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EM & Vector calculus #4Physical Systems, Tuesday 13 Feb. 2007, EJZ
Vector Calculus 1.4: Curvilinear Coordinates
• Quick review of quiz and homework
• Review Cartesian coordinates, unit vectors, and dl
• Spherical and cylindrical
• Coordinates, unit vectors, dl, and vector derivatives
Ch.3b: Finding potentials using separation of variables
• Quick review of quiz and homework
• Example 3.3
• Worksheets for Problems 3.12 and 3.23
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Vector calculus HW & Quiz review
Online solutions at http://192.211.16.13/curricular/physys/0607/solns/
HW: VCsoln31.pdf, EMsoln3a.pdf, EMsoln3b.pdf
Quiz: VCMidSoln.pdf, EMmodMidSoln.pdf
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Cartesian Coordinates
The infinitesimal displacement vector from (x,y,z) to (x+dx, y+dy, z+dz) is dl:
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Cylindrical Coordinates
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Spherical Coordinates
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Cylindrical Coordinates
Derive these (Problem 1.41):
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Spherical Coordinates
Derive these (Problem 1.37):
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Vector calculus HW due next week:
Ch.1.4 Problems: 1.37, 1.38, 1.41, 1.42
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E&M Ch.3b: Separation of variables
Quick review of quiz and homework
When to use separation of variables?
• In charge-free regions
• With well-specified boundary conditions
• Without sufficient symmetry to use 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
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Poisson and Laplace equations
Gauss: Potential:
combine to get Poisson’s eqn:
Laplace equation holds in charge-free regions:
Last week we found the general solutions to Laplace’s eqn. in spherical and cylindrical coordinates for the case where V depends only on r (Prob.3.3, p.116) →
0
E
VE
2
0
V
2 0V
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Solving Laplace w/ Separation of Variables
2 0V
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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.