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3.5 Two dimensional problems. Cylindrical symmetry Conformal mapping. Laplace operator in polar coordinates. Example: Two half pipes. Conformal Mapping. Is there a simple solution?. iy. x. Examples:. For two-dimensional problems complex analytical function - PowerPoint PPT Presentation
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Laplace operator in polar coordinates
01
)(1
2
2
2
V
rr
Vr
rr
)()( rRV
,....2,1,0sincos)( nnDnC nn
)ln()(,...2,1)( rBArRnrBrArR nn
nn
For two-dimensional problems complex analytical functionare a powerful tool of much elegance.
ieiyxz
x
iy
)(
),(),()(
zFivuw
yxihyxgzF
Maps (x,y) plane onto (u,v) plane.
For analytical functions the derivative exists.
Examples: zezzz znn ln,,sin,, /1
Analytical functions obey the Cauchy-Riemann equations
which imply that g and h obey the Laplace equation,
, and x
h
y
g
y
h
x
g
.0 and 02
2
2
2
2
2
2
2
y
h
x
h
y
g
x
g
If g(x,y) fulfills the boundary condition it is the potential.
If h(x,y) fulfills the boundary condition it is the potential.
g and h are conjugate. If g=V then g=const gives the equipotentialsand h=const gives the field lines, or vice versa.
If F(z) is analytical it defines a conformal mapping.
A conformal transformation maps a rectangular grid onto a curvedgrid, where the coordinate lines remain perpendicular.
Cartesian onto polar coordinates:
.lnln,,ln izezzw i
.,, iyxz eewiyxzew
Example
Polar onto Cartesian coordinates:
i2 Fullplane
z w
A corner of conductors
oiVAzzF 2)(
constyxAyxg
VAxyyxV o
)(),(:lines field
2),( :potential22
xAx o 2)( :plate horizontal on thedensity charge
Edge of a conducting plane
oiVAzzF 2/1)(
2/122
2/122
)(2
),(
)(2
),(
xyxA
yxg
VxyxA
yxV o
equipotentials
field lines