5 Curl-free elds and electrostatic potential
Mathematically, we can generate a curl-free vector eld E(x, y, z ) as E = ( V V V , , ), x y z
by taking the gradient of any scalar function V (r) = V (x, y, z ). The gradient of V (x, y, z ) is dened to be th
6 Circulation and boundary conditions
Since curl-free static electric elds have path-independent line integrals, it follows that over closed paths C (when points p and o coincide)
C
z y dS C S o=p
lj Ej
E dl = 0,
where the C E dl is called the circulatio
7 Poissons and Laplaces equations
Summarizing the properties of electrostatic elds we have learned so far, they satisfy the constraints D = and in addition E = V as a consequence of E = 0. Combining the equations above, we can re-write Gausss law as D= fr
8 Conductors, dielectrics, and polarization
We have so far been examining static eld congurations of charge distributions assumed to be xed in free space, in the absence of materials (solid, liquid, or gas) composed of neutral atoms and molecules. In pra
9 Static elds in dielectric media
Summarizing important results from last lecture: within a dielectric medium, displacement D= E=
oE
+ P,
and if the permittivity = r o is known, D and E can be calculated from free surface charge s or volume charge in the
10 Capacitance and conductance
Parallel-plate capacitor: Consider a pair of conducting plates with surface areas A separated by some distance d in free space (see margin). The plates are initially charge neutral, but then some amount of electrons are tran
11 Lorentz-Drude models for conductivity and permeability and polarization current
In this lecture we will describe simple microscopic models for conductivity and electric susceptibility e of material media composed of free and bound charge carriers. The
12 Magnetic force and elds and Amperes law
Pairs of wires carrying currents I running in the same (opposite) direction are known to attract (repel) one another. In this lecture we will explain the mechanism the phenomenon is a relativistic1 consequence of
13 Current sheet, solenoid, vector potential and current loops
In the following examples we will calculate the magnetic elds B = oH established by some simple current congurations by using the integral form of static Amperes law.
B
Js = z Js z
y
L
C
B =
14 Faradays law and induced emf
Michael Faraday discovered (in 1831, less than 200 years ago) that a changing current in a wire loop induces current ows in nearby wires today we describe this phenomenon as electromagnetic induction: current change in the
15 Inductance solenoid, shorted coax
Given a current conducting path C , the magnetic ux linking C can be expressed as a function of current I circulating around C .
z
3
2
1
0 1
If the function is linear, i.e., if we have a linear ux-current relation =
16 Charge conservation, continuity eqn, displacement current, Maxwells equations
Total electric charge is conserved in nature in the following sense: if a process generates (or eliminates) a positive charge, it always does so as accompanied by a negative
17 Magnetization current, Maxwells equations in material media
Consider the microscopic-form Maxwells equations D = B = 0 Gausss law
B Faradays law t D H = J+ , Amperes law t E = where D = oE B = oH. Direct applications of these equations in material med
18 Wave equation and plane TEM waves in sourcefree media
With this lecture we start our study of the full set of Maxwells equations shown in the margin by rst restricting our attention to homogeneous and non-conducting media with constant and and zero . O
19 dAlembert wave solutions, radiation from current sheets, Poynting theorem
dAlembert wave solutions of Maxwells equations for homogeneous and source-free regions obtained in the last lecture having the forms z ) E, H f ( t v are classied as uniform pla
20 Monochromatic wave solutions and phasor notation
Recall that we reached the traveling-wave dAlembert solutions z E, H f ( t ) v via the superposition of time-shifted and amplitude-scaled versions of f (t) = cos( t), namely the monochromatic waves z )]
3 Gausss law and static charge densities
We continue with examples illustrating the use of Gausss law in macroscopic eld calculations:
Example 1: Point charges Q are distributed over x = 0 plane with an average surface charge density of s C/m2 . Determine
2 Static electric elds Coulombs and Gausss laws
Static electric elds are produced by static (i.e., non-time varying) distribution of charges in space. At the most elementary level, each stationary point charge (electron or proton) Q is surrounded by its r
ECE 329 Lecture Notes Summer 09, Erhan Kudeki
1 Vector elds and Lorentz force
Interactions between charged particles can be described and modeled1 in terms of electric and magnetic elds just like gravity can be formulated in terms of gravitational elds o
23 Signal transmission, circular polarization
Since in perfect dielectrics the propagation velocity vp = v and the intrinsic impedance are frequency independent (i.e., propagation is non-dispersive), dAlembert plane wave solutions of the form f (t z ) z v
24 Wave reections, standing waves, radiation pressure
In this lecture we will examine the phenomenon of plane-wave reections at an interface separating two homogeneous regions where Maxwells equations allow for traveling TEM wave solutions. The solutions
25 Guided TEM waves on TL systems
An x polarized plane TEM wave propagating in z direction is depicted in the margin. A pair of conducting plates placed at x = 0 and x = d would not perturb
E H EH
x
z
the elds except that charge and current density varia
26 Introduction to distributed circuits
Last lecture we learned that voltage and current variations on TLs are governed by telegraphers equations and their dAlembert solutions the latter can be expressed explicitly as z z V (z, t) = V +(t ) + V (t + ) v
27 Bounce diagrams
Last lecture we obtained the implulse-response functions z z 2l V (z, t) = g [ (t ) + L (t + )] v v v and Source matched to line:
Zo I (z, t)
+ + -
g z z 2l I (z, t) = [ (t ) L (t + )] Zo v v v for the voltage and current in the TL cir
28 Multi-line circuits
In this lecture we will extend the bounce diagram technique to solve distributed circuit problems involving multiple transmission lines. One example of such a circuit is shown in the margin where two distinct l TLs of equal lengths
29 Periodic oscillations in lossless TL ckts
Lossless LC circuits (see margin) can support source-free and co-sinusoidal voltage and current oscillations at a frequency of 1 = LC known as LC resonance frequency. Lossless TL circuits can also support sour
30 Input impedance and microwave resonators
The input impedance and admittance of the series and parallel LC resonators shown in the margin are, respectively, 1 1 ) and Yp = j ( C ), Zs = j ( L C L both of which vanish at the common resonance frequency o
31 TL circuits with half- and quarter-wave transformers
Last lecture we established that phasor solutions of telegraphers equations for TLs in sinusoidal steady-state can be expressed as V +ej d V ej d V (d) = V e + V e and I (d) = Zo in a new coordinate
34 Line impedance, generalized reection coefcient, Smith Chart
Consider a TL of an arbitrary length l terminated by an arbitrary load ZL = RL + jXL. as depicted in the margin. Voltage and current phasors are known to vary on the line as V (d) = V e
+ j d