An Integral Formula for Two-Dimensional Fields

The complex analytic function treatment previously given [J. Appl. Phys. 37, 2568 (1966)] for two‐dimensional magnetic fields outside conductors is extended to regions of constant current density, σ, inside conductors. Separate analytic functions for field points Z = X+iY inside and outside such conductors are given by the integral formula $\mathrm{F(Z)=i\sigma \oint z*dz/(z-Z)}$ , where z* is the complex conjugate of z = x+iy on the conductor circumference. The complex field, H = H_Y+iH_X, is then F(Z) + 2πσZ* for points Z inside the conductor and F(Z) for Z outside. If we replace σ by −iρ we obtain the field E = E_Y+iE_X in the analogous electrostatic case with constant charge density ρ, e.g., for a uniform beam of charged particles. The results are illustrated by evaluating the integral for an elliptical contour.