ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 5 Solutions 1. Spherical shell with frozen-in polarization P = kr r . First, lets nd the bound charge densities,
b = P n =
ka, inner surface kb, outer surface 1d 2 r Pr = 3k r2 d
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 7 Solutions 1. Average potential over a sphere with point charge q outside. Consider the geometry of the problem shown below:
The average potential over the spherical surface is,
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 8 Solutions 1. Magnetic elds of current distributions. (a) The hairpin turn can be considered as half the superposition of two innite staight-line currents that border a loop of c
ECE 3209 - Electromagnetic Fields University of Virginia
Fall 2009 Homework # 9 Solutions 1. Vector potential of small rectangular loop. Since the loop is small (compared to the distance to point P ), we can approximate the vector potential due to each si
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 10 Solutions 1. Cheng, P. 6-39. Mutual inductance between a long wire and loop. The B -eld due to a long straight wire (from Amperes Law) is,
B (r ) =
Thus, the ux through the loo
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 As you study and learn about the dierent specialized areas of electrical engineering, many of you might notice that only a few of your courses make explicit reference to electromagnetic elds
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 1 Review of Vector Analysis Due: Friday, September 4
1. Cheng P.2-5 2. Cheng P.2-8 3. The height of a certain hill (in feet) is given by
h(x, y ) = 10(2xy 3x2 4y 2 18x + 28y + 12)
ECE 3209 - Electromagnetic Fields University of Virginia Fall 2009
Homework # 2 - Vector Calculus and Electric Fields Due: Friday, September 11
1. Verify Gauss's Theorem for the function,
^ ^ F = r2 sin ^ + 4r2 cos + r2 tan r
using the volume of the "ice
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 3 Electric Fields and Scalar Potential Due: Friday, September 18
1. Cheng, P. 3-11. 2. Cheng, P. 3-15. 3. Two spheres, each of radius R and carrying uniform charge densities + and
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 4 Conductors, Polarization, and Dielectrics Due: Friday, October 2
1. Consider two perfectly conducting spheres of radii a and b, respectively, that are connected by a long, thin
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 5 Dielectrics, Capacitance, and Energy Due: Friday, October 9
1. A thick spherical shell (inner radius a, outer radius b) is lled with dielectric material with a frozen in polariz
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 6 Laplaces Equation and the Method of Images Due: Friday, October 16
1. Cheng, P.4-7 2. Cheng, P.4-13 3. Cheng P.4-23 Two innite insulated conducting planes maintained at potentia
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 7 Steady Electric Currents and Ohms Law Due: Friday, October 23
1. In class, we noted that the solution to Laplaces equation at any particular point is the average of the potentia
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 8 Magnetic Fields Due: Friday, November 6
1. Using the Biot-Savart Law, nd the magnetic ux density (B ) at point P for each of the steady current congurations given below: (a) Che
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 9 the Magnetic Vector Potential and Magnetization Due: Friday, November 13
1. Cheng P. 6-19. For a small rectangular loop with sides a and b that carries current I , (a) Find the
ECE 3209 Electromagnetic Fields University of Virginia Fall 2009
Homework # 10 Faradays Law, Maxwells Equations, and Plane Waves Due: Friday, December 4
1. Cheng, P. 6-39. Find the mutual inductance between a very long straight wire and a conducting loop
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 6 Solutions 1. Cheng P.4-7. For a point charge Q placed a distance d above a large grounded conducting plane, we found the potential in class to be,
(x, y, z ) =
Q 4 0
1 x2 + y2 +
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 4 Solutions
1. Two spherical conductors separated by a long wire.
If the spheres are separated by enough distance, then the charge distribution on one will not inuence that on the
ECE 3209 Electromagnetic Fields University of Virginia
Fall 2009 Homework # 3 Solutions 1. Cheng, P.3-11 Consider a spherical distribution of charge,
= 0 1
R2 b2
in the region 0 R b, surrounded by a conducting shell with inner radius Ri and outer radius
ECE 309 Electromagnetic Fields University of Virginia
Fall 2008 Homework # 8 Solutions 1. Magnetized cylinder. Take the cylinder (and direction of magnetization) to be along the z -axis,
Thus, the bound current densities are
Jb =
M = 0,
Kb = M n = M0 (
ECE 309 Electromagnetic Fields University of Virginia
Fall 2008 Homework # 9 Solutions 1. From either Gausss Law or Laplaces equation, the eld between the inner and outer conductors of a cylindrical capacitor is,
E=
V r r ln(b/a)
where V is the voltage ac
ECE 309 Electromagnetic Fields University of Virginia
Fall 2008 Homework # 10 Solutions 1. Propagation in Moist Earth With the given constitutive parameters, = 0.01 S/m, r = 30, and r = 1, we rst check to see if moist earth can be considered a good conduc
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 1 Review of Vector Analysis and Electric Fields Due: Wednesday, September 3
Note: The rst six problems of this homework are meant as a review of vector analysis and multivariable c
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 2 Electric Fields and Gausss Law Due: Wednesday, September 10
1. Find the force between a charged circular disc of radius R0 and uniform charge density and a point charge Q located
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 3 The Scalar Potential Due: Wednesday, September 17
1. Find the potential, , at a distance z above the center of the charge distributions shown below:
(a)
(b)
(c)
In each case, nd
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 4 Energy, Conductors, and Polarization Due: Wednesday, September 24
1. Find the energy stored in a uniformly charged solid sphere of radius R and total charge q . Find the energy i
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 5 Dielectrics and Capacitors Due: Wednesday, October 1
1. A thick spherical shell (inner radius a, outer radius b) is lled with dielectric material with a frozen in polarization, P
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 6 Laplaces Equation, the Method of Images, Current, and Magnetic Fields Due: Wednesday, October 22
1. Find the average potential over a spherical surface of radius R due to a point
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 7 Magnetic Fields and the Vector Potential Due: Wednesday, October 29
1. A thin conducting wire is bent into the shape of a regular polygon of N sides. A current I ows in the wire.
ECE 309 Electromagnetic Fields University of Virginia Fall 2008
Homework # 8 Magnetization and Faradays Law Due: Friday, November 7
1. An innitely long circular cylinder carries a uniform magnetization of M parallel to its axis. Find the bound current den