# What does the path of least resistance really mean?

1. May 19, 2009

### parsec

What does "the path of least resistance" really mean?

A lot of physical phenomena tend to be explained (at least on a superficial level) by referring a current or particle flow's tendency for it's trajectory to follow the "path of least resistance".

I've always found this explanation peculiar. How does a particle or electron "know" or have forewarning that surrounding adjacent paths are more resistive and that it should not venture down them?

For simple conductors this seems obvious. The conductor is filled with charge carriers, so as soon as a potential is applied across it, the potential gradient distributes itself across the length of the conductor and the electrons lose energy in all points simultaneously in the conductor. If multiple conductors are in parallel then current flows through all conductors at different rates proportional to the conductivity of each conductors. A very similar scenario applies to water in parallel pipes.

This got me thinking about electrical discharges and breakdown in a gaseous dielectric. How can these discharges follow a path of least resistance? Surely a local fluctuation (or the presence of impurities) which results in the resistivity of the dielectric being lowered locally can only be exploited if the path of ionization happens to wander through it. I can't reconcile this with the explanation I've been given that "lightning follows the path of least resistance".

Imagine an experiment featuring a sharp high voltage cathode suspended above an infinite plate which is grounded. The cathode is subjected to many thousands of voltage impulses and the results are recorded using a triggered camera. One would expect that when all of the images are overlayed, the result would be a purple discharge cone that is darkest in the centre (the shortest path between the electrode and plate) and symmetrically becomes less dark towards its edges (let's ignore the fact that because we're imaging a 3d cone in a 2d plane the centre is going to be darker anyway. pretend we've digitally removed that effect).

I have done some reading into gaseous ionization processes involved in avalanche (fast) breakdowns (i.e. not coronas). As far as I can tell, the electric field distribution within the gas doesn't seem to affect the probability of charge carriers being produced.

Now imagine that a length of conductor is (isolated and) suspended in the discharge area outside of the cone such that it provides a path of relatively low resistance but is in an area where it is statistically unlikely that the ionization random walk processes are going to encounter it. Will the current somehow find its way to this piece of conductor? Will it induce an electric field locally within the gas which causes charge carriers to be attracted to it?

Do lightning towers actually actively attract stepped leaders, or do they guide lightning discharges simply by providing a path for upward positive leaders to meet with downward negative leaders?

Sorry, I know it's a bit hard to isolate a single question in this post. I'm just after a general discussion on the topic to shed some light on it.

2. May 19, 2009

### diazona

Re: What does "the path of least resistance" really mean?

Wow, well... I think the "path of least resistance" is just something that was invented to justify to laypeople (i.e. curious non-physicists) why objects behave the way they do. Like many analogies, it works if you don't think about it too closely, but once you start asking detailed questions (as you have) the "path of least resistance" principle falls flat on its face - it's just not able to provide any sort of rigorous justification for anything. I suppose it could have been inspired by the minimization of potential energy, i.e. to explain why, when you do a sit-up, your upper body rises instead of your legs (or maybe the other way around, for many people): it increases your potential energy less to raise the lighter half of your body.

It does remind me of Hamilton's principle of action minimization. If you're not familiar with it, basically the action is defined as
$$S = \int L \mathrm{d}t$$
where L is the Lagrangian (kinetic minus potential energy in classical mechanics) and the integral is taken along some path; the path that minimizes the value of S is the one that actually occurs in reality. This is actually the basis of all of modern physics; you can describe anything just by coming up with the right expression for L. Now, I suppose you could very roughly associate S with "resistance" in some metaphysical sense, but it's quite a stretch. I really couldn't see the "path of least resistance" idea that the public is familiar with coming from that equation. But who knows :?

3. May 20, 2009

### krishna mohan

Re: What does "the path of least resistance" really mean?

Hi...

I do not know much about electric discharges...but, as diazona has already pointed out...the path of least resistance is conceptually very close to the path of least action used in mechanics, optics and even in quantum field theories.

I invite you to read Chapter 19: The principle of least action from Feynman lectures Volume 2.
In particular, see how he explains how the particle can follow the path of least action without having to know all the paths from the initial to the final point but by just following the law for each small step.
Also watch out for the exceptions. For example, diffraction is an exception. Just by making the hole through which light travels smaller and smaller, one can make the light to forget the rule of straight line propagation.

I hope this chapter will give you some ideas as to how to tackle your problem.

4. May 20, 2009

### parsec

Re: What does "the path of least resistance" really mean?

I was more or less aware that the phrase is a crock, it's just I haven't heard a better explanation for the formation and propagation of discharges.

I know of the hamiltonian formulation, but have never thought to apply it to anything other than rigid body mechanics.

I'll dig up the feynmann lectures and give that chapter a read.

Thanks for the tips.

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