THE UNIVERSITY OF BRITISH COLUMBIA
 
Physics 401 Assignment # 9:
 
LOOSE ENDS & COAX CABLES
 
Wed. 8 Mar. 2006 - finish by Wed. 15 Mar.

1. Cutoff Frequencies: Explain in words why there is a lower limit on the frequencies of EM waves that will propagate freely either through a tenuous plasma or down a rectangular waveguide. Why is there no cutoff frequency (neither upper nor lower) for wave propagation down an ideal coaxial cable?

2. Rectangular Waveguide with Dielectric: Show that if a hollow rectangular waveguide of the type shown in Griffiths' Figure 9.24 is completely filled with a dielectric of permeability $\epsilon$, its cut-off frequency is lower than if it were empty, by a factor of $\sqrt{\epsz/\epsilon}$:
   
   $${\omega_{mn}^{\rm dielectric} \over \omega_{mn}^{\rm vacuum}} = \sqrt{\epsz \over \epsilon} \; .$$
   
   
   Thus, for a given operating frequency, a dielectric filled waveguide can be smaller than an empty one.

3. (p. 412, Problem 9.31) - Coax Cable:¹
   1. Show directly that Eqs. (9.197) satisfy MAXWELL'S EQUATIONS (9.177) and the boundary conditions (9.175).
   2. Find the net charge per unit length, $\lambda(z,t)$, and the net current, I(z,t), on the inner conductor.

4. Coax Impedance:

   

   In class, we derived the electric and magnetic fields in a coaxial transmission line. From those we deduced the characteristic impedance of a coaxial cable:

   ${\displaystyle Z = {V(z,t) \over I(z,t)} = {\ln (b/a) \over 2\pi} \sqrt{\mu \over \epsilon} = 60\;\Omega \cdot \ln (b/a) }$

   where a is the radius of the inner coax line and b is the radius of the outer coax cylinder, as shown.

   In general, the characteristic impedance of a transmission line is given by

   ${\displaystyle Z = \sqrt{ {\cal L} \over {\cal C} } \; , }$

   where ${\cal L}$ and ${\cal C}$ are the inductance and capacitance per unit length, respectively.

   Show that the characteristic impedance of this coax line satisfies this definition by calculating ${\cal L}$ and ${\cal C}$ explicitly, and then Z.