Coulombs law

1
Coulombs law

• force (a VECTOR quantity!) between two simple
  charged objects is directly along the line that
  joins the two objects
• force is attractive for oppositely-charged
  objects
• force is repulsive for like-charged objects
• the magnitude of the force decreases as the
  square of the distance between the charges
• the magnitude of the force is proportional to the
  product of the two charges
• in free space

• in vector form

2
The electric field

• consider a single point charge Q fixed in space
  at point A
• introduce a single unit test charge that we can
  move around, always monitoring the magnitude and
  direction of the force on the test particle

• the right hand side depends on
• the inverse of the square of the distance from
  the real charge Q to our test point charge
• direction from the real charge to the test point
• the left hand side is a vector electric field E
• for a point charge located at the origin this is

3
Charge distributions

• volume rv, units Coul / (length)3
• surface rS, units Coul / (length)2
• line rL, units Coul / length

4
Field due to a charge distribution

• volume charge distribution

5
Field due to a charge distribution

• so to get the total filed we need to integrate
  all the charge
• integrate over the primed coordinates

6
Electric flux density D and Gausss law

• vector quantity D
• aka the displacement density, displacement
  flux density
• Gausss law
• using Gausss law
• trick is to look for symmetries and pick the
  right surface
• you want to pick a surface such that D (or
  equivalently, E) is either
• constant
• or zero

7
Infinite charged sheet

• from symmetry
• the only field component must be perpendicular to
  sheet
• for a fixed perpendicular distance above the
  sheet the magnitude of the field should be
  constant as you move around in x and y
• gaussian surface any shape with top and bottom
  parallel to plane, sides perpendicular to plane
• lets just use a simple box
• top area element dxdy field is perpendicular
  to surface, same direction as the surface normal,
  constant magnitude ? D?dS Ddxdy ? ?D?dS
  Darea
• bottom area element dxdy field is
  perpendicular to surface, same direction as the
  surface normal, constant magnitude ? D?dS
  Ddxdy ? ?D?dS Darea
• sides field is parallel to surface,
  perpendicular to surface normal ? D?dS 0 !
• total flux 2Darea
• charge enclosed rSarea
• 2Darea rSarea ? D rS/2 ? E
  rS/2eo

8
Differential form of Gausss law

• for the vector field is D, the flux density,
  then
• if the volume dv contains a volume charge density
  rv
• then the charge enclosed is dQ rvdv
• and Guasss law tells us that flux equals charge
  enclosed

• or written more compactly we have the
  differential form of Gausss law

9
Potential for a charge distribution

• since Coulombs law is linear the potential of
  many charges should add
• charge m is located at position rm
• and for a charge distribution
• you integrate over the region of space where
  there is charge, i.e., over r
• this is somewhat easier to evaluate than the
  vector sum/integral needed to directly calculate
  the field from the charge

10
Potential difference for a charge distribution

• the potential difference between two points is
  then

11
Equipotential surfaces

• an equipotential surface (or more simply, an
  equipotential) is such that if the beginning and
  end points fall on that surface, the potential
  difference is zero
• the electric field is always perpendicular to an
  equipotential
• example equipotential surfaces
• single point charge spheres centered at the
  charge
• uniform spherical charge distributions spheres
  centered on the distribution
• infinitely long uniform line charge cylinders
  centered on the line charge

12
Electric field as the gradient of potential

• we define the gradient as the set of
  differential operations on a scalar function that
  produces a vector function
• the interpretation is
• magnitude is the maximum rate of change of the
  scalar function at the point of observation
• the direction is the direction you must travel to
  see this maximum rate of change
• or, for convenience, we have the vector operator
  grad
• the electric field E is equal to the negative of
  the gradient of the scalar potential V

13
Laplaces Equation

• if the charge density is zero we get the special
  case of Laplaces equation
• refs
• http//hyperphysics.phy-astr.gsu.edu/hbase/electri
  c/laplace.html
• http//www.physics.arizona.edu/restrepo/475B/Note
  s/source/node48.html
• note we can now change our method of solution
  from an integral approach to one involving the
  solution of a differential equation
• once we have V(r), we have everything since we
  can get E from V

14
Laplace equation solution coax

• example coaxial cable aligned along the z-axis
• cylindrical coordinates, no ? variations
• solution varies only with r
• now apply BCs
• V Vout _at_ r b
• V Vin _at_ r a

15
Summary page Electrostatic boundary conditions
for dielectrics

• at the interface between two dielectrics
• the component of the electric field tangent to
  the interface between two dielectrics is
  continuous
• Dtan is discontinuous
• the component of the D field normal to the
  interface between two dielectrics is continuous
• if there is no free charge at the interface
• if there is a surface charge
• Enorm is discontinuous
• material properties

16
Ohms law and resistance

• consider a block of uniform conducting material
  with perfect electrical contacts on each end
• the current density from Ohms law J s E
• but the normal form of Ohms law is I R V
• here the total current I is just the current
  density integrated across a cross section
• here the field is uniform and hence so is the
  current density
• so I (cross sectional area) J w t s E
• the voltage difference V between one end and the
  other is just the line integral of the electric
  field
• again, since everything is uniform this is easy
• V E l
• so we should have (w t s E) R E l
• or

17
Summary page Electrostatic boundary conditions
for conductors

• in the absence of current flow we have the
  following conditions for a conductor
• the electric field (and D as well) inside is
  identically zero
• at the surface of a conductor, the field is
  everywhere normal to that surface
• the conductor is an equipotential
• at the surface of a conductor, any normal
  component of the field induces a surface charge
  that is proportional to the field strength

18
Conductor/dielectric example parallel plates

• now what happens if we carefully add a slab of
  dielectric that completely fills the gap between
  the plates?
• the charge on the two plates does not change
• so D does not change since rsurf Q/(plate area)
  did not change
• but E DOES change, in fact it is REDUCED by er
  compared to the original value
• with eo E E rsurf/eo
• with dielectric E (rsurf/eo)/er REDUCED as
  expected

conductor
D ereoE
ereo
d
conductor
gaussian surface ? DDS rsurfDS ? D rsurf
? E rsurf/ereo
19
Capacitance

• for two oppositely charged conductors the ratio
  of charge to potential difference is the
  capacitance of the system
• the capacitance is a function only of the
  geometry and the dielectric constant(s)

20
Given the location of charge

• given the locations of all the charges in a
  system (or equivalently rv(r), rs(r), or rl(r))
  you can
• find the electric field E, or electric flux
  density D
• directly using Coluombs law
• you integrate over the locations of all the
  charges, adding all the VECTOR fields produced by
  each charge
• LOOK FOR SYMMETRY FIRST!!!!!
• look for cancellation of various vector
  components
• indirectly using Gausss Law
• probably only helps you find the fields if the
  symmetry tells you what shape the gaussian
  surface should be
• look for a shape that gives you either
• constant field perpendicular to the surface (so
  dot product is easy)
• zero field
• field that is everywhere parallel to the surface
  (so the dot product is zero)
• you could also find the potential
• this requires an integration over only the
  distance between the charges (the sources) and
  your observation point

21
Given the vector electric field

• given the full vector electric field (i.e., the
  magnitude and direction of E at all points in
  space) you can
• find the potential difference VAB between two
  points by performing a line integral of E along
  ANY path that connects B to A
• find the free charge distribution that produce
  the electric field using the divergence

22
Given the potential function

• given the scalar potential (voltage) you can
• find the electric field using the negative of the
  gradient
• find the charge by taking the divergence of the
  field
• i.e., take the divergence of the negative of the
  gradient of the potential
• find the capacitance from the ratio of charge to
  voltage

23
Notes on symmetry

• for a single point object or spherically
  symmetric charge distribution, it seems likely
  that spherical symmetry will hold
• for an arbitrary collection of points or charge
  distribution, youre helpless
• for an infinitely long line or charge
  distribution that is uniform along the length
  (lets call it the z-axis)
• no matter where you stand in z, the line still
  looks the same
• so it seems likely that the z coordinate
  shouldnt really matter
• if you walk around the line, keeping the same
  perpendicular distance from it, the line still
  looks the same
• so it seems likely that the ? coordinate
  shouldnt really matter
• use cylindrical coordinates
• if you walk in a direction perpendicular to the
  line (towards or away) it may look different
• so it seems like the only variable should be r

24
Notes on scale

• for any finite object, if you stand far enough
  away, it looks like a point
• so for a charged object, approximately what does
  the electric field look like from far away?
• for any line segment, if you stand close enough,
  it looks like its infinitely long
• for any surface, if you stand close enough, it
  looks like its infinite

25
Biot-Savart (bE'O-su-vär) Law

• magnetic equivalent of Coulombs Law
• a short element of a current carrying line
  contributes to the B-field
• refs
• hyperphysics
• Fitzpatricks page over in UT-Physics
• mo is the permeability of free space,
• units Henry / meter
• value 4px10-7 H/m
• issue current must always flow in a closed loop
• so you can never have a single points worth of
  magnetic flux density

26
Biot-Savart (bE'O-su-vär) Law

• we also use a version without the m
• the magnetic field intensity H, or more simply
  the magnetic field
• units of H amperes per meter
• for a short segment of current, dH is
• issue current must always flow in a closed loop
• you never have a single points worth of
  magnetic field
• need to do a full path integral to find the
  actual magnetic field
• integral form
• integration path is along the current, which
  always forms a closed path, since current can
  only flow in a closed loop

27
Amperes Circuital Law

• similar in form to Gausss law
• the line integral of the magnetic field H about
  any closed path is equal to the direct current
  enclosed by the path

I
28
Example Amperes Circuital Law and finite radius
wire

• for an infinitely long straight current carrying
  wire on the z-axis, symmetry suggests
• field should not vary with z or ?
• field should be constant in magnitude for fixed r
  
• to get direction of H field
• direction of current z
• direction to observation point r
• direction of field z ? r ?
• lets use circles of constant radius r so dl is
  the same direction as H, and H is constant in
  magnitude!
• dot product is a constant!
• for r gt a, exactly the same as for the filament

29
Example Amperes Circuital Law and finite radius
wire

• what about INSIDE the wire?
• symmetries are all still the same!
• lets use circles of constant radius r so dl is
  the same direction as H, and H is constant in
  magnitude!
• dot product is a constant!
• for r lt a, the only difference is the amount of
  current inside the path

z
a
y
r
x
30
Example Amperes Circuital Law and coaxial cable

• so the full solution is

31
The Curl operator

• these limits occur frequently in physics, so they
  get a special name the curl
• the curl of a vector field F is a vector function
  whose component in a particular direction (âi) is
  found
• by first orienting an infinitesimal patch
  normal to the desired direction (i.e., the area
  dSi) ,
• then finding the line integral of F around the
  patch,
• and then finding the ratio of the line integral
  to the patch area
• the component of the curl in direction âi is
• or in short hand
• and the curl of H is J !

32
Magnetic flux through a closed surface

• wed get the total flux through a closed surface
  by doing the integral
• but there are no magnetic charges, and the net
  flux through a closed surface is always zero
• equal flux in and out
• or using the divergence theorem

33
Vector magnetic potential

• turns out the scalar magnetic potential is of
  only limited usefulness
• there is another possibility
• recall for the magnetic field the divergence is
  always zero
• but for any vector field A the divergence of the
  curl of A is always zero
• so were ok if we set
• (this is the Coulomb gauge)
• A is the vector magnetic potential

34
Finding A from the current distribution

• the magnetic vector potential can be found from
• note we dont have a cross product in this!!!!
• does this make sense??
• recall
• combining

35
Magnetostatic boundary conditions

• for many materials the relative permeability is
  very close to 1
• nothing to worry about here, just use mo
  everywhere
• for the cases where mr is not one, we have BCs
  very similar to what we had for dielectrics
• method of derivation also looks similar, using
  Biot-Sarvat law and Amperes law this time
• at the interface between two magnetic materials
• the component of the B field normal to the
  interface between the two materials is continuous
  (use Biot-Sarvat law to get this one)
• the component of the H field tangent to the
  interface is equal to the surface current density
  at the interface (use Amperes law to get this
  one)
• if theres no current, Htan is continuous

36
Summary of magnetostatics

• for dc currents, in the absence of permanent
  magnets, summarizing everything we have so far

37
Inductance

• in circuits, we already know the magnetic energy
  should be
• but the energy using field concepts is
• so the inductance is given by
• volume must include ALL the fields

38
Special case coax with perfect conductors

• if the conductivity of our metals was infinite
  (i.e., a perfect conductor) then all the
  current would have to on the surface
• why?
• because Ohms law would give us infinite current
• because inside a perfect conductor ALL fields
  (electric and magnetic) must be zero
• in this case we dont have to do any energy
  integrals for the inductance inside the
  conductors, so all we have left is the part form
  the gap between inner and outer conductors
• what was the capacitance?
• interesting note product of inductance per unit
  length and capacitance per unit length

39
Summary of electromagnetics Maxwells equations

• summarizing everything we have so far, valid even
  if things are changing in time
• plus material properties

40
Maxwells equations, time harmonic form

• so assuming that none of the materials change in
  time, time independent fields Es and Hs that
  satisfy
• when multiplied by exp(jwt), the product will
  satisfy the full time dependent set of Maxwell
  equations

41
Maxwell, time harmonic, transverse-to-z

• collecting all the terms,
• assuming time harmonic solutions
• using Ohms law
• assuming there is no component of either E or H
  in the z direction
• Maxwells equations reduce to
• things to notice
• Ey is connected to Hx via d/dz and w
• Ex is connected to Hy via d/dz and w
• Hy is connected to Ex via d/dz and w
• Hx is connected to Ey via d/dz and w
• Ey and Ex are connected via d/dx and d/dy
• Hy and Hx are connected via d/dx and d/dy

42
Uniform plane wave solution to Maxwells equations

• the complete, time harmonic solution is
• E and H are perpendicular to each other
• g is called the complex propagation constant
• direction of propagation

43
Lossless, uniform plane wave solution to
Maxwells equations

• so in a region of space that has zero
  conductivity, one possible solution to Maxwells
  equations is
• this is called a uniform transverse
  electromagnetic plane wave
• a wave because it is periodic in time and space
• a plane wave because a surface of constant
  phase is a flat plane, and is propagating in
  the z direction
• electromagnetic because E and H are intimately
  connected
• transverse because the E and H fields are
  contained completely in the x-y plane, transverse
  to the direction of propagation
• uniform because the magnitude of the field is
  constant wrt x and y

44
Lossless, uniform plane wave summary

• the direction of propagation is given by
• the propagation constant is
• the wavelength is
• inversely dependent on frequency
• inversely dependent on square root of m and e
• the phase velocity is
• independent of frequency
• inversely dependent on square root of m and e
• and the fields are

45
Poynting vector in phasor form

• the actual power flow would be
• where H is the complex conjugate of H
• the imaginary part of H is replaced by its
  negative
• notes
• the complex conjugate of a product is the product
  of the complex conjugates
• the complex conjugate of exp(ajb) is exp(a-jb)

46
Finite conductivity and loss tangent

• clearly the relative magnitudes of s and we are
  critical in determining whats going to happen
• lets take a quick look back to the original
  Maxwell equation that mixed conductivity,
  dielectric constant, and frequency
• tan q is called the loss tangent tan q
  s/(we)

47
Low loss material behavior in a good
dielectric

• lets consider different limits for the loss
  tangent (or equivalently, different limits for s
  compared to we)
• good dielectric
• s ltlt we or equivalently w gtgt s/e , i.e.,
  high frequency
• tan q ltlt 1

48
Good dielectric tan q ltlt 1 (w gtgt s/e)

• so the complex propagation constant in the low
  loss limit is
• note that the phase constant in this limit is the
  same as we got for the zero conductivity case
• or another way to write the same thing
• the attenuation constant is proportional to the
  conductivity
• the attenuation constant is proportional to the
  (frequencyloss tangent) product
• since in this limit s ltlt we ? a ltlt b

49
Good conductor wave propagation

• good conductor s gtgt we , low frequency w gtgt
  s/e
• in the limit s gtgt we well drop the small terms
  inside the bracket, so we have the propagation
  constant in a good conductor
• note that the real part (the attenuation
  constant) and the imaginary part (the phase
  constant) are identical, proportional to the
  square root of (frequencyconductivity)
• pick the sign on gamma to make sure the wave
  DECAYS, not grows!

50
Regardless of whether we call it a conductor or
a dielectric

• g is the complex propagation constant
• E and H are perpendicular to each other,
  direction of propagation is given by E x H
• what happens if s is not zero?
• since g2 is complex, so will be the square root
• lets write the real and imaginary parts of g as
• so a represents attenuation a is the
  attenuation constant
• b is still the phase (propagation) constant
• applet showing attenuation with finite
  conductivity (try s 0.05)

51
Plane waves and boundaries

• as a result of the incident wave there will be a
  transmitted wave and a reflected wave

x
medium 1 er1 mr1 s1
y
medium 2 er2 mr2 s2
z
52
Fields at the interface

• the total fields at the interface between the two
  materials (i.e., at z0) are
• infinitesimally to the left of the interface
• infinitesimally to the right of the interface
• since the fields are tangential to the interface,
  and were assuming there is no surface current in
  this problem, the fields must be CONTINUOUS
  across the interface
• so we have two unknowns, Ereflect (Ex1o-) and
  Etransmit (Ex2o)

53
Reflection coefficient

• we now have simple relation that gives the ratio
  of the reflected electric field to the incident
  electric field
• the reflection coefficient G is
• for our assumed coordinate system the sign of G
  will tells us which way the reflected electric
  field points
• Eincident pointed in the x direction
• if G is positive, then Ereflect also points in
  the x direction
• if G is negative, then Ereflect points in the -x
  direction

54
Transmission coefficient

• recall we had the equation from continuity of
  total tangential electric field at the interface,
  and we also have the reflection coefficient, so
• we define the transmission coefficient t to be
  the ratio of the transmitted electric field to
  the incident electric field

55
Power flow reflected power

• Poynting vectors at the interface (z 0)
• so the ratio of the incident power to the
  reflected power is

56
Power flow transmitted power

• Poynting vectors at the interface (z 0)
• so the ratio of transmitted power to incident
  power is

57
Generalized transmission line impedance

• relationships between the traveling wave
  quantities V- and V
• lets assume that we connect a load with
  impedance ZL at z 0
• what is Z seen at z -l?

58
T-Lines load reflection coefficient

• at z 0 the load voltage reflection
  coefficient is
• more generally, we define a reflection
  coefficient at the location z -l

59
Input impedance

• the impedance Z at the location z -l depends on
• the load impedance ZL
• the transmission line characteristic impedance Zo
  
• set mostly by the geometry dielectric constant
  of the wire-pair connected to the load
• the distance l between you and the load
• and the propagation constant g
• which depends on the T-line characteristics AND
  FREQUENCY!

60
L-C transmission line summary

• lossless L-C transmission line

61
The reflection coefficient for a lossless lines

• lets plot in the complex plane
• example
• ZLn 0.5 j0.5
• you go all the way around the circle when l is
  such that 2bl 2p
• recall b 2p/l, so you go all th way around
  when l l/2 !

62
Notes on the reflection coefficient on lossless
lines

• the load reflection coefficient behavior is given
  
• as r and x vary, what exactly does GL do?
• lets plot G in the complex plane
• for any non-negative value of Re(Z) (i.e., r 0)
  and any value of Im(Z), G falls on or within the
  unit circle in the complex G plane

• pick a value of x, then vary r

• pick a value of r, then vary x

63
Plot of reflection coefficient in complex plane

• the Smith Chart is a plot of the reflection
  coefficient in the complex plane, with contours
  of constant load resistance and load reactance
  superimposed

64
The Smith Chart

• the Smith chart also has a phase angle scale
  measured using distance in units of wavelengths
• link to high res smith chart
• why? because the angle of G is

• direction of increasing l, or equivalently
• away from load
• towards generator

65
Plot of reflection coefficient in complex plane
Im(G)

• example
• ZLn 0.5 j0.5
• you go all the way around the circle when l is
  such that 2bl 2p
• recall b 2p/l, so you go all the way around
  when l l/2 !
• recall the input impedance is
• tangent is periodic with period p, so you repeat
  every time bl mp
• recall b 2p/l, so you get back to the same
  impedance when l ml/2 !
• you can find Z(l) directly from the Smith chart!
• applet

-1
1
0
Re(G)