1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.6 • TWO- DIMENSIONAL ELECTROSTATICS 465

6. Find the electrostatic potential</> (x , y ) in t he infinite strip 0 < x < ~ that satisfies
   the following boundary values (shown in Figure 11.44). Hint: Use w = sinz.

¢ (0, y) = 100 ,

"'G· v) = o,

¢(0,y) = - 100 ,

y

?= 100

0

?= - 100

Figure 11.44

for y > O;

for all y;

for y < 0.

7. Consider t he conformal mapping w = S (z) =
   2
   z -
   3

6

.

z +

(a) Show that S (z) maps the domain D that is the portion of the right

half-plane Re (z) > 0 that lies exterior to the circle lz - 51 = 4 onto

the annulus 1 < lwl < 2.

(b) Find the electrostatic potential <P (x, y) in the domain D that satisfies

the boundary values shown in Figure 11.45:

</> (0, y) = 100,

<f>(x, y) = 200

Figure 11.45

for all y;

when lz - 51 = 4.

z-10

8. Consider the conformal mapping w = S (z) =
   2
   z _
   5
   ·

u

(a) Show that S (z) maps the domain D that is the portion of the disk

lzl < 5 that lies outside the circle lz - 21 = 2 onto the annulus defined

by 1 < lwl < 2.