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F16 PH 333 3.1.4-3.1.6

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Richard Datwyler

on 16 October 2018

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Transcript of F16 PH 333 3.1.4-3.1.6

fix this!!
Laplace's Equation in 3D & Uniqueness
3.1.4 3D Laplace equation
3.1.5 B.C. and uniqueness
3.1.6 Conductors and more uniqueness
Same two properties
Average value
No local max or min
only at boundary points
This is the potential a distance z away (at the center of this sphere)
Boundary Conditions
V is not enough, we need BC
1 D is easy
either value or derivative
3D much harder
complete determined by BC
only one - unique- solution
Uniqueness Theorem
The solution to Laplace's equation in some volume is uniquely determined if the potential is specified on the boundary surface to that volume.
Uniqueness proof
consider two solutions to the same boundary conditions
Define third such that:
Consider V on the boundaries of our surface
But if V is zero on all boundary then it is also zero everywhere.
Second uniqueness theorem
In a volume V surrounded by conductors and containing a specified charge density, the electric field is uniquely determined if the total charge on each conductor is given
Harder proof, with trick
Full transcript