## Coulomb’s Law in (1+2) dimensions

The idea for this post came from an unsolved problem in Tony Zee’s book on QFT. We begin with the ‘mutual’ energy:

Define . Transforming to cylindrical polar coordinates, this is

The integral produces a Bessel Function of order 0, i.e. , so that

I’m lazy, so I used Mathematica to get the final result, which is

where denotes the modified Bessel function of the second kind, of order 0. A plot of as a function of is shown below.

The graph in red is that of the usual 1/r potential. So, in 2 spatial dimensions, the potential energy actually falls off faster than 1/r.

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It’s great to see another engineer following Zee’s book! 😀

The potential energy only falls of faster than 1/r due to the mass of the exchanged particle (it’s analogous to the exponential factor in the 3D Yukawa potential). If you do the integral for m = 0 using r = 1 as the reference point for the energy, the result is the usual logarithmic potential.

Best wishes!

mchouzaDecember 19, 2010 at 11:39

Hi, thanks for your comment. Have a look at a sequel to this post as well.

VivekDecember 20, 2010 at 23:19