ECE 329
Homework 2
Due: Tuesday September 8, 2015, 5PM
• Homeworks are due Mondays on the due date indicated at 5 p.m.
• Each student must submit individual solution for each homework. You may discuss homework prob-
lems with other students registered in the course. If you use any source outside of class materials that
we've provided, you must cite every source that you use.
• Write your name, netID, and section on each homework.
• Homework are to be turned in to homework boxes on 3rd oor of ECEB. Please put your homework
into the appropriate box for the section you are registered in:
Section E - 1pm (Lynford Goddard): Box 29
Section X - 12 noon (Wei He): Box 30
Section P - 2pm (Dan Wasserman): Box 31
Section A - 9am (Arne Fliet): Box 32
Section C - 11am (Zhi-Pei Liang): Box 33
• Unstapled homeworks will not be accepted.
• For each homework, a problem will be randomly selected and graded in detail, and will count for 20
points. The rest of the problems will be worth 10 points each, and will be graded on eort:
10 points if excellent/good eort is made to solve the problem completely
5 points if only fair eort is made to solve the problem
0 points if it is clear that there was little or no eort to solve the problem.
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.
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1. Gauss' Law for displacement ux density D states that S D · dS = V ρdV over any closed surface
S enclosing a volume V in which electric charge density is specied by ρ(x, y, z) C/m3 . Note that
D = 0 E in free space.
¸
a) What is the displacement ux D · dS over the surface of a cube of volume V = L3 centered
on the origin, if ρ(x, y, z) = 4 C/m3 within V ?
b) Repeat (a) for ρ(x, y, z) = x C/m3 .
2. An innitely long, cylindrical wire of radius R = 1 cm is centered along the ẑ axis and carries a
uniform volumetric charge density ρ0 . As with a uniform, innitesimally thin line charge along ẑ
(see Example 2 in Lecture 2 online notes), the electric eld associated with this charge distribution is
everywhere oriented radially (with no azimuthal or ẑ components), owing to the cylindrical symmetry
of the charge distribution.
a) Given that Er = − 310 V/m at a distance r = 3 cm, use D = 0 E and Gauss' Law with a
cylindrical surface of radius r = 3 cm and length L to determine the charge density ρ0 of the
wire.
b) Calculate the strength of the electric eld E inside the wire, where r = 31 cm. Hint: How does
the total charge enclosed in a cylindrical Gaussian surface of radius r < R vary with r?
c) Show that the electric eld strength is continuous at the surface of the wire, where r = R, and
sketch the magnitude of the electric eld as a function of r for 0 < r < 3R.
3. Two unknown charges Q1 and Q2 are located along
the ŷ axis at (x, y, z) = (0, 1, 0) and at (x, y, z) =
´
(0, −1, 0), respectively. The displacement ux D · dS through the entire x̂ẑ plane at y = 0 in the
−ŷ direction is 8 C, while the displacement ux through the ŷẑ plane at x = 1 in the x̂ direction is 2
C.
a) Determine the magnitude of the two charges by solving a pair of algebraic equations relating the
given displacement uxes ´to the unknown charges. Hint: What are the respective contributions
of Q1 and Q2 to the ux yz−plane D · (x̂dydz)? See example 5 in Lecture 3 online.
b) Sketch the eld lines in the entire x̂ŷ plane using 1 eld line for each Coulomb of charge (i.e. if
the charge is +3 C, draw 3 eld lines initially emanating radially from the charge). Be sure that
your drawing is consistent with the given displacement ux of 8 C in the −ŷ direction at y = 0
and that very far away from the origin, the eld looks like the one produced by a single point
charge with charge Q1 + Q2 . Hint: Draw the eld lines in the far away region rst and then
connect the lines from near each charge to these distant eld lines and/or to the other charge.
Following is an example of two +8C charges:
4. Consider two innite parallel slabs having equal widths W and equal charge densities. Slab 1 is
parallel to the xy -plane and extends from z = −2W to z = −W while slab 2 extends from z = W to
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.
z = 2W . Both slabs have a negative uniform charge density −ρ0 C/m3 . The charge density is zero
everywhere else.
a) Determine and sketch the electric eld component Ez in terms of ρ0 over the region −3W < z <
3W by using shifted and scaled versions of the static eld conguration of a single charged slab
(See Example 4 in Lecture 3 online). Be sure to label both axes of your plot and mark the eld
value at each break point.
b) Verify that your eld from part (a) satises Gauss' Law in dierential form, ∇ · D = ρ in each
slab as well as in the free space regions.
5. Curl and divergence exercises:
a) On a 25-point graph consisting of x and y coordinates having integer values -2, -1, 0, 1, 2, sketch
the vector eld F = xx̂ + 2ŷ and calculate ∇ × F (curl of F) and ∇ · F (divergence of F).
b) Repeat (a) for F = yx̂ + 2ŷ .
c) (Select one): If ∇ × F 6= 0, the eld strength varies along or across the direction of the eld.
d) (Select one): If ∇ · F 6= 0, the eld strength varies along or across the direction of the eld.
6. Given that E = cos(y)x̂ + sin(y)ŷ V/m in free space,
a) determine ∇ × E and ∇ × ∇ × E
b) determine ρ such that Gauss' Law is satised
7. Given an electrostatic potential V (x, y, z) = 3z 2 − 2 V in a certain region of free space, determine the
corresponding
a)
b)
c)
d)
electrostatic eld E = −∇V
curl ∇ × E
divergence ∇ · E
charge density ρ in the region
8. Bonus Question:
a) On a 25-point graph consisting of x and y coordinates having integer values -2, -1, 0, 1, 2, sketch
Q
the vector eld, v = 4πr
2 r̂ where r is the distance from the origin. You can exclude the origin
from your sketch.
b) Show that ∇ · v = Qδ(r) and that ∇ × v = 0.
c) In a few sentences, give both a physical interpretation and a mathematical interpretation of the
results from parts (a) and (b). Hint: think about Gauss' Law for displacement ux density D.
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.