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F16 PH333 3.1.1-3.1.3
Transcript of F16 PH333 3.1.1-3.1.3
3.1.2 Equation in 1D
3.1.3 Equation in 2D
Electrostatics: Goal, find the electric field - (Coulomb's law)
Problem: Nice geometries only, or bad integrals
Solution: Solve differential form, and use B.C.
Tools: Potentials, Gauss's law
Further: often we want the E field where there are no charges.
Its solutions are harmonic functions
Which just means they are second differentiable and continuous
There are some fundamental principles that must be obeyed
Laplace Equation in 1 D
What are solutions to this equation?
2 unknowns, set by boundary conditions
straight line, no local maxima/minima
smooth and continuous, points on solution are close to their neighbors, specifically they are the average
Let us have a potential in 1D set by B.C. specifying that the potential needs: V(x=2)=0 and V(x=4)=3
be a solution to Laplace's equation?
A It doesn't satisfy the B.C.
B It is not smooth and continuous
C There are local maxima/minima
D Each point is not an average of its neighbors
E It IS a solution!
Check the math!
Think Straight lines
Set by boundary conditions
Summary 1 D
Each point is an average of its neighbors (this means linear in 1D)
From above, no local maxima or minima
No general solutions - not just a plane
Not specified by 2 or any number of constants
Harmonic Functions (solutions)
Drum head / soap bubble
Key feature remain
Each point is still an average of its neighbors
Method of relaxation
Still no local maxima or minima
no local min or max
determined by B.C.
"What are harmonic functions? Are they just the result of laplacians, or are they a type of function which laplacians fall into?"
"I'm not quite sure how the relaxation method works."
"I just want some mathematical examples that make geometrical sense"
"Why did the average for the laplacian have a x-a and an x+a? that doesnt really seem like an average?"