11.6 • Two-DIMENSIONAL ELECTROSTATICS 4 6 1y
w=logzvxFigure 11.36 The electrical field in a coaxial cylinder, where U2 < U 1.x > 1, y = 0 onto the vertical strip - 2 " < u. < I · The new problem is to find
the potential i1> ( u, v) that satisfies the boundary values
i1> ( ~?T, v) = - 300 and i1> (~, v) = 300, for all v.
From Example 11.1,
600
il>(u., v) = - u..
1T
As in the discussion of Example 11.17, the solution in the z plane is
(x, y) =
600
Re(Arcsinz)
1T
600. V(x+l)2+y2- J(x- l)2+y2
= - Arcsm....:.....-------'---- -
rr 2Several equipotential curves are shown in Figure 11.37.
- EXAMPLE 11. 21 Find the electrical potential</> (x, y) in the disk D: lzl < 1
that satisfies the boundary values
(x, y) = 80,
for x + iy = z on Ct = { z = eiO : 0 < e < ~} ;
for x + iy = z on C2 = { z = eiO : ~ < e < 21T}.
(1 - i) (z - i)Solution The mapping w = S (z) = is a one-to-one conformal
z-1
mapping of D onto the upper half-plane Im (w) > 0 with the property that C1