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EE 543 Theory and Principles of Remote Sensing

EE 543 Theory and Principles of Remote Sensing. Topic 3 - Basic EM Theory and Plane Waves. Outline. EM Theory Concepts Maxwell’s Equations Notation Differential Form Integral Form Phasor Form Wave Equation and Solution (lossless, unbounded, homogeneous medium)

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EE 543 Theory and Principles of Remote Sensing

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  1. EE 543Theory and Principles of Remote Sensing Topic 3 - Basic EM Theory and Plane Waves

  2. Outline • EM Theory Concepts • Maxwell’s Equations • Notation • Differential Form • Integral Form • Phasor Form • Wave Equation and Solution (lossless, unbounded, homogeneous medium) • Derivation of Wave Equation • Solution to the Wave Equation – Separation of Variables • Plane waves O. Kilic EE543

  3. E, H J EM Theory Concept The fundamental concept of em theory is that a current at a point in space is capable of inducing potential and hence currents at another point far away. O. Kilic EE543

  4. Introduction to EM Theory • In remote sensing we are interested in the interactions of em waves with the medium and target of interest. • The existence of propagating em waves can be predicted as a direct consequence of Maxwell’s equations. • These equations satisfy the relationship between the vector electric field, E and vector magnetic field, H in time and space in a given medium. • Both E and H are vector functions of space and time; i.e. E (x,y,z;t), H (x,y,z;t.) O. Kilic EE543

  5. What is an Electromagnetic Field? • The electric and magnetic fields were originally introduced by means of the force equation. • In Coulomb’s experiments forces acting between localized charges were observed. • There, it is found useful to introduce E as the force per unit charge. • Similarly, in Ampere’s experiments the mutual forces of current carrying loops were studied. • B is defined as force per unit current. O. Kilic EE543

  6. Why not use just force? • Although E and B appear as convenient replacements for forces produced by distributions of charge and current, they have other important aspects. • First, their introduction decouples conceptually the sources from the test bodies experiencing em forces. • If the fields E and B from two source distributions are the same at a given point in space, the force acting on a test charge will be the same regardless of how different the sources are. • This gives E and B meaning in their own right. • Also, em fields can exist in regions of space where there are no sources. O. Kilic EE543

  7. Maxwell’s Equations • Maxwell's equations give expressions for electric and magnetic fields everywhere in space provided that all charge and current sources are defined. • They represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. • These set of equations describe the relationship between the electric and magnetic fields and sources in the medium. • Because of their concise statement, they embody a high level of mathematical sophistication. O. Kilic EE543

  8. Notation: (Time and Position Dependent Field Vectors) O. Kilic EE543

  9. Notation: Sources and Medium O. Kilic EE543

  10. Two Forms of Maxwell’s Equations • Differential form • This is the most widely used form. • Theydescribe the relationship between the electric and magnetic fields and sources in the medium at a point in space. • Integral form • Integral form of Maxwell’s equations can be derived from the differential form by using Stoke’s theorem and Divergence theorem. • These set of equations describe the field vector relations over an extended region in space. • They have limited use. Typically, they are applied to solve em boundary value problems with symmetry. O. Kilic EE543

  11. Maxwell’s Equations – Physical Laws • Faraday’s Law Changes in magnetic field induce voltage. • Ampere’s Law  Allows us to write all the possible ways that electric currents can make magnetic field. Magnetic field in space around an electric current is proportional to the current source. • Gauss’ Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. • Gauss’ Law for Magnetism The net magnetic flux out of any closed surface is zero. O. Kilic EE543

  12. Differential Form Faraday’s Law: (1) Ampere’s Law: (2) Gauss’ Law: (3) (4) O. Kilic EE543

  13. Some Observations (1) How many scalar equations are there in Maxwell’s equations?Answer = 8 ?? Three scalar equations for each curl (3x2 = 6) 1 scalar equation for each divergence (1x2 = 2). O. Kilic EE543

  14. Some Observations (2) But, the divergence equations are related to the curl equations. This is known as the conservation of charge. O. Kilic EE543

  15. Conservation of Charge If we take the divergence of both sides of eqn. 2 and use the vector identity: (5) Using eqn. (3) in eqn. (5), we obtain the continuity law for current: (6) Rate of decrease of charge in the volume Flow of current out of a differential volume O. Kilic EE543

  16. Some Observations (3) Thus, the divergence equations(eqn. 3, 4)are dependent on the curl equations(eqn. 1, 2). So, Maxwell’s equations represent6 independent equations. O. Kilic EE543

  17. Some Observations (4) How many variables are there in Maxwell’s equations? Answer:12, three for each component of E, H, D, and B vectors. Therefore, the set of Maxwell’s equations is not sufficient to solve for the unknowns. We need 12-6 = 6 more scalar equations. These are known as constitutive relations. O. Kilic EE543

  18. Constitutive Relations Constitutive relations provide information about the environment in which electromagnetic fields occur; e.g. free space, water, etc. permittivity (7) permeability (8) Free space values. O. Kilic EE543

  19. Duality Principle • Note the symmetry in Maxwell’s equations O. Kilic EE543

  20. Integral Form of Maxwell’s Equations Using Stoke’s theorem and Divergence theorem: (9) HW #2.1 Prove by applying Stoke’s and Divergence theorems to Maxwell’s eqn in differential form O. Kilic EE543

  21. Time Harmonic Representation - Phasor Form • In a source free ( ) and lossless ( ) medium characterized by permeability m and permittivity e, Maxwell’s equations can be written as: (10) O. Kilic EE543

  22. Time Harmonic Fields • We will now assume time harmonic fields; i.e. fields at a single frequency. We will assume that all field vectors vary sinusoidally with time, at an angular frequency w; i.e. In other words: (11) Note that the E and H vectors are now complex O. Kilic EE543

  23. Time Harmonic Fields (2) • The time derivative in Maxwell’s equations becomes a factor of jw: O. Kilic EE543

  24. Phasor Form of Maxwell’s Equations Maxwell’s equations can then be written in phasor form as: Phasor form is dependent on position only. Time dependence is removed. O. Kilic EE543

  25. The Wave Equation (1) If we take the curl of Maxwell’s first equation: Using the vector identity: And assuming a source free, i.e. and lossless; i.e. medium: O. Kilic EE543

  26. The Wave Equation (2) Define k, which will be known as wave number: O. Kilic EE543

  27. Wave Equation in Cartesian Coordinates where O. Kilic EE543

  28. Scalar Form of Wave Equation For each component of the E vector, the wave equation is in the form of: Denotes different components of E in Cartesian coordinates O. Kilic EE543

  29. Solution to the Wave Equation – Separation of Variables Assume that a solution can be written such that O. Kilic EE543

  30. Separation of Variables This decomposition is arbitrarily defined at this point. Will depend on the medium and boundary conditions. Determines how the wave propagates along each direction. O. Kilic EE543

  31. Separation of Variables + O. Kilic EE543

  32. Possible Solutions to the Wave Equation Energy is transported from one point to the other Standing wave solutions are appropriate for bounded propagation such as wave guides. When waves travel in unbounded medium, traveling wave solution is more appropriate. HW 2.2: Show that the above are solutions to the wave equation by plugging the solution on the differential eqn on the previous page O. Kilic EE543

  33. The Traveling Wave • The phasor form of the fields is a mathematical representation. • The measurable fields are represented in the time domain. Let the solution to the a-component of the electric field be: Then Traveling in +x direction O. Kilic EE543

  34. Traveling Wave As time increases, the wave moves along +x direction O. Kilic EE543

  35. Standing Wave Then, in time domain: O. Kilic EE543

  36. Standing Wave Stationary nulls and peaks in space as time passes. O. Kilic EE543

  37. To summarize • We have shown that Maxwell’s equations describe how em energy travels in a medium • The E and H fields satisfy the “wave equation”. • The solution to the wave equation can be in various forms, depending on the medium characteristics O. Kilic EE543

  38. The Plane Wave Concept • Plane waves constitute a special set of E and H field components such that E and H are always perpendicular to each other and to the direction of propagation. • A special case of plane waves is uniform plane waves where E and H have a constant magnitude in the plane that contains them. O. Kilic EE543

  39. Example 1 (1/5) Assume that the E field lies along the x-axis and is traveling along the z-direction. wave number We derive the solution for the H field from the E field using Maxwell’s equation #1: Intrinsic impedance; 377 W for free space O. Kilic EE543

  40. Example 1 (2 of 5) Thus the wave equation (Page 26) simplifies to: Where as before O. Kilic EE543

  41. z y E, H plane x Example 1 (3 of 5) direction of propagation E and H fields are not functions of x and y, because they lie on x-y plane O. Kilic EE543

  42. Example 1 (4 of 5) phase term *** The constant phase term j is the angle of the complex number Eo O. Kilic EE543

  43. Example 1 (5 of 5) Wavelength: period in space kl = 2p O. Kilic EE543

  44. Velocity of Propagation (1/3) • We observe that the fields progress with time. • Imagine that we ride along with the wave. • At what velocity shall we move in order to keep up with the wave??? O. Kilic EE543

  45. Velocity of Propagation (2/3) E field as a function of different times Constant phase points kz O. Kilic EE543

  46. Velocity of Propagation (3/3) In free space: Note that the velocity is independent of the frequency of the wave, but a function of the medium properties. O. Kilic EE543

  47. Example 2 A uniform em wave is traveling at an angle q with respect to the z-axis. The E field is in the y-direction. What is the direction of the H field? O. Kilic EE543

  48. x E q z y Solution: Example 2 The E field is along the unit vector: The direction of propagation is along y. Because E, H and the direction of propagation are perpendicular to each other, H lies on x-z plane. It should be in the direction parallel to: O. Kilic EE543

  49. Plane Wave Characteristics amplitude frequency phase polarization Wave number, depends on the medium characteristics Direction of propagation O. Kilic EE543

  50. Example 3 Write the expression for an x-polarized electric field that propagates in +z direction at a frequency of 3 GHz in free space with unit amplitude and 60o phase. + z-direction = 2p*3*109 x =1 60o O. Kilic EE543

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