Yes, I mentioned that perspective in my previous article:

• Parallel line masses and Marden’s theorem.

where I used Newtonian gravity instead of electrostatics. I needed both “line masses” whose potential goes like log(r) and also point masses whose potential goes like -1/r.

]]>The way I was taught this bit of electrostatics is not as 2d but as a cylindrical geometry in 3d. The contributions of field strength further along the cylinder axis generate the 1/r dependence radially outward from the axis.

]]>Remember, I’m doing electrostatics in 2d space in this post.

In 3-dimensional space the potential of a point charge is proportional to 1/*r*, but in 2-dimensional space the potential is proportional to the logarithm of distance.

The reason for this is that in 3-dimensional space the electric field of a point charge is proportional to 1/*r*^{2}, while in 2-dimensional space the electric field is proportional to 1/*r*.

And the reason for this is that in 3-dimensonal space the area of a sphere is proportional to *r*^{2}, while in 2-dimensional space the circumference of a circle is proportional to *r*.

Thanks! That makes sense.

]]>Then

]]>Remember that in 2d space a point charge at *a _{i}* creates a potential proportional to

so a collection of equal point charges at points *a _{1}*, … ,