Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESyxFigure 25.1 The equipotentials (dashed circles) and field lines (solid lines)for a line charge perpendicular to the z-plane.of solving some two-dimensional physical problems describable by a potentialsatisfying ∇ 2 φ = 0. The general method is known as that of complex potentials.We also found that if f = u + iv is an analytic function of z then any curveu = constant intersects any curve v = constant at right angles. In the context ofsolutions of Laplace’s equation, this result implies that the real and imaginaryparts of f(z) have an additional connection between them, for if the set ofcontours on which one of them is a constant represents the equipotentials of asystem then the contours on which the other is constant, being orthogonal toeach of the first set, must represent the corresponding field lines or stream lines,depending on the context. The analytic function f is the complex potential. Itis conventional to use φ and ψ (rather than u and v) to denote the real andimaginary parts of a complex potential, so that f = φ + iψ.As an example, consider the functionf(z) = −q ln z (25.1)2πɛ 0in connection with the physical situation of a line charge of strength q per unitlength passing through the origin, perpendicular to the z-plane (figure 25.1). Itsreal and imaginary parts areφ = −q ln |z|, ψ = −q arg z. (25.2)2πɛ 0 2πɛ 0The contours in the z-plane of φ = constant are concentric circles and those ofψ = constant are radial lines. As expected these are orthogonal sets, but in additionthey are, respectively, the equipotentials and electric field lines appropriate to872