Tutorial: Cylindrical waveguide and the TEM mode

  1. (0 pts) Study the video lecture on cylindrical waveguides.
  2. (3 pts) Verify Eq. (8.26b) in Jackson, which appears in the video at 18:15, as explained there.
  3. (3 pts) Verify the expressions for the transverse fields of a TE mode, which appear at 31:18 in the video. Compare with (8.31-33) in Jackson.
  4. (1 pts) As mentioned in the video, a waveguide may have a special TEM mode in which the axial components of \( \mathbf{E} \) and \( \mathbf{B} \) are both zero while \( k=\sqrt{\epsilon\mu}\omega \). Verify that this does not contradict the expressions for the transverse fields (step #3). [You will find out in the following steps that a TEM mode can only exist in a waveguide whose cross-section is not simply connected.]
  5. (2 pts) Use Eq. (8.23-25) [also see 13:15 in the video] to show that the purely transverse \( \mathbf{E} \) and \( \mathbf{B} \) fields for the TEM mode have zero two-dimensional curl and divergence.
  6. (2 pts) Using the result of step #4, show that \( \mathbf{E} \) and \( \mathbf{B} \) for the TEM mode satisfy the two-dimensional Laplace equation.
  7. (1 pts) Use (8.19) to show that the result of step #5 requires that \( k=\sqrt{\epsilon\mu}\omega \), in agreement with step #4. Note that this is the same linear dispersion law that a plane wave has in infinite medium (without any metallic walls), independent of the shape of the waveguide’s cross-section.
  8. (1 pts) Show that Eq. (8.28) holds for the TEM mode.
  9. (3 pts) Consider the 2D Laplace equation for the \( \mathbf{E} \) field of a TEM mode. Assume that the cross-section is simply connected, i.e., its boundary is a single closed contour. Using the appropriate boundary condition for \( \mathbf{E} \), explain why the \( \mathbf{E} \) field [i.e., the factor \( \mathbf{E}(x,y) \) in \( \mathbf{E}(x,y) e^{ikz-iωt} \)] is the same as the solution of an electrostatic problem inside the waveguide with a constant scalar potential on its wall.
  10. (2 pts) Argue why the solution of the auxiliary electrostatic problem in step #9 must vanish identically. What is the conclusion about the TEM mode in a waveguide with a simply connected cross-section?
  11. (2 pts) Explain why the arguments of step #8 do not apply to a waveguide with a doubly connected boundary, such as a coaxial cable.
  12. (5 pts) By solving the electrostatic problem of step #9, find the electric field configuration for the TEM mode in a perfectly conducting coaxial cable, using the notation of problem 8.2 from Jackson. [First, establish the boundary conditions, then use Gauss’s law.]
  13. (2 pts) Using the results of step #12 and step #8, find the magnetic field configuration for the TEM mode in the coaxial cable.
  14. (2 pts) Sketch the E and B field configurations found in steps #12-13.
  15. (3 pts) Integrate the axial component of the Poynting vector to find the time-averaged power flow along this coaxial cable (this is problem 8.2 part (a)).
  16. (0 pts) Review the discussion of the power loss due to incomplete reflection at an imperfectly conducting metallic surface (Eq. 8.12 or 8.15).
  17. (0 pts) Convince yourself that Eq. (8.57-88) can be used to find the attenuation constant defined in Eq. (8.56).
  18. (5 pts) Use the result of step #13 to find the attenuation constant for the TEM mode in a coaxial wire (see steps #10-11). This is Jackson problem 8.2(b).
  19. (2 pts) Revise the solution of step #14 if the conductivities for the central wire and the outer cylinder are different.
  20. (10 pts) Solve Jackson problem 8.3 (only find \( P \) and \( \gamma \)).

Total: 49 points.