Show that if w = (z + 3)/(z - 3), w = u + iv, z = x + iy, the circle u2 + v2 = k

Question

Show that if w = (z + 3)/(z - 3), w = u + iv, z = x + iy, the circle u2 + v2 = k2 in the w plane is the image of the circle x2 + y2 + 6 1+k2/1-k2 x + 9 = 0 (k2 1) in the z plane. The long cylindrical wires, each of radius 4 mm, are placed parallel to each other with their axes 10mm .apart, so that their cross-section appears as in figure 1. The wires c any potentials 2V0 as shown. Show that the potential V(x, y) at the point (x, y) is given by V = V0/ln4 {ln[(x + 3)2 +y2] - ln[(x - 3)2 + y2]} Figure 1 Cylindrical wires.

Explanation / Answer

Question Details

Show that if w = (z + 3)/(z - 3), w = u + iv, z = x + iy, the circle u^2 + v^2 = k^2 in the w plane is the image of the circle x^2+y^2+6(1+k^2/1-k^2)(x)+9=0 (k^2 not equal 1) in the z plane.

w = (z + 3)/(z - 3) =U+IV

U^2+V^2=K^2...................................................................1

IMPLIES

|W|= =K

HENCE WE GET

|[(Z+3)/(Z-3)]| =K

|(Z+3)|/|(Z-3)|=K

|Z+3]=K|Z-3|

z = x + iy

|X+IY+3|=K|X-IY-3|

|(X+3)+IY|=K|[(X-3)+IY]|

=

SQUARING

FOR K^2 NOT EQUAL TO ONE WE GET ON DEVIDING WITH 1-K^2....

......................................................2

HENCE the circle GIVEN BY EQN.1 ...VIZ.....

u^2 + v^2 = k^2..............................................1

in the w plane is the image of the circle GIVEN BY EQN.2

in the z plane. ...FOR K^2 NOT EQUAL TO ONE .

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