3 16. An alternating voltage was connected between the plates of a parallel plate capacitor. The
resulting electric field between the plates 15 given by
E = 10 cosan a2
(a) If the medium between the plates 15 air (so) determine the total current crossing
808. F orce and torque on a static electric dipole. An electric dipole has moment p = p2.
Find the force and torque on the dipole for the following conditions:
(a) It is in a eld E = E02 and is located on the z-axis at z = R .
(b) It is in a E = E02 and i
3 1. The electric field E = 322 y cos(108 t) ax is applied to a dielectric material (Lucit) of e, = 2.56.
Determine the following:
(a) The polarization P
(b) The induced polarization charge density p,. Explain physically the reason for the zero
value of p
3 , 3. An N turn toroid of rectangular cross section is shown in Figure P3.3. The core consists of
three regions: region 1, an iron core of relative permeability ii,1 = 3000; region 2, an in
#3 = #0 ' [Jo '
#2 = 11,2 #0
#1 =m1 no
$06. Conductivity of silicon. Review what you know about the conductivity of silicon,
perhaps from EE306.
(a) What charged species are present in pure silicon? What are their concentrations [cm'
3] and mobilities [cmZV'lsl]? Convert to SI units [H13] and
J , 7. In Figure P3.7, a cylindrical conductor of radius a carries a direct current that exponentially
varies throughout the cross section of the conductor. If the current density is given by
' J = Keg-(15) a2 A/m2
and if the conductor is made of magnetic
3_ 12. Determine the frequency range for which the conduction current exceeds the displacement
current by a factor of at least 100 in sea water (0- =_ 4 S/m, 5.1., = 1, and e, = 81).
f S 23100505.-
3, 5. Consider the two solid concentric cylinders shown in Figure P3.5. The current density in the
inside cylinder is given by '
J1= 0.521 A/mz (p s a)
whereas the current density in the outside cylinder is given by
J2 = a-(az) A/m2 a < p s b
3, 6. The cylindrical surface p = a is charged with a surface charge density p: = 0.2 X 106 C/mz.
The electrical properties of the materials inside 9 s a and outside p > a the cylindrical
"surface are given by
INSIDE THE CYLINDER (p S a) OUTSIDE THE CYLIN
SOl. Maxwells equations. Write out from memory Maxwells equations in both integral
and differential form, and also in words. Make sketches to illustrate your verbal
description. Include all the conventions you need, such as the relation between surface
3 _ 21. A magnetic eld given in the spherical coordinates by
. 1 ' _
H = (r2 Slna, + F cosG cosdaae + r2 smBm)
is applied to the following magnetic materials:
Aluminum u, = 1.000021
Cobalt u, = 250
Nickel p, = 600
Iron is, = 2 X 105
Determine the induced
'5. 10. The electric and magnetic fields inside a rectangular wave guide made of apyroximately
perfectly conducting material are given by
A . a . 17x
E = ]wuHO Sinay
IT a .
. . a . 17x Trx
H = jBHO SlIlax + H0 cosa:
w a a
where u), u, 1-10, and [3 are all
3,8 (b) A spherical conductor of radius a is charged with a total positive charge Q. If the
conductor is coated with two different dielectric materials of radii r1 and r2 as shown
in Figure P3.8, determine
(i) The electric ux density D, the electric field
S02. Force on a currentcarrying rod at rest in a uniform magnetic eld. The Lorentz
force law in continuum form is fem = pVE + J x B , giving the force per unit volume.
Starting with this, nd the the total force F6m on a straight metal wire of length L and
.3 , 9. The interface between regions 1 and 2 is charged with a surface charge density p3 = 0.2 C/mz.
Region 1 (z > 0) is air, whereas region 2 (z < 0) is a material with e; = 260 and 5L2 = 3.15.140.
If the electric ux density in region 1 is given by D1 =
3 , 2. A coaxial power cable has a core (conductor) of radius a. The region between the inner and
outer conductors is filled with two concentric layers of dielectrics 61 = 1-5o and 52 = 4.560
as shown in Figure P32. If the outer conductor is grounded whil
803. Conductivity ofsalt waterionic mobility. A mole of any substance is the number
of grams of the substance equal to its molecular number, i.e. the total number of neutrons
and protons in one molecule of the substance (approximately, with some minor
3.20. A nonuniform time-varying electric field given in the cylindrical coordinates by
is applied to the following homogeneous, isotropic dielectric materials:
E = (3p2 cotdmp + 34,) sin3 X 1081 Vim I i
Teon p. = no, 6 = 2.120, and o = 0
Figure P7.27 The double-stub matching device available in the labora-
tory. Determine which position, 1 or 2, you would select to locate the
second stub to achieve matching.
28. TWO transmission lines are connected