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Electromagnetics
Final examination
June 20, 2003
1. Please derive (from Maxwell’s eqs.) impedance Zo of free space and express it in
terms of the constants of free space  o and  o . Prove that E wave and H wave
are in phase in the free space. Also prove that the energy densities of E and H
waves are equal. (15%)
2. Please give the equations of D, B and J of HILS medium with constants of  ,  ,
and  . Give the Maxwell’s equations of this medium. Derive the wave equations
of E and H in this medium. (15%)
3. What is plane wave? Is a plane wave transverse? Why? Please prove that the plane
wave propagates in a conductive medium is transverse. (20%)
4. The Poynting vector is defined as S = E x H. For a wave propagating in a
conductive medium, whether the electromagnetic energy flows into the volume 
is equal to the increase in the electric and magnetic energies in the volume  per
unit time? Why? Please prove it and explain each term in the expression you
obtained. (15%)
5. Please prove that in good conductors (homogeneous, isotropic and linear), as long
as the currents satisfy Ohm’s law, the free charge density  f can only decrease.
Estimate the time constant of free charge decreasing for copper by using copper’s
conductivity  = 5.8 x 107 siemens/m and approximating  r as 1. (15%)
6. Inside a superconductor E = 0 and B = 0, under ideal conditions. This is true
(except for a thin region, of the order of a tenth of a micron thick, near the
surface), even if the material is cooled in a magnetic field. (a) Show that, just
outside, B is tangential to the surface. (b) Show that Jf = 0 inside. Therefore
currents must flow very close to the surface. Assume steady-state conditions and
set the magnetization M equal to zero. (c) Does there exist a relation between the
tangential B and the surface current density? (d) What can you say about the
magnetic forces? (e) Show that the magnetic flux linking a curve C which is
entirely situated inside a pure superconducting material must be constant. (f) What
happens if a superconducting ring is moved from a region where the magnetic
induction is B1 to another where it is B2? (20%)
7. Copper has a conductivity  of 5.80 x 107 mho/meter. Then, at 1 megahertz,
what is the Q value? (  r  1) Calculate the skin depth  ? Give the meaning of
skin depth? Derive to see whether it is equal to radian length  ? Calculate the
phase velocity? Finally, calculate its impedance? (20%)
8. A thin, air-insulated parallel-plate capacitor has circular plates of radius R,
separated by a distance s. A constant current I charges the plates through thin
wires along the axis of symmetry. (a) Find the value of E between the plates as a
function of the time. Assume a uniform E. Show the direction of E on a figure. (b)
Find the value of H between the plates by utilizing the displacement current. Show
also the direction of H on the figure of (a). (c) The magnetic field H of (b) is the
sum of two terms, Hw, related to the current in the wire, and Hp, related to the
current in the plates. The latter current deposits charges on the inside surface of
the plates. Find Hw and Hp. (d) Find E x H. (25%)