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F16 PH333 3.2.4-3.3.1
Transcript of F16 PH333 3.2.4-3.3.1
3.2.4 Final method of images
3.3.1 Separation of variables (Cartesian)
by symmetry, it must be on the axis.
we want the sphere to be zero, so the point in front and behind must be zero
Separation of variables
actual solution to Laplace equation
solutions are products of functions of only one coordinate
find V using boundary conditions for either V or sigma (charge density)
start with Cartesian then move to spherical
note, we need boundary conditions to specify the unknowns.
V=0 at infinity kills of A
the rest depends on the object
V(y=0)=0 kills off D
V(y=a)=0 specifies k
Last boundary condition
use Fourier and multiply by
completeness and orthogonality
Repeating boundary conditions
If a set of functions can express ANY other function then it is complete
Orthogonality says that the integral of the product of any two different members of this set is zero
this is a such a set of functions!
now we cannot delete A
introduce hyperbolic functions
We can handle k and D the same way
x=b V = V
This works in 3 D as well.
positive vs. negative constants. (negative constants are sinusoidal, repeating
"I'm not clear why a non-zero potential works for a sphere."
use V(x=0) = V