Consider a 2D Euclidean plane having two distinct sources distributions, each along the curves $S_1$ and $S_2$. The density of sources along both curves are identical and let it be equal to $m$. If I'm not mistaken, the problem can be defined in terms of a potential function at any given point $z=x+iy$ in a complex plane. It is given by: $V(z) = \int_{S_1}\frac{m}{2 \pi} \log{(z-\zeta)d\zeta} + \int_{S_2} \frac{m}{2 \pi} \log{(z-\zeta)d\zeta}$. This being the case,
(a) what is the general condition to find the boundary curve that divides the space between $S_1$ and $S_2$?
(b) Consider the simplest case where the curves $S_1$ and $S_2$ becomes two points. The solution is straight forward and intuitive. The boundary, which can be defined as the curve along which the effect due to the distributions on either side is absent, should be along a line perpendicular to the line segment joining the point sources ($S_1 S_2$) and passing through its midpoint. The equation $\frac{dV}{dz}=0$ apparently gives only the midpoint. Is it possible to get the equation of this line from this condition?
(c) Is there any change if $S_1$ and $S_2$ becomes more complicated but non-self intersecting curves?