= Solution
For the insulating wall, the image instead has the same strength $q$. With the distances defined above,
$$
\boxed{V_N(x,y)=\frac q{2\pi}(\log r+\log r_*)+C,\qquad x<b.}
$$
It has the required source in the physical domain. Its normal <derivative> is
$$
\partial_xV_N=\frac q{2\pi}\left[\frac{x}{x^2+y^2}+\frac{x-2b}{(x-2b)^2+y^2}\right].
$$
At $x=b$ the terms are $b/(b^2+y^2)$ and $-b/(b^2+y^2)$, so they cancel. The <Neumann boundary condition> leaves the additive constant $C$ free.
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