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This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of that question.

I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. (See e.g. Wikipedia.)

However, the problems seem to have very little "real life" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world.

To see this problem concretely, the electrostatic force is goes like $F\sim \frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d.

And this is my question, is there any physical application of this technique? Or is it completely irrelevant?

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Yes and no.

As Muphrid wrote in math.se, 2D problems in "real life" are really 3D problems where one dimension has been ignored because the system has translational symmetry along it. Thus a single point charge in 2D corresponds to an infinitely long line charge in three dimensions - which has indeed got a force scaling as $1/r$.

This is a very useful physical system and its understanding is aided greatly by the use of complex analysis. However, it has the very real drawback that it is not a real physical system: infinitely long rods are hard to come by in laboratory cupboards. As such, all 2D problems can only ever be approximations to a 3D system where the variations along the third dimension are over length scales much greater than the other two, and that is not an unreasonable thing to ask.

The same holds for 2D fluid dynamics simulations: if an airplane wing is long, it is reasonable to model it as a 2D flow over its cross section, particularly if it will get us a good understanding of the physics. While the 2D model can never be a very exact representation of actual systems - except perhaps for very restricted wind tunnel experiments - the physical insight we get from it can very often be transported to far more complex situations.

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    $\begingroup$ +1: I was thinking of how to answer this question; you did a better job than I would have. Great, simple line charge example, and great analogy with fluid dynamics! $\endgroup$ – joshphysics Aug 26 '13 at 23:02

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