Electrostatics: What we have so far

1
Electrostatics What we have so far

• for the vector field D, the flux density, we
  have
• interpretation if the volume dv encloses charge
  dQ rvdv then the net flux emerging from the
  point is the charge density rv
• we also have the electric field as the gradient
  of potential
• so combining

if e is constant wrt space
2
Laplace and Poissons Equations

• for a uniform dielectric, the relation between
  the potential and the charge density is given by
  Poissons equation
• if the charge density is zero we get Laplaces
  equation (a special case of Poisson)
• we can now change our method of solution from an
  integral approach to one involving the solution
  of a differential equation, then apply BCs
• once we have V(r), we have everything since we
  can get E from V

if e is constant wrt space
3
Whats next History of magnetics

• from http//history.hyperjeff.net/electromagnetis
  m.html by Jeff Biggus
• 1st cent BCE
• Chinese fortune tellers begin using loadstone to
  construct divining boards, eventually leading to
  the first compasses. (Mentioned in Wang Ch'ung's
  Discourses weighed in the balance of 83 B.C.)
• 1st cent CE
• South-pointing divining boards become common in
  China
• ca 271 True compasses come into use by this date
  in China.
• 6th century (China) Discovery that loadstones
  could be used to magnetize small iron needles.
• 11th century (China) Iron magnetized by heating
  it to red hot temperatures and cooling while in
  south-north orientation.
• 1086 Shen Kua's Dream Pool Essays make the first
  reference to compasses used in navigation.
• 1155-1160 Earliest explicit reference to magnets
  per se, in Roman d'Enéas.
• 1190-1199 Alexander Neckam's De naturis rerum
  contains the first western reference to compasses
  used for navigation.

4
19th century

• 1820 Oersted notes deflection of a magnetic
  compass needle caused by electric current
  demonstrates effect is reciprocal initiates
  unification of electricity and magnetism.
• 1820 Biot and Savart deduce the formula for
  the strength of the magnetic effect produced by a
  short segment of current carrying wire.
• 1827 Ohm formulates the relationship between
  current to electromotive force and electrical
  resistance
• 1835 Gauss formulates separate electrostatic
  and electrodynamical laws, including Gauss's law
  
• 1838 Faraday explains electromagnetic induction,
  electrochemistry and formulates his notion of
  lines of force.
• 1838 Weber and Gauss apply potential theory to
  the magnetism of the earth.

5
Observations on magnetic fields

• there are no magnetic point charges
• we never see a force relationship that is exactly
  analogous to Coulombs force law for a single
  point charge
• bar magnets, for instance, always have force
  loops that start and end on the bar
• you always have both a south pole and a north
  pole
• if there are no permanent magnets present
• if there is no current flowing in a circuit,
  there is no magnetic field
• if there is current flow, there is always a
  magnetic field produced

6
Origin of magnetic fields

• simple experiment
• two parallel dc-current carrying wires, carrying
  current in the same direction, are attracted to
  one another
• this is NOT due to coloumbic attraction
• the wires are space-charge neutral, and hence
  there is no electric field outside the wires
• a sheet of floating metal placed between the
  wires has no effect on the force
• the magnitude of the force between the wires is
  proportional to
• the product of the two currents
• the inverse of the square of the distance between
  the wires
• if the directions of current flow are opposite,
  the force is repulsive
• the rail gun page http//home.insightbb.com/jmen
  gel4/rail/rail-intro.html
• magnetic fields cannot be explained solely on the
  basis of Coulombs law
• but when combined with special relativity, the
  existence of a field that is related to moving
  charge appears

7
Magnetic force

• the force on a moving charge in a magnetic field
  is perpendicular to both the velocity and the
  magnetic flux density (or magnetic induction) B
• it obeys the relationship
• in general, this looks to be trickier than
  Coulombic force, since we immediately have two
  vectors involved

• from http//hyperphysics.phy-astr.gsu.edu/hbase/hf
  rame.html

8
Cross products

• cross product vector result
• direction set by right hand rule

• cross product is not commutative
• A x B - B x A

• in rectangular coordinates

run cross product applet http//www.phy.syr.edu/c
ourses/java-suite/crosspro.html
9
Example the Hall effect

• example bar of conducting material, carrying a
  current I due to an applied electric field E,
  electrons as carriers

e-

• if apply magnetic field to drifting carriers a
  Lorentz force is generated

• note Lorentz force is proportional to velocity,
  direction depends on carrier type (sign of q)

10
Lorentz forces and the Hall effect

• side view again

• top view of sample

11
Magnetic flux density B

• SI unit of magnetic flux density (magnetic
  induction)
• Tesla Weber/m2 kg / (secAmp)
• weber volt-second
• One tesla is defined as the field intensity
  generating one newton of force per ampere of
  current per meter of conductor
• unit force / (lengthamp)
• nice dimensional and units tables
• http//www.sizes.com/units/index.htm
• http//www.udel.edu/mvb/units.html
• http//hyperphysics.phy-astr.gsu.edu/hbase/units.h
  tmluni4
• B often given in the non-SI unit of gauss 1T
  104 gauss
• earths magnetic field is 1/2 gauss 5x10-5 T
• 1T is 10,000 x earths magnetic field
• hair dryer B 10-7 10-3 T (0.2 to 20x earth's
  B field)
• color TV 10-6 T (2 earth's B field)
• small bar magnet will produce B 10-2 T
• sunspot B 0.3 T
• MRI body scanner magnet B 2T
• National High Magnetic Field Lab at Los Alamos
  (LANL) 60 T Long Pulse Magnet
• powered using a 1.4 gigaWatt inertial storage
  motor-generator

12
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

13
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

14
Example magnetic field of a line current

• infinitely long, straight current filament,
  current flowing from ? to ?
• by symmetry, there should be no variation with
  respect to z or ?

z
z
15
Example magnetic field of a line current

• infinitely long, straight current filament

• dL x r12 is INTO the page here

16
Example magnetic field of a line current

• infinitely long, straight current filament,
  current flowing from ? to ?

17
Example magnetic field of a line current

• infinitely long, straight current filament,
  current flowing from ? to ?

note direction is ?, points around the wire
18
Another line current circular loop

• assume a line current forms a circle in the x-y
  plane centered at the origin, radius ro
• lets find the magnetic field for points on the
  z-axis
• by symmetry, the field at points on the z axis
  should not depend on angle ?
• lets look in the y-z plane

z
z
field _at_ point 2
y
ro
y
ro
ro
1
rod?
x
current out of page
source _at_ point 1 current into page
19
circular loop line current

• line current forms a circle in the x-y plane
  centered at the origin, radius ro

dH

• BUT as we go around the loop of current the rho
  directed components will cancel
• for points on z-axis H points only in the
  z-direction

20
Circular loop line current H-field on z-axis

• for points on z-axis H points in the z-direction
  only
• to get total H field still need to integrate dHz
  over ? from 0 ? 2?

21
Circular loop (ring) current asymptotic behavior

• for z gtgt ro (far away)
• for z ltlt ro

22
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
23
Example Amperes Circuital Law and a long
straight wire

• just like when trying to use Gausss law, we need
  to look for symmetries to pick the best path
  along which to evaluate the line integral
• youd like the dot product to either give a
  constant or zero
• for an infinitely long straight current filament
  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

z
r
?
I
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Example Amperes Circuital Law and a long
straight wire

• for an infinitely long straight current filament
  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!

25
Example Amperes Circuital Law and finite radius
wire

• for an infinitely long straight current carrying
  wire of radius a 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
  !

26
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
27
Example Amperes Circuital Law and finite radius
wire
z
a
y
r
x

• what happens at r a?
• so H is continuous here as you move from inside
  to outside the wire

28
Example Amperes Circuital Law and coaxial cable

• finite radius wire carrying current I in z
  direction, surrounded by finite thickness
  shield carrying the return current I in the z
  direction
• 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 b, the same as for the finite radius wire
  since current contained is the same

29
Example Amperes Circuital Law and coaxial cable

• what happens for r gt b?
• for r gt c its really easy
• Ienclosed 0 ? H 0
• for b gt r gt c ,we just need to find out the net
  current enclosed

30
Example Amperes Circuital Law and coaxial cable

• so the full solution is

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Differential form of Amperes law

• Amperes law is an integral form
• is there a differential form as well?
• recall we did something like that with Gausss
  law
• here lets look at a generic H field, and a path
  as shown, with the path small enough that the
  field changes only slowly on it

32
Differential form of Amperes law

• generic H field, and a path as shown, with the
  path small enough that the field is changes
  slowly on it

33
Differential form of Amperes law
z

• we still need the value of H along each side of
  the path
• assuming we know the value at the center Ho, we
  can estimate the value along the sides using a
  one term Taylor series approximation
• for path 1-2 we are located a distance ½dx from
  our central point, and the rate of change of H we
  need is its change wrt x

dx/2
1
2
x
34
Differential form of Amperes law

• using the same approach

35
Differential form of Amperes law

• combining

36
Differential form of Amperes law

• but by Amperes law this path integral over H
  should equal the current passing through the
  closed path
• here our path is in an x-y plane, so the current
  passing through it would have to be in the
  z-direction
• assume the current density is Jz, then the
  current would be
• Ienclosed Jz(area) Jz(dxdx)

37
Differential form of Amperes law

• letting the path become smaller and smaller the
  approximations become exact, so we have
• doing the same thing for surfaces oriented in the
  x-z and y-z planes gives the same basic results

38
Amperes law at a piont

• so we have found a relationship between the
  vector current density J and the vector magnetic
  field H

39
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 !

40
Curl in rectangular coordinates

• from Amperes law weve actually found the curl
  in rectangular coordinates

41
Curl in other coordinates

• cylindrical coordinates
• spherical coordinates

42
What does Curl mean?

• the paddle wheel analogy in water flow (applet)
• examples

vector field analyzer
43
What does Curl mean?

• the paddle wheel analogy in water flow (applet)
• examples

vector field analyzer
44
Curl of H from a wire

• For a circular wire we found

45
Curl of H from a wire

• but its a lot easier in cylindrical coordinates!

46
Curl of H from a wire

• For a circular wire we found

47
Curl of H from a wire

• For a circular wire we found

vector field analyzer
48
Curl applets

• http//www.ee.surrey.ac.uk/Teaching/Courses/EFT/dy
  namics/html/curl.html
• http//math.la.asu.edu/kawski/myjava/vfanalyzer/

49
some math and vector results we may need

• Stokes Theorem
• you can get this using the curl concepts
  discussed above
• and we had the Divergence Theorem from before
• and a couple general vector identities

from Lorrain and Corson
50
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

51
Amperes Law from the curl equation

• we can reverse the derivations to get back to
  integrals
• apply Stokes theorem to H
• but
• so
• which says that the path integral of H along a
  closed path equals the current flowing through
  the surface (area) bounded by the path

52
Potential functions for magnetic fields

• recall that in electrostatics it was sometimes
  easier to find the potential function, then get
  the electric field from the gradient
• we could either integrate over charge density to
  get V, or solve a differential equation (Poisson
  or Laplace) subject to boundary conditions
• is there a potential function for H, and is it
  easier to find??
• lets assume there is a scalar magnetic potential
  Vm that gives H from the negative gradient (just
  like in electrostatics)
• what conditions, if any, are there on Vm?