The function  is a complex potential of a two-dimensional electrostatic field.(a) Show tha...

The function  is a complex potential of a two-dimensional electrostatic field.

(a) Show that the equipotential curves φ(x, y) = c1 and the lines of force ψ(x, y) = c1 are, respectively

Observe that the equipotential curves and the lines of force are both families of circles.

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(b) The centers of the equipotential curves in part (a) are (coth c1, 0). Approximately, where are the centers located when c1 → ∞? When c1 → −∞? Where are the centers located when c1 → 0+? When c1 → 0−?

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(c) Verify that each circular line of force passes through z = 1 and through z = −1.

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(d) Use a CAS or graphing software to plot representative circles from each family on the same coordinate axes. If you use a CAS do not use the contour plot application.