Finding electrostatic potential Griffiths Ch.3: Special Techniques week 3 fall EM lecture, 14.Oct.2002, Zita, TESC

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Finding electrostatic potentialGriffiths Ch.3
Special Techniquesweek 3 fall EM lecture,
14.Oct.2002, Zita, TESC

• Review electrostatics E, V, boundary
  conditions, energy
• Homework and quiz
• Ch.3 Techniques for finding potentials Why
  and how?
• Poissons and Laplaces equations (Prob. 3.3
  p.116), uniqueness
• Method of images (Prob. 3.9 p.126)
• Separation of variables (Prob. 3.12 Cartesian,
  3.23 Cylindrical)
• Your minilectures on vector analysis (choose one
  prob. each)

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Review of electrostatics
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Electrostatic BC and energy
Boundary conditions across a surface
charge potential and E are continuous
discontinuity in E? equal to
Electrostatic energy Homework and quiz
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Ch.3 Techniques for finding electrostatic
potential V

• Why?
• Easy to find E from V
• Scalar V superpose easily
• How?
• Poissons and Laplaces equations (Prob. 3.3
  p.116 last week)
• Guess if possible unique solution for given BC
  
• Method of images (Prob. 3.9 p.126)
• Separation of variables

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Poissons equation

• Gauss
  Potential
• combine to get Poissons eqn
• Laplace equation holds in charge-free regions
• V(r) average of V of neighboring points
• no local max or min in V(r)
• NB proof of shell theorem in Section 3.1.4,
  p.114

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Uniqueness theorems
(1) The solution to Laplaces eqn. in some
volume is uniquely determined if V is specified
on the boundary surface. (cf Fig.3.5 p.117) (2)
In a volume surrounded by conductors and
containing a speciried charge density, the
electric field is uniquely determined if the
total charge on each conductor is given. (cf
Fig.3.6 p.119) Elegant proof in Prob.3.5 p.121.
(cf Z.34)
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Solution V depends on boundary conditions
has solutions V(x)
mxb specify two points
or point slope Dirichlet and von Neumann BC
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Method of images

• A charge distribution r in space induces s on a
  nearby conductor.
• The total field results from combination of r and
  s.
• 
  -
• Guess an image charge that is equivalent to s.
• Satisfy Poisson and BC, and you have THE
  solution.
• Prob.3.9 p.126 (cf 2.2 p.82)

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Separation of variables
Guess that solution to Laplace equation is a
product of functions in each variable. If that
works, the diffeq is separable, and boundary
conditions will determine the unknown
constants. Cartesian coordinates
Prob.3.12 (worksheet) Cylindrical coordinates
Prob.3.23 (worksheet)
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1.1.3 Triple Products, by Andy
Syltebo 1.2.11.2.2 Ordinary derivatives
Gradient, by Don Verbeke 1.2.3 Del operator,
by Andrew White