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ECE 2317 Applied Electricity and Magnetism. Spring 2014. Prof. David R. Jackson ECE Dept. Notes 2. Notes prepared by the EM Group University of Houston. Statics. Definition: No time variation. In terms of frequency, f = 0 [ Hz ] .

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ECE 2317

Applied Electricity and Magnetism

Spring 2014

Prof. David R. Jackson

ECE Dept.

Notes 2

Notes prepared by the EM Group University of Houston

Statics

Definition: No time variation. In terms of frequency, f= 0 [Hz]

The electromagnetic field splits into two independent parts:

Electrostatics: (q, E) charges produce electric field

Magnetostatics: (I, B) current produces magnetic field

The static approximation is usually accurate for d << 0

(d is the dimension of the circuit or device).

Statics (cont.)

Example: f = 60 [Hz]

Note: This is an exact (defined) value since 1983.

0 = c / f

c = 2.99792458  108 [m/s]

f = 60 [Hz]

This gives: 0 = 4.9965106 [m]

= 4,996.5 [km]

= 3,097.8 [miles]

Clearly, most circuits fall into the static-approximation category at 60 [Hz]!

Statics (cont.)

The following are special cases of electromagnetics at low frequency:

• Circuit theory (e.g., ECE 2300)

• Electronics

• Power engineering

• Magnetics (design of motors, generators, transformers, etc.)

Examples of high-frequency systems that are not modeled by statics:

• Antennas

• Transmission lines

• Microwaves

• Optics

ECE 3317

Charge

e

Atom

p

proton: q= 1.602  10-19 [C]

Ben Franklin chose the convention of positive and negative charges.

electron: q= -1.602  10-19 [C]

1 [C] = (1 / 1.602 x10-19) protons = 6.242 x 1018 protons

Ben Franklin

Charge Density

1) Volume charge density v[C/m3]

a) Uniform (homogeneous) volume charge density

+ + + +

+ + + +

+ + + +

v

Example: protons floating in space.

V

Uniform cloud of charge density

Q

Charge Density (cont.)

b) Non-uniform (inhomogeneous) volume charge density

+ + + +

+ + + +

+ + + +

v(x, y, z)

dV

Non-uniform cloud of charge density

dQ

Example: protons are closer together as we move to the right.

Charge Density (cont.)

v(x, y, z)

dV

dQ

so

Hence

Charge Density (cont.)

2) Surface charge density s[C/m2]

Example: protons are sprayed onto a sheet of paper.

S

s(x, y, z)

Non-uniform sheet of surface charge density

Q

+ + + +

+ + + +

+ + + +

Non-uniform

Uniform

Charge Density (cont.)

S

s(x, y ,z)

Q

so

Hence

Charge Density (cont.)

+

+

+

+

+

+

+

+

+

+

+

+

+

+

l

l (x, y, z)

Q

3) Line charge density l[C/m]

Example: protons are sprayed onto a thread.

Non-uniform line charge density

Uniform

Non-uniform

Charge Density (cont.)

+

+

+

+

+

+

+

+

+

+

+

+

+

+

l

l (x, y, z)

Q

so

Hence

z

v= v0=10 [C/m3]

a

y

x

Example

Find: Q

Note: This is a uniform charge density.

Example

z

v = 2r [C/m3], r < a

A separable integrand with fixed limits of integration.

Note: This is a non-uniform (inhomogeneous) charge density.

a

Find: Q

y

x

dV

Separable integrand

Example: Find the EquivalentSurface Charge Density for a Slab of Volume Charge Density

y

x

z

Example (cont.)

Equivalent surface charge density:

y

x

z

Example (cont.)

Compare: