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: