- (0 pts) Study the video lecture on cylindrical waveguides.
- (3 pts) Verify Eq. (8.26b) in Jackson, which appears in the video at 18:15, as explained there.
- (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.
- (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.]
- (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.
- (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.
- (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.
- (1 pts) Show that Eq. (8.28) holds for the TEM mode.
- (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.
- (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?
- (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.
- (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.]
- (2 pts) Using the results of step #12 and step #8, find the magnetic field configuration for the TEM mode in the coaxial cable.
- (2 pts) Sketch the E and B field configurations found in steps #12-13.
- (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)).
- (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).
- (0 pts) Convince yourself that Eq. (8.57-88) can be used to find the attenuation constant defined in Eq. (8.56).
- (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).
- (2 pts) Revise the solution of step #14 if the conductivities for the central wire and the outer cylinder are different.
- (10 pts) Solve Jackson problem 8.3 (only find \( P \) and \( \gamma \)).
Total: 49 points.