1. EEE 431 Computational Methods in Electrodynamics Lecture 2 By Rasime Uyguroglu

2. Boundary Conditions Maxwell’s Equations are partial differential equations, Boundary conditions are needed to obtain a unique solution, Maxwell’s differential equations do not apply on the boundaries because the fields are discontinuous Our target is to determine the electric and magnetic fields in a certain region of space due to excitations satisfying the problem’s boundary conditions

3. Finite Conductivity Case

4. Finite Conductivity Case (Cont.) Applying Faraday’s Law we get As the RHS vanishes and we get; Or

5. Finite Conductivity Case (Cont.) It follows that the tangential component of the electric field is continuous.

6. Finite Conductivity Case (Cont.) Similarly, starting with the modified Ampere’s law (no current J at the interface), we get: Or

7. Finite Conductivity Case (Cont.) It follows that the tangential component of the magnetic field intensity is continuous if there are no boundary electric currents.

8. Assuming there are no surface charges, Gauss’s law gives It follows that

9. Finite Conductivity Case (Cont.) Or alternatively But as,

10. Finite Conductivity Case (Cont.) Normal components of the electric field are discontinuous across the interface.

11. Finite Conductivity Case (Cont.) Similarly, by applying Gauss’s law for magnetic fields we get It follows that

12. Finite Conductivity Case (Cont.) But as Normal components of the magnetic fields are discontinuous.

13. Finite Conductivity Case (cont.) no interface surface magnetic currents no interface surface electric currents

14. Finite Conductivity Case (cont.) no interface surface electric charges, when perfect dielectric materials are assumed (zero conductivity): no interface magnetic surface charges

15. Boundary Conditions with Sources Boundary conditions must be changed to take into account the existence of surface currents and surface charges.

16. Boundary Conditions with Sources Applying the modified Ampere’s law we get: As

17. Boundary Conditions with Sources is the surface current density in (A/m)

18. Boundary Conditions with Sources We see that: Tangential components of the magnetic field intensity are discontinuous if surface electric current density exists.

19. Boundary Conditions with Sources (Cont.) If medium 2 is a perfect conductor: Similarly, starting with Faraday’s Law:

20. Boundary Conditions with Sources (Cont.) For a perfect conductor

21. Boundary Conditions with Sources (Cont.) Applying Gauss’s law for the shown cylinder we have

22. Boundary Conditions with Sources (Cont.) Gauss’s Law:

23. Summary of the Boundary Conditions We have,

24. Time Harmonic Electromagnetic Fields If sources are sinusoidal and the medium is linear then the fields everywhere are sinusoidal as well. The field at each point is characterized by its amplitude and phase (Phasor).

25. Time Harmonic Electromagnetic Fields (Cont.) Example

26. Time Harmonic Electromagnetic Fields (Cont.) Similarly for all field quantities we can write:

27. Time Varying Electromagnetic Fields (Cont.) Maxwell’s equations for the time- harmonic case are obtained by replacing each time vector by its corresponding phasor vector and replacing

28. Time Harmonic Electromagnetic Fields (Cont.) Maxwell’s equation’s in time form:

29. Time Harmonic Electromagnetic Fields (Cont.) Maxwell’s Equation’s in phasor forms:

30. Time Harmonic Electromagnetic Fields (Cont.) Apply the same boundary conditions.

31. Wave Equation’s Maxwell’s Equation’s are coupled first order differential equations which are difficult to apply when solving boundary value problems. Wave equation is decoupled second order differential equation which is useful for solving problems.

32. Wave Equations For a linear, isotropic, homogeneous, source free medium:

33. Wave Equation We obtain the wave equation for H:

34. Wave Equation (Cont.) Each of the vector equations has three components: Each component of the wave equations has the form in free space:

35. Wave Equation (Cont.) Wave equation in phasor form: