Name: _____________________
GTID: _____________________
ECE 3025: Electromagnetics (section of Professor Wenshan Cai)
Final Exam, 2:50 pm – 4:50 pm, May 4, 2012
Please read this cover sheet carefully before continuing with the exam.
Please remove everything from your desk except for this test itself, writing instruments, a
timepiece, and your one-page handwritten cheat sheet (letter size, front only).
All pages are numbered at the bottom center of the page. Make sure that you have all 12 pages
including this cover page (p.1), a blank page (p.10), and two formula sheets (p.11-12).
Work all problems in the spaces below the problem statement. If you need more room, use the
blank page (p.10).
Carefully tear out the formula sheets (p.11-12). Do not remove the staple or tear out any other
pages.
You can use the back side of the pages for scratch, but I will not grade answers written on the
back side.
The exam will last until 4:50 p.m.
I will not grade your exam if you fail to sign on the line below.
I acknowledge the above terms for taking this exam. I commit to uphold the ideals of honor and
integrity by refusing to betray the trust bestowed upon me as a member of the Georgia Tech
community. I pledge my honor that I have not violated the Honor Code during this examination.
Student’s signature: ___________________________
Page 1 of 12
1. True or False (2 points × 7 = 14 points). Please circle your answer.
a) When a dc current forms a circular loop, the generated magnetic field is maximum at the
center of the loop.
True
False
b) All electric field lines begin and terminate on electric charge, and they never form closed
loops.
True
False
c) When a charged particle is ejected into a uniform magnetic field, the particle will move in
a circular path at a constant speed.
True
False
d) At the boundary between two arbitrary materials, the normal component of the magnetic
flux density B must be continuous, regardless whether the field is static or time-varying.
True
False
e) When a plane wave travels in a dielectric material with an attenuation coefficient α, the
power density of the wave falls off as along the propagation direction z of the wave.
True
False
f) The expression , = 10 cos10 − [V/m] may represent the electric field of a
uniform plane wave propagating in the free space.
True
False
g) When a plane wave enters from vacuum into glass, the frequency of the wave remains
unchanged, and the wavelength becomes shorter.
True
False
Page 2 of 12
2. Parallel-plate capacitor (5 × 4 = 20 points)
(a) An air-filled parallel-plate capacitor with plate separation d and plate area A is
connected to a constant source of voltage. Determine how C, E, D, Q, and WE change
as a dielectric slab (size × ) of = 2 is inserted between the plates.
(note: “changes by a factor of 1” means no change)
The capacitance, C, changes by a factor of __________.
The electric field between the plates, E, changes by a factor of __________.
The electric flux density, D, changes by a factor of __________.
The amount of charge on each plate, Q, changes by a factor of __________.
The energy stored in the capacitor, WE, changes by a factor of __________.
(b) An air-filled parallel-plate capacitor with plate separation d and plate area A is
charged and then disconnected from the voltage source. Determine how C, E, D, V,
and WE change as the same dielectric slab of = 2 is inserted between the plates.
The capacitance, C, changes by a factor of __________.
The electric field between the plates, E, changes by a factor of __________.
The electric flux density, D, changes by a factor of __________.
The voltage difference between the two plates, V, changes by a factor of __________.
The energy stored in the capacitor, WE, changes by a factor of __________.
(See next page for parts c-d of this problem)
Page 3 of 12
(c) An air-filled parallel-plate capacitor has plate separation d and plate area A. By what
factor does the capacitance change if a dielectric slab ( = 2 ) with the size of
/2 × is inserted between the plates?
(d) If the air-filled parallel-plate capacitor with plate separation d and plate area A is
completely discharged and then connected to a voltage source = / , where
τ is a constant, at = 0. Find the displacement current as a function of .
Page 4 of 12
3. Current density, magnetic field, and magnetic force (5 × 3 = 15 points)
A long, cylindrical conductor with cross-sectional radius a is placed along the z axis. The
magnetic field is found to be = !/2"#$ %& within the conductor ( ! < # ) and
= /2"!%& for ! > #.
a)
Find the current intensity, J, within the conductor (! < #).
b)
Find the current intensity, J, outside the conductor (! > #).
c)
Now we place in the space another infinitely long current line, which is parallel to
the z axis and intersects with the xy plane at (2a, 0, 0). The line carries a current
% . Find the force on this current line per unit length.
Page 5 of 12
4.
Magnetic field from two current sources (9 + 3 + 3 = 15 points)
z
2d
A current sheet lies in the z = 0 plane and
carries a surface current density ) = * %
I=?
•d
A/m. A current filament I is parallel to the x
axis and intersects with the z axis at z = 2d.
y
x
K0
a)
Determine the magnitude and direction of the current I such that the total
magnetic field H is zero at the point (0, 0, d ).
b)
(Continuing from part a) Are there any other positions (except for = ) along
the z axis where the magnetic field H is zero?
c)
(Continuing from part a) Is the force between the two current sources repulsive or
attractive?
Page 6 of 12
5. emf of a circuit in motion (6 + 10 = 16 points)
z
(See figure) A rectangular loop of the size
1 m × 2 m moves toward the origin at a
v
R
constant velocity of 10 m/s in a magnetic
2m
field 1 = 5 3 % (T). The loop contains a
resistor R = 15 Ω.
y
1m
x
a)
Determine the direction (clockwise or counterclockwise as seen from above) of
the induced current. Explain the reason for your answer.
b)
Find the induced current at the instant when the leading side (the left side) of the
loop reaches the position of y = 1 m.
(These numerical values may be useful: + , 0.4; $ , 0.1)
Page 7 of 12
6.
Reflection of uniform plane wave (20 points)
Region 1
x
Region 2
5+ = 45+ = 1
A uniform plane wave in air, ;<= = 10 >? %
(V/m), is normally incident on another material
A
@+
= 10 >?
in region 2. The boundary is located at z = 0.
=0
a)
z
Assume that the material in region 2 is a perfect dielectric with 45$ = 1 and
5$ > 1 being a real number. Measurements of the electric field in region 1 show
that the first maximum occurs 0.5 meter from the interface, with the amplitude
being |+7 |89 = 15 V/m.
Find the numerical values of : and 5$ .
(See next page for part b of this problem)
Page 8 of 12
b)
Assuming that the material in region 2 is a perfect conductor with the intrinsic
impedance B$ = 0, find the expressions for the total electric field +7 and the
total magnetic field C+7 in region 1, and sketch the distributions of the
magnitudes |+7 | and |C+7 |.
Sketch |+7 | and |C+7 | here:
Region 1
5+ = 45+ = 1
Perfect
conductor
z
=0
Page 9 of 12
(This page is intentionally left blank)
Page 10 of 12
Cylindrical coordinates
Differential surfaces: , , and. Differential volume: .
Spherical coordinates
Differential surfaces: , sin , and
sin . Differential volume: sin .
Dot product and cross product
in rectangular coordinates:
∙ ∙ ; in rectangular coordinates:
Similarly, in cylindrical coordinates, , etc.
Divergence
Gradient
Curl
Laplacian
Maxwell’s equations in point form and in integral form
Wave equation and solutions
Poynting’s theorem in point form and in integral form
Poynting vector
∙ ∙ #
" $ ∙ % ! #
" ∙ %
! Lorentz force equation: ' ( ) & W/m2