ECE 329
Homework 6
Due: Tuesday October 6, 2015, Noon
• Homeworks are due Tuesdays at noon.
• Each student must submit individual solution for each homework. You may discuss homework problems 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 floor 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 Fliflet): 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 effort:
– 10 points if excellent/good effort is made to solve the problem completely
– 5 points if only fair effort is made to solve the problem
– 0 points if it is clear that there was little or no effort to solve the problem.
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.
1. An infinite current sheet with uniform current density Js produces a magnetostatic B that has a
magnitude µ02Js Wb/m2 in free space. The field is oppositely directed on either side of the sheet in
accordance with the right hand rule and the Biot-Savart Law. Use the principle of superposition of
this result to determine the magnetic field intensity H = µB0 A/m at the origin for the following
current sheet configurations:
a) A pair of current sheets in the y = 1 and y = −1 planes with equal current densities Js = 2ẑ
A/m.
b) The same current sheets but with equal and opposite current densities where Js = −2ẑ A/m on
the y = −1 plane.
c) A pair of current sheets in the z = 1 and z = −1 planes with equal current densities Js = 2x̂+4ŷ
A/m.
2. An electricially neutral slab (having equal densities of positive and negative charge carriers) of thickness W extends infinitely in the x̂ and ŷ dimensions in the region |z| < W
2 m. The positive and
negative charges inside the slab move in opposite directions along ŷ such that the slab conducts a
uniform current density of J = −2ŷ A/m2 . Outside the slab, charge and current densities are zero.
a) Using the right-hand rule and the Biot-Savart Law, discuss why the current slab should generate
equal and oppositely directed magnetic fields in the ±x̂ directions above and below the slab,
respectively.
b) Based on part (a), what is B on the z = 0 m plane?
c) Make use of Ampere’s Law in integral form, along with the results of parts (a) and (b) above,
to find Bx (z) above the slab. Note that it should be possible to evaluate the line integral of
B = Bx x̂ around a suitably defined rectangle just using simple multiplications.
d) Use Ampere’s Law in integral form to find Bx (z) at a distance z within the current slab (i.e.,
0 < z < W/2).
e) Plot Bx (z) as a function of z over the interval −W < z < W
H
3. Gauss’ Law for B states that S B · dS = 0 over any closed surface S enclosing a volume V. Given
2
that B = B0 (−3x̂ + 2ŷ − πR
L ẑ) Wb/m , determine the magnetic flux through the partial cone surface
shown in the following figure:
4. A very very long cylindrical shell with radius a and negligible thickness has a uniform surface charge
density ρs . The primary axis of the shell is the z-axis and the shell rotates around this axis with
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.
rotational speed ω.
a) Decide whether there is a displacement current and explain why or why not.
b) Write down the expression for the current per unit length of the cylinder, J (magnitude in A/m
and direction).
c) Away from the cylinder edges, the magnetic field intensity, H outside the cylindrical shell goes
to zero in the limit of an infinitely long cylindrical shell. Provide a brief physical explanation.
d) By considering the rectangular loop drawn in the picture, find H (magnitude and direction)
inside the shell (r < a) away from the cylinder edges.
2
5. Given the time-varying magnetic field B = B0 ((x − 2)sin(ωt)x̂ − (y + 2)sin(ωt)ŷ + (y − 2)e−t ẑ)
Wb/m2 , find the emf E for the following closed paths C:
a) C is a rectangular path going from (0, 0, 0) to (0, 0, 1) to (1, 0, 1) to (1, 0, 0) back to (0, 0, 0), with
distance along the path measured in meters.
b) C is a rectangular path having the same coordinates as defined in part (a) above, but with the
path direction reversed.
c) C is a triangular path going from (0, 0, 0) to (0, 1, 0) to (0, 1, 1) back to (0, 0, 0), with distance
along the path measured in meters.
6. A square loop of wire of finite resistance R and 4 m2 surface area is located with a region of a constant
magnetic field B = −B0 ẑ, where B0 = 0.25 Wb/m2 . Although the loop is initially positioned in the
x̂ŷ-plane, it is free to rotate out of that plane around the x̂-axis, as illustrated in the following
perspective view diagram.
a) What is the magnetic flux Ψ through the loop when its angle of orientation with respect to the
x̂ŷ-plane, denoted by θ, is θ = 0◦ ? In your flux calculation, make use of the dS orientation
shown in the diagram on the right.
b) What is the flux Ψ as a function of angle θ (use the same sign convention for dS as in part a)?
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.
c) Assuming that angle θ is time varying at a rate of
loop when θ = 45◦ ?
dθ
dt
= 2π rad/s, what is the emf E around the
d) In what direction will a positive induced current flow around the loop at the same instant?
Explain why.
7. A conducting wire of resistance R = 1 Ω forms a circular loop of radius r = 10 cm and moves with a
velocity v = 2ŷ m/s in a region where the background magnetostatic field is described by
B = 25 × 10−6 (1 +
y
)ẑ Wb/m2
L
where L = 1000 m. The center of the loop coincides with the origin (x, y, z) = (0, 0, 0) at t = 0 and
the plane of the loop coincides with the z = 0 plane.
a) Write an expression for the induced emf E(t) of the loop in motion for t > 0. Since r L, the
magnetic field across the loop can be considered nearly constant at each instant in time.
b) What is the magnitude of the loop current?
Interesting fact: The strength of the earth’s magnetic field is just about 25 × 10−6 T at equatorial
latitudes. However, the scale length L associated with the spatial variation of earth’s magnetic field
is much longer than 1000 m.
8. There is no bonus question this week.
c 2015 Lynford Goddard. All rights reserved. Redistributing without permission is prohibited.