Physics 743 Fall 2000

Michael Fowler

Homework # 3

 

  1. (Jackson 5.29) A transmission line consists of two parallel perfect conductors of arbitrary but constant cross section. Current flows down one conductor and returns via the other. Show the product of inductance per unit length L and capacitance per unit length C is equal to me for the medium surrounding the conductors.

 

  1. Suppose you have a cylindrical solenoid of circular cross section that is so long that in analyzing the field near one end you can take it to be semi-infinite.  First, use the principle of superposition (adding another such solenoid) to show that the total magnetic flux out of the end of the solenoid is just half of the flux deep inside the solenoid.  Sketch the field. For a field line at distance d from the axis as it exits the solenoid, find how far it was from the axis deep inside the solenoid.  Now find an exact expression for the field by summing over the loops that make up the solenoid.  Next, suppose a small paramagnetic sample is constrained to move along the axis of the solenoid. At which point (how far from the end of the solenoid) does it experience the greatest force?

 

 

  1.   Draw the field lines for both electric and magnetic fields for (a) TEM radiation in a coaxial cable (b) the lowest TM mode in a rectangular waveguide (c) the lowest TE mode in a rectangular waveguide.

 

  1. (Jackson, 8.6.) A resonant cavity of copper consists of a hollow, right circular cylinder of inner radius R and length L, with flat end faces. 

 

(a)   Determine the resonant frequency of the cavity for all types of waves.  With (1/Ö(me)R) as a unit of frequency, plot the lowest four resonant frequencies of each type as a function of R/L for 0 < R/L < 2.  Does the same mode have the lowest frequency for all R/L? 

(b)   If R = 2cm, L = 3cm, and the cavity is made of pure copper, what is the numerical value of Q for the lowest resonant mode?

 

  1. You know that a magnetic field can never do work on a charge, since the force is always perpendicular to the motion of the charge. Yet if you use a magnet to pick up a pin, work is done, it seems by the magnet. Explain.