Electrostatics Applications

Complex Analysis · Axiom Academy

Electrostatics with Conformal Maps An electric field is just the orthogonal net of an analytic function — so a conformal map can carry a field you can t solve onto a shape you can. In flat 2-D, the voltage obeys Laplace s equation , so it is the real part of an analytic complex potential — and a conformal map reshapes the region without breaking the field. Three moves: read it , let a conductor bend it , then map a hard corner to an easy wall . An electric field is an orthogonal net A point charge has complex potential . Its real part gives equipotentials (rings of equal voltage) and its imaginary part gives the field lines — and the two families always cross at right angles. Drag the charge and watch the net tighten. Drop in a conductor and the field bends Start with a uniform field and add the right image term: . A grounded conducting cylinder of radius R appears — its surface r=R becomes an equipotential , the field lines bend to meet it head-on (at right angles), and the field doubles at the poles facing the field. Grow the cylinder and watch the lines part. Map a hard corner to an easy wall

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