EEE 241Spring 2016
HW 4Due February 9
3-10 Assuming that the electric field intensity is E = 100xax (V/m), find the total
electric charge contained inside (a) a cubical volume 100 mm on a side
centered symmetrically at t
1.
EEE 241Spring 2016
HW 6 Due February 23
Work problem 3-34 from the textbook.
The coordinate system is cylindrical coordinates, and we can take the surface
charge on the inner coaxial cylinder as
Q
.
s =
2 ri L
Then, betwee
NAME: ; SID #:
EEE 433, ASU
Spring 2015, David R. Allee
Midterm
18 March, 2015, Noon to 1:15pm
Closed Book & Closed Notes
No Calculator
Show Your Work
Put Your Final Answer in the S ace Provided and Box Your Answer
/
NAME:
(LAST) (FIRST)
BEE 241: Sprin
P.4-1
The upper and lower conducting plates of a large parallel-plate capacitor are separated by a distance d and maintained at potentials V0 and 0, respectively. A dielectric slab of dielectric constant 6.0 and uniform thickness 0.8d is placed over
EEE241 Fundamentals of Electromagnetics Spring 2007
Solutions to Homework 3
P3.5: Two point charges, Q1 and Q2 are located at (1,2,0) and (2,0,0) respectively. Find the relation between Q1 and Q2 such that the total force on a test charge at the p
=
=
+
( )
=0
+
=
= (T)
=
0
4
The vector field + is curl-free (irrotational) so it can be
expressed as the gradient of a scalar field V
Note 1: in the static case,
=
1
40
= 0, =
Note 2: solutions valid only for slowly varying () and ()
2
2 = +
Solution of Wave Equations for Potentials due to a point charge at the origin: =
2
2 =
2
1
2
, =
2
2 = 0
2
1. V is radial:
, , = ()
2. This equation is valid everywhere except at the origin
1
(, )
2
2
2 = 0
2
Any twice-differentiable functi
Special case of time-dependent Maxwells equations
Sources are periodic time functions
can be expanded in Fourier sums
of sinusoidal of cosinusoidal
functions
Maxwells equations are linear
sum of solutions is still a solution
Maxwells equations with sinuso
A uniform plane wave characterized by = propagating in the+ direction has associated with
it a magnetic field = . Thus and are perpendicular to each other, and both are transverse
to the direction of propagation. It is a particular case of a transverse el
Idea: Model the magnetic response of all atoms or molecules in a
material as a collective property of the material.
= 0
1
= +
0
=
= +
=
0
=
A1
0
=
( ) =
A2
Magnetic field intensity
( ) =
Stokess theorem:
=
=
=
()
=
Amperes circuit
Source-free equation in free space:
1 2
2 2 = 0
A plane wave is a solution of Maxwells equations with
assuming the same direction, same magnitude, and
same phase in infinite planes perpendicular to the
direction of propagation. Similarly for .
Freque
(, , )
2
Such configuration is a magnetic dipole.
Because of the far field approximation
( ), we use the spherical coordinate
system, i.e. we assume that the circular
loop is point-size from a large distance.
1
Idea: study the magnetic flux density
gener
=
= =
= 0
0
=
4
0
A=
4
homogeneous current flowing along a thin wire
of length and cross section S
1
the current must flow in a close circuit
0
= A=
4
swap integral and curl
B=
0
4
B =
0
4
=
=
0
4
1
=
probe (field point)
1
, =
1
+
=
z
Postulates of Electrostatics:
Coulomb Force:
=
B
F (q<0)
=0
=
q
(N)
(N)
Magnetic Force:
=
Lorentzs Force:
= q( + )(N)
x
F (q>0)
y
integral form
differential form
= 0
integrate over volume V
= 0
= 0
divergence theorem
= 0
= 0
integrate over
= +
=
= ( )
=
= = 2
1
1 2
=
=
=
2
2
1
1
=
=
=
2
2
2
=
1 2 1 2
+ 2
2
2
rate of decrease of electric and magnetic energy stored in V
minus ohmic power dissipated in V
divergence theorem
=
=
1 2 1 2
+
2
2
2
equals power l
(m/s)
for plane waves in a lossless medium = is a linear function of
phase velocity
=1
=
is independent of frequency
However, in many cases, is not a linear function of
If a signal is made of several plane waves, each wave will travel with its own p
2 + 2 = 0
Homogeneous wave equation, in frequency domain
=
Easy solution: all info is in = +
Issue: how to account for material response
= =
(m1 )
= + = 1 +
= + =
1
1
1
2
solution:
= =
2
Note: in a lossless medium
2 2 = 0
=
Traditional solut
=
one direction
linear polarization
= + ()= 0 0
two directions, same amplitude and phase
linear polarization
http:/youtu.be/oDwqUgDFe94
10 and 20 are denoting the amplitudes of the two
= 1 + 2 () = 10 20
, = Re
linearly polarized waves
1 + 2 () =
System: polarized dielectric (p ) plus free charges ( )
=
( + p )
=
0
0
p =
(0 + ) =
D=(0 + )
D =
(/2 )
= Electric Flux Density, or Electric Displacement, D
1) Info about bound charges has been moved inside D
2) Same form of first postulate, with
Dielectrics contain bound charges affecting the external electric field
External electric fields cause small displacements in positive and
negative bound charges within the dielectric material
The displacements create or modify existing dipoles that ar
System: one point charge Q above a grounded conducting plane
2 2
2
Formal approach: = 2 + 2 + 2 = 0
2
Conditions for the solution:
1. At all points on the ground conducting
plane, the potential is zero:
(0, , 0)
Grounded plane
conductor
, 0, = 0
2. At