cruz-Pol, Electromagnetics UPRM

8. Maxwell Equations in General Form Cruz-Pol, Electromagnetics UPRM

9. Maxwell’s Eqs. • Also the equation of continuity • Maxwell added the term to Ampere’s Law so that it not only works for static conditions but also for time-varying situations. • This added term is called the displacement current density, while J is the conduction current. Cruz-Pol, Electromagnetics UPRM

10. S1 S2 Maxwell put them together • And added Jd, the displacement current I L At low frequencies J>>Jd, but at radio frequencies both terms are comparable in magnitude. Cruz-Pol, Electromagnetics UPRM

11. Moving loop in static field • When a conducting loop is moving inside a magnet (static B field, in Teslas), the force on a charge is Cruz-Pol, Electromagnetics UPRM Encarta®

12. Who was NikolaTesla? • Find out what inventions he made • His relation to Thomas Edison • Why is he not well know? Cruz-Pol, Electromagnetics UPRM

13. Special case • Consider the case of a lossless medium • with no charges, i.e. . The wave equation can be derived from Maxwell equations as What is the solution for this differential equation? • The equation of a wave! Cruz-Pol, Electromagnetics UPRM

14. Phasors & complex #’s Working with harmonic fields is easier, but requires knowledge of phasor, let’s review • complex numbers and • phasors Cruz-Pol, Electromagnetics UPRM

15. COMPLEX NUMBERS: • Given a complex number z where Cruz-Pol, Electromagnetics UPRM

16. Review: • Addition, • Subtraction, • Multiplication, • Division, • Square Root, • Complex Conjugate Cruz-Pol, Electromagnetics UPRM

17. For a time varying phase • Real and imaginary parts are: Cruz-Pol, Electromagnetics UPRM

18. PHASORS • For a sinusoidal current equals the real part of • The complex term which results from dropping the time factor is called the phasor current, denoted by (s comes from sinusoidal) Cruz-Pol, Electromagnetics UPRM

19. To change back to time domain • The phasor is multiplied by the time factor, ejwt, and taken the real part. Cruz-Pol, Electromagnetics UPRM

20. Advantages of phasors • Timederivative is equivalent to multiplying its phasor by jw • Timeintegral is equivalent to dividing by the same term. Cruz-Pol, Electromagnetics UPRM

21. Time-Harmonic fields (sines and cosines) • The wave equation can be derived from Maxwell equations, indicating that the changes in the fields behave as a wave, called an electromagnetic field. • Since any periodic wave can be represented as a sum of sines and cosines (using Fourier), then we can deal only with harmonic fields to simplify the equations. Cruz-Pol, Electromagnetics UPRM

22. Maxwell Equations for Harmonic fields Cruz-Pol, Electromagnetics UPRM * (substituting and )

23. A wave • Start taking the curl of Faraday’s law • Then apply the vectorial identity • And you’re left with Cruz-Pol, Electromagnetics UPRM

24. A Wave • Let’s look at a special case for simplicity • without loosing generality: • The electric field has only an x-component • The field travels in z direction • Then we have Cruz-Pol, Electromagnetics UPRM

25. To change back to time domain • From phasor • …to time domain Cruz-Pol, Electromagnetics UPRM

26. Ejemplo 9.23 • In free space, • Find k, Jdand H using phasors and maxwellseqs. Cruz-Pol, Electromagnetics UPRM

27. Several Cases of Media • Free space • Lossless dielectric • Lossy dielectric • Good Conductor Recall: Permittivity eo=8.854 x 10-12[ F/m] Permeability mo= 4px 10-7 [H/m] Cruz-Pol, Electromagnetics UPRM

28. 1. Free space There are no losses, e.g. Let’s define • The phase of the wave • The angular frequency • Phase constant • The phase velocity of the wave • The period and wavelength • How does it moves? Cruz-Pol, Electromagnetics UPRM

29. 3. Lossy Dielectrics(General Case) • In general, we had • From this we obtain • So , for a known material and frequency, we can find g=a+jb Cruz-Pol, Electromagnetics UPRM

30. Intrinsic Impedance, h • If we divide E by H, we get units of ohms and the definition of the intrinsic impedance of a medium at a given frequency. *Not in-phase for a lossy medium Cruz-Pol, Electromagnetics UPRM

31. Note… • E and H areperpendicular to one another • Travel is perpendicular to the direction of propagation • The amplitude is related to the impedance • And so is the phase Cruz-Pol, Electromagnetics UPRM

32. Loss Tangent • If we divide the conduction current by the displacement current http://fipsgold.physik.uni-kl.de/software/java/polarisation Cruz-Pol, Electromagnetics UPRM

33. Relation between tanq and ec Cruz-Pol, Electromagnetics UPRM

34. 2. Lossless dielectric • Substituting in the general equations: Cruz-Pol, Electromagnetics UPRM

35. Review: 1. Free Space • Substituting in the general equations: Cruz-Pol, Electromagnetics UPRM

36. 4. Good Conductors • Substituting in the general equations: Is water a good conductor??? Cruz-Pol, Electromagnetics UPRM

37. Summary Cruz-Pol, Electromagnetics UPRM

38. Skin depth, d We know that a wave attenuates in a lossy medium until it vanishes, but how deep does it go? • Is defined as the depth at which the electric amplitude is decreased to 37% Cruz-Pol, Electromagnetics UPRM

39. Short Cut … • You can use Maxwell’s or use where k is the direction of propagation of the wave, i.e., the direction in which the EM wave is traveling (a unitary vector). Cruz-Pol, Electromagnetics UPRM

40. Waves • Static charges > static electric field, E • Steady current > static magnetic field, H • Static magnet > static magnetic field, H • Time-varying current > time varying E(t) & H(t) that are interdependent > electromagnetic wave • Time-varying magnet > time varying E(t) & H(t) that are interdependent > electromagnetic wave Cruz-Pol, Electromagnetics UPRM

41. EM waves don’t need a medium to propagate • Sound waves need a medium like air or water to propagate • EM wave don’t. They can travel in free space in the complete absence of matter. • Look at a “wind wave”; the energy moves, the plants stay at the same place. Cruz-Pol, Electromagnetics UPRM

42. Exercises: Wave Propagation in Lossless materials • A wave in a nonmagnetic material is given by Find: • direction of wave propagation, • wavelength in the material • phase velocity • Relative permittivity of material • Electric field phasor Answer: +y, up= 2x108 m/s, 1.26m, 2.25, Cruz-Pol, Electromagnetics UPRM

43. Power in a wave • A wave carries power and transmits it wherever it goes The power density per area carried by a wave is given by the Poynting vector. See Applet by Daniel Roth at http://fipsgold.physik.uni-kl.de/software/java/oszillator/index.html Cruz-Pol, Electromagnetics UPRM

44. Poynting Vector Derivation • Start with E dot Ampere’s • Apply vector identity • And end up with: Cruz-Pol, Electromagnetics UPRM

45. Rearrange Poynting Vector Derivation… • Substitute Faraday in 1rst term Cruz-Pol, Electromagnetics UPRM

46. Rate of change of stored energy in E or H Ohmic losses due to conduction current Total power across surface of volume Poynting Vector Derivation… • Taking the integral wrt volume • Applying Theorem of Divergence • Which means that the total power coming out of a volume is either due to the electric or magnetic field energy variations or is lost in ohmic losses. Cruz-Pol, Electromagnetics UPRM

47. Power: Poynting Vector • Waves carry energy and information • Poynting says that the net power flowing out of a given volume is = to the decrease in time in energy stored minus the conduction losses. Represents the instantaneous power density vector associated to the electromagnetic wave. Cruz-Pol, Electromagnetics UPRM

48. Time Average Power • The Poynting vector averaged in time is • For the general case wave: Cruz-Pol, Electromagnetics UPRM

49. Total Power in W The total power through a surface S is • Note that the units now are in Watts • Note that power nomenclature, P is not cursive. • Note that the dot product indicates that the surface area needs to be perpendicular to the Poynting vector so that all the power will go thru. (give example of receiver antenna) Cruz-Pol, Electromagnetics UPRM

50. Exercises: Power 1. At microwave frequencies, the power density considered safe for human exposure is 1 mW/cm2. A radar radiates a wave with an electric field amplitude E that decays with distance as |E(R)|=3000/R [V/m], where R is the distance in meters. What is the radius of the unsafe region? • Answer: 34.64 m 2. A 5GHz wave traveling In a nonmagnetic medium with er=9 is characterized by Determine the direction of wave travel and the average power density carried by the wave • Answer: Cruz-Pol, Electromagnetics UPRM