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WOLFRAM NOTEBOOK
Electrostatic Fields Using Conformal Mapping
Electrostatic Fields Using Conformal Mapping
A conformal mapping produces a complex function of a complex variable, , so that the analytical function maps the complex plane into the complex plane. This technique is useful for calculating two-dimensional electric fields: the curve in the plane where either or is constant corresponds to either an equipotential line or electric flux. This Demonstration shows 10 examples of electrostatic fields often encountered in high voltage applications. The electric field is shown in the - plane (or the plane, where ). The electrodes correspond to either or , where (). The 10 examples are:
f
w=f(z)
f
z
w
w
Re[z]
Im[z]
u
v
w
w=u+iv
x=,
x
1
x
2
y=,
y
1
y
2
z=x+iy
x,y,u,v∈
• concentric circles:
f(z)=
-iz
e
• ellipses:
f(z)=sin(z)
• hyperbolas:
f(z)=sin(z)
• parabolas:
f(z)=
2
z
• bipolar circles:
f(z)=icot(z)
• Cassinian ovals:
f(z)=+1
-iz
e
• elliptical pairs:
f(z)=sn(iz)
• blade to plate shape:
f(z)=cn(z)
• Maxwell curves:
(z+1+)/π
z
e
• square edge using a function derived by the Schwarz–Christoffel method
Three options give slightly different boundary conditions or electrode potentials or . The parameters are shown on the right. The calculated electric fields are shown by color, normalized to the average field =|-|/g or , where is the smallest distance between two electrodes. If you select option 1, the local field is high in the vicinity of sharp electrode edges. When selecting option 2 or 3, the values are reduced owing to blunted edge conditions. The white lines indicate the flux line and the dashed lines are the equipotential lines for constant or constant . Those two families of curves are orthogonal.
x=,
x
1
x
2
y=,
y
1
y
2
E
ave
x
1
x
2
|-|/g
y
1
y
2
g
x
y