Transcript EM3

Electromagnetism Ch.3
Methods of Math. Physics, Thus. 11 Feb. 2011, EJZ
Special Techniques in Electrostatics
• Review HW?
• 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, 3.23)
Techniques for finding V (potential, voltage)
Why?
• Easy to find E from V
• Scalar V superpose easily
How?
• Poisson’s and Laplace’s equations (Prob. 3.3 p.116)
• Guess if possible: unique solution for given BC
• Method of images (Prob. 3.9 p.126)
• Separation of variables (Spherical harmonics, Fourier series…)
Poisson’s equation
Gauss:
  
E 
0
Potential:

E   V
combine to get Poisson’s eqn:
Laplace’s equation holds in charge-free regions:
d 2V
 0 in 1-D (x):
Solve
2
dx
Laplace’s equation
V 0
2
→ ∂V =____
→ ∂V =____
Method of images
A charge distribution  induces s on a nearby conductor.
The total field results from combination of  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)
Method of Images: #3.9 p.126
Now put this line of charge
above a conducting plane
Guess that the induced charge
on the plane has the same field
as a line of (-) charge the same
distance BELOW the sheet.
By symmetry, V=0 at the conducting sheet. This satisfies BC.
Simplify the ln terms, and write in terms of z, d, and y…
Simplify, and write up your solution for homework.
break time …
Separation of Variables
When to use separation of variables?
• In charge-free regions
• With well-specified boundary conditions (BC)
• Without sufficient symmetry for Gauss’ law
How to use separation of variables?
• Guess form of solutions based on BC
• Separate variables, insert guessed solutions with constants
• Apply BC and solve for constants
Review Poisson and Laplace equations
Gauss:
  
E 
0
combine to get Poisson’s eqn:

E   V
Potential:
V
2

0
Laplace equation holds in charge-free regions:
 2V  0
We have found the general solutions to Laplace’s eqn. in
spherical and cylindrical coordinates when V=V(r) …
Solving Laplace w/ Separation of Variables
 V 0
2
Worksheets for Problems 3.12 (136), 3.23 (145)
Homework due next week: work through Ex.3.3, do
3.12 and 3.23. Extra credit: #13, 24.
Other good problems: #16, 17, 18 (p.144)