Introduction to Electromagnetism

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Finding electrostatic potential
Griffiths Ch.3: Special Techniques
week 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?
• Poisson’s and Laplace’s 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)
Review of electrostatics
Electrostatic BC and energy
Boundary conditions across a surface charge:
potential and E|| are continuous; discontinuity in E equal to  
0
Electrostatic energy:
W
1
2
  d 
Homework and quiz
0
2
2
E
 d
Ch.3: Techniques for finding
electrostatic potential V
Why?
• Easy to find E from V
• Scalar V superpose easily
How?
• Poisson’s and Laplace’s 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
Poisson’s equation
Gauss:
  
E 
0
Potential:
combine to get Poisson’s 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

E   V
Uniqueness theorems:
(1) The solution to Laplace’s 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)
Solution V depends on boundary conditions:
V
 V 0 2
x
2
2
has solutions V(x) = mx+b
specify two points
Dirichlet and von Neumann BC
or point + slope
Method of images
A charge distribution  in space induces  on a nearby conductor.
The total field results from combination of  and .
+
-
• Guess an image charge that is equivalent to .
• Satisfy Poisson and BC, and you have THE solution.
Prob.3.9 p.126 (cf 2.2 p.82)
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.
V=0
y=a
V=?
y=0
V=0
Cartesian coordinates: Prob.3.12 (worksheet)
Cylindrical coordinates: Prob.3.23 (worksheet)
x
1.1.3 Triple Products, by Andy Syltebo
1.2.1+1.2.2: Ordinary derivatives + Gradient, by Don Verbeke
1.2.3: Del operator, by Andrew White

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