Prof. D. WiltonECE Dept.
Notes 2
ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism
Notes prepared by the EM group,
University of Houston.
Charge!
Definitions: Statics – frequency f = 0 [Hz]
Quasi-statics – slow time variation, f << ? [Hz]
The electromagnetic field splits into two independent parts:
Electrostatics: (q, E) Static charge
Magnetostatics: (I, B) Constant current w.r.t. time
StaticsStatics
/t d c d t The (quasi-)static approximation is generally valid if a signal's "travel time"
across a circuit of dimension is much smaller than , the time for
a significant change in the signal (e.g. "ris
-,
- 1 /1,
- /
t t
ftd
t T fTc
f c c f
e time" of a pulse, or "period" of a
sinusoid) to occur. E.g., ("implies")
frequencya pulse
period =a sinusoid
wavelength
Example:
Statics (cont.)Statics (cont.)
in
in
sin(2 )
sin 2
V f tt
VT
circuit
T t
t
d c t
out
out
sin 2 ( )( )
sin 2
V f t tt t
VT
tt
1dt T d
c f c
Example: f = 60 [Hz]
0 = c / f
Clearly, most circuits fall into the static-approximation category at 60 [Hz]!
c = 2.99792458 108 [m/s]
f = 60 [Hz]
This gives: 0 = 4.9965106 [m] = 4,996.5 [km] = 3,097.8 [miles]
Statics (cont.)Statics (cont.)
The following rely on electro(quasi-)static and magneto(quasi-)static field theory:
• circuit theory (e.g, ECE 2300)
• electronics
• power engineering
• magnetics
Examples of high-frequency systems that are not modeled by statics:
• antennas
• transmission lines
• microwaves
• optics
ECE 3317
Statics (cont.)Statics (cont.)
ECE 2317
proton: q = -e = +1.602 x 10-19 [C]
electron: q = e = -1.602 x 10-19 [C]
1 [C] = 1 / 1.602 x10-19 protons = 6.242 x 1018 protons
ChargeCharge
atom
e
p Ben Franklin
1) Volume charge density v [C/m3]
[ ]v V
Q 3C / m
uniform cloud of charge density
v
V
Q
Charge DensityCharge Density
0
, , limv V
Q dQx y z
V dV
non-uniform cloud of charge density
v (x,y,z)
dV
dQ
non-uniform (inhomogeneous) volume charge density
Charge Density (cont.)Charge Density (cont.)
0
, , limv V
Qx y z
V
v (x,y,z)
dV
dQ
, ,v
Qx y z
V
, ,vQ x y z V
so , ,vdQ x y z dV
or
, ,v
V
v
V
Q x y z dV
dV
Charge Density (cont.)Charge Density (cont.)
2) Surface charge density s [C/m2]
0lim [ ]s S
Q dQ
dSS
2C / m
non-uniform sheet of charge density
s (x,y,z)
Q
S
[ ]s
Q
S
2C/m
non-uniform uniform
Charge Density (cont.)Charge Density (cont.)
, ,sdQ x y z dS
, ,s
S
s
S
Q x y z dS
dS
[ ]s
dQ
dS 2C/m
Charge Density (cont.)Charge Density (cont.)
s (x,y,z)
dQ
dS
3) Line charge density l [C/m]
0lim [ ]l l
Q dQ
dll
C / m
non-uniform line charge density
non-uniform uniform
[ ]l
Q
l
C/m
l (x,y,z)
Q
l+
+ + ++ + +++++ + + +
Charge Density (cont.)Charge Density (cont.)
, ,ldQ x y z dl
, ,l
C
l
C
Q x y z dl
dl
Charge Density (cont.)Charge Density (cont.)
l (x,y,z)
dQ
dl+
+ + ++ + +++++ + + +
, ,
10
10
v
V
V
V
Q x y z dV
dV
dV
Find: Q
3
10
4=10
3
Q V
a
340[ ]
3Q a C
Example – Find the total charge Example – Find the total charge in the spherical distributionin the spherical distribution
x
y
z
av = 10 [C/m3]
2
0
4
, ,
2
2
v
V
V
a
Q x y z dV
r
r dV
rr d
Find: Q3
0
4
8
84
a
Q r dr
a
42 [ ]Q a C
Example – Find the total chargeExample – Find the total charge in the in the non-uniformnon-uniform spherical distribution spherical distribution
z
x
y
av = 2r [C/m3]
r
Example – Find the Equivalent Surface Example – Find the Equivalent Surface Charge Density for a Thin Slab of ChargeCharge Density for a Thin Slab of Charge
2 ,
0
v xyz
z z
3C/m
x
y
z
z
SV
0
2
0 0, , , 2
v
V S S
z z
S
z
v S
v
Q dV dS dS
x
d
y x y z dz x
z
yzdz xy z
2C / m
Example – Find the Equivalent Line Charge Example – Find the Equivalent Line Charge Density for a Thin Cylinder of ChargeDensity for a Thin Cylinder of Charge
22 ,
0 , 0
v x yz
x x y y
3C/m
/2
/2
2 23 2
3 2
3 2
0
2
0 0
/ 2 / 22
3 2 2
2 2
3 2 2
( ) lim3
2y xz
v
V z
Q dV dx yz dxdy z
z zx y
x y z
Q x yz z
C / m
x
y zV
dxdy
/2
/2
z
z
dz
Summary of Conversion to Equivalent Summary of Conversion to Equivalent Charge DensitiesCharge Densities
, ,v
z
z x y z dxdy Cross sectionof cylinder at fixed
C/m
,
, , ,S v
x y
x y x y z dz 2
Cross sectionof slab at fixed
C/m
Slab lying in a constant z-plane:
Cylinder lying parallel to the z-axis: