3.5 Two dimensional problems

• Cylindrical symmetry

• Conformal mapping

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

1

)sincos)((

)ln(),(

nnn

nn

nn nDnCrBrA

rBArV

Example: Two half pipes

Conformal Mapping

Is there a simple solution?

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

Parallel Plate Capacitor

zwwwd

z

Fwwi

VzF o

map )]1(2

1[ln

map ln)(

2

]2sin2

1[ln

)]2cos1(2

1[ln

ln,

0,0,

2

2

dy

dx

VhVV

ew

oo

i