Well I kind of get what you want, but you do seem to have a singularity at the origin when r=0.
p = 1; sig = 1; % constants that are still a mystery.
V = @(r,theta) min(max(p*cos(theta)./(4*pi*sig*r.^2),-1),1); % Need thresholding here.
npts = 1000;
r = linspace(-1,1,npts)';
phi = linspace(0,2*pi, npts)';
[R, Phi] = meshgrid(r,phi);
E = V(R,Phi);
[X,Y] = pol2cart(Phi,R);
contourf(X,Y,E)
colorbar
and this gives 
