What we have so far

1
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
The Laplacian operator

• weve written things symbolically, but what are
  the actual derivatives?
• lets try things in rectangular coordinates

3
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)
• 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

if e is constant wrt space
4
various coordinates

• rectangular
• cylindrical
• spherical

5
What do we really need to know?

• how much of the potential do I need to know in
  order to find the potential everywhere else?
• if I only know the potential or its normal
  derivative in certain regions of space
• can I use that to find the potential somewhere
  else?
• if I can, do I know if what I get is the ONLY
  answer?
• i.e., is the solution UNIQUE?
• Laplace and Poisson equations are examples of
  elliptic partial differential equations
• for this type of differential equation, if you
  can find a solution that satisfies specified
  boundary conditions, then uniqueness is assured!!
• this is REALLY important if you can find one
  solution you dont have to keep looking!

6
Math stuff

• Dirichlet problem for Laplace's equation
• consists of finding a solution f inside a region
  of space such that on the boundary f is known to
  equal some given function.
• example the Laplacian operator appears in the
  heat equation
• physical interpretation
• fix the temperature on the boundary of the domain
  and wait until the temperature in the interior
  doesn't change anymore
• the temperature distribution in the interior
  will then be given by the solution to the
  corresponding Dirichlet problem
• Neumann boundary conditions for Laplace's
  equation
• rather than knowing the values of f itself on the
  boundary, the normal derivative of f is known
  instead
• physically, this corresponds to the construction
  of a potential for a vector field whose effect is
  known at the boundary alone
• solutions to Laplace's equation which are twice
  continuously differentiable are called harmonic
  functions such functions are all analytic

7
Boundary Conditions

• Dirichlet (values specified at boundary) V
  b1 on dW
• Neumann (normal derivative specified at
  boundary) ??/?n n???  b2 on dW
• Robin mix of the two above
• Only 1 boundary condition may be specified on
  each element of the surface dW. But can have
  Neumann for one portion of dW and Dirichlet for
  the other.
• Physically Poisson equation describes many
  things
• example steady state-temperature distribution of
  an object occupying W, with heat sources and
  sinks represented by
• Dirichlet boundary conditions represent the
  situation when the temperature is specified at
  boundary
• Neumann would be the case where the heat flux is
  specified at boundary. For example, if ??/?n
  n??? 0 we would have a perfectly insulating
  boundary

8
Laplace equation examples

• problems that have conductors on the boundaries
  are usually easy to do
• conductors are equipotentials, i.e., V constant
  on a metal surface
• hence we have the Dirichlet boundary conditions
  (values specified at boundary) for these problems
• lets try to solve the pde for some simple cases
• assume we know (hopefully by symmetry) that the
  solution varies with only one coordinate
• example coaxial cable aligned along the z-axis,
  uniform dielectric, no free charge
• cylindrical coordinates, no ? variations
• solution varies only with r

9
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

10
Laplace equation solution conducting wedge

• example two metal planes, almost touching along
  the z-axis, separated by angle ? a
• cylindrical coordinates, no r variation in V
• solution varies only with ?

• now apply BCs
• V Vbottom _at_ ? 0

• V Vtop _at_ ? a

11
Laplace equation solution conducting wedge

• example two metal planes, almost touching along
  the z-axis, separated by angle ? a
• cylindrical coordinates, no r variation in V
• solution varies only with ?
• example a p/2 , Vb 0, Vt 1 volt

• equipotentials arestraight lines from origin

? stream lines are circles
12
Conducting wedge charge
w

• example two metal planes, almost touching along
  the z-axis, separated by angle ? a
• cylindrical coordinates, no r variation in V
• solution varies only with ?

l
blows up at r 0
still blows up!
13
Laplace and Poissons Equations

• 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)
• refs
• http//hyperphysics.phy-astr.gsu.edu/hbase/electri
  c/laplace.html
• http//www.physics.arizona.edu/restrepo/475B/Note
  s/source/node48.html
• 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

if e is constant wrt space
14
More general solution methods for Laplaces
equation

• separation of variables
• lets guess that the solution is a product of
  three functions, each a function of only one of
  the coordinates

15
Separation of variables for Laplaces equation

• lets guess that the solution is a product of
  three functions, each a function of only one of
  the coordinates

• similar for Y(y) and Z(z)

• to solve a specific problem all we have to do
  is find the constants
• apply boundaries conditions!

16
Example

• consider a long metal pipe with rectangular
  cross-section
• three sides are grounded, V 0
• one side is held at V Vo
• only variations in V in x and y

17
rectangular cross-section pipe with conducting
walls

• lets do y first
• first BC V 0 at y 0
• ? B 0
• second BC V 0 at y b
• ? a mp/b

18
rectangular cross-section pipe with conducting
walls

• now lets work on x
• first BC V 0 at x 0
• ? C - D

19
rectangular cross-section pipe with conducting
walls

• this solution will satisfy the three sides with V
  0
• lets check
• but what about x d??
• ? its not Vo !!

20
rectangular cross-section pipe with conducting
walls

• we still need to satisfy the BC at x d V(d, y)
  Vo
• there is no way to satisfy this BC as y varies!
• lets try superposition
• can we satisfy the BC now?

21
Fourier Series

• let f(x) be a function defined and integrable on
  the interval -L, L
• the Fourier series of f(x) is
• reference on Fourier series
• applets http//www.falstad.com/fourier/
• so we need to extend our function so it looks
  periodic

22
rectangular cross-section pipe with conducting
walls

• using superposition of our solutions to satisfy
  the BC on the right
• at x d, 0 lt y lt b V Vo
• this looks just like a Fourier series!
• we need to find the constants in the infinite sum
  necessary to satisfy the boundary condition at x
  d

23
Fourier Series for x d

• so here the Fourier coefficients are
• so the series for the x d boundary is

24
Fourier Series for x d

• so the series for the x d boundary is
• applets http//www.falstad.com/fourier/

25
Full solution

• at the x d boundary

• but from the Fourier solution

• comparing the two results at x d we find
• and so we have the complete solution for the
  potential inside

26
Full solution

• the complete solution for the potential inside

27
Full solution electric field

• what about the electric field inside?

28
Electric field

• does this make sense?
• BCs at metal Etan 0
• bottom y 0 tan Ex
• top y b tan Ex
• left x 0 tan Ey
• right x d tan Ey

29
Electric field

• field lines are perpendicular to the
  equipotentials
• the fields are VERY big near the corners on the
  right side!

30
Free charge density

• what about the charge inside?

31
Conductor surface charge

• what about the surface charge on the conductors?
• negative if D points into the surface
• positive if D points out of the surface
• bottom Dnorm Dy(x, 0)
• top Dnorm Dy(x, b)
• left side Dnorm Dx(0, y)
• right side Dnorm Dx(d, y)

32
Full solution conductor surface charge

• surface charge on the bottom Dnorm Dy(x, 0)
• surface charge on the top Dnorm Dy(x, b)
• when you do the dot product to get rsurf, the
  extra minus sign will cancel since surface
  normal on bottom is y, but surface normal on
  top is y
• rsurf top rsurf bottom

33
Full solution conductor surface charge

• surface charge on the left side Dnorm Dx(0, y)
• surface charge on the right side Dnorm Dx(d,
  y)

• link to calculations

34
Applets showing some vector fields

• 2-d view http//www.physics.orst.edu/tevian/micr
  oscope/
• 3-d view http//www.falstad.com/vector/
• fields available http//www.falstad.com/vector/fu
  nctions.html
• 1/r single line electric field around an
  infinitely long line of charge. It is inversely
  proportional to the distance from the line.
• 1/r double lines field around two infinitely
  long conductors. The distance between them is
  adjustable.
• 1/r2 single field associated with gravity and
  electrostatic attraction gravitational field
  around a planet and the electric field around a
  single point charge.
• This is a two-dimensional cross section of a
  three-dimensional field.
• In three dimensions, the divergence of this field
  is zero except at the origin in this cross
  section, the divergence is positive everywhere
  (except at the origin, where it is negative).
• 1/r2 double field associated with gravity and
  electrostatic attraction. gravitational field
  around two planets and the electric field around
  two negative point charges are similar to this
  field.