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I'm reading Griffiths' Electrodynamics and having a bit of a problem to understand when we can use infinity as a reference point and when it cannot be used.

One question asked to calculate the potential a distance s from an infinitely long wire that carries a uniform line charge. According to the solution manual, infinity cannot be used as reference point because as he puts it "the charge itself extends to infinity". What does that mean? And is there a general rule for when we can use infinity as a reference point and when it cannot be used?

A previous post (Potential when charge distribution is to infinity) explained it somewhat from the point of view that the potential blows up if we use infinity as a reference point, but Griffiths draws that conclusion before he calculates the potential, and that is what I don't understand how he sees it.

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  • $\begingroup$ Depends upon the equations. If the evaluation of the equation at $\infty$ blows up - or becomes infinite - it becomes a physical singularity - which is a bad thing. $\endgroup$ Jul 13, 2019 at 20:55
  • $\begingroup$ If you think about, e.g., an infinite line of charge in spherical coordinates, can you imagine the potential on a surface of constant radial coordinate $r$ tending to a constant as $r \rightarrow\infty$? $\endgroup$ Jul 13, 2019 at 21:29

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The general answer is that infinity is a valid (and convenient) reference point whenever the system you are considering is finite. When you deal with infinite systems (which is just a mathematical convenience and not a physical thing) you run into trouble. If it is an infinite wire, you can see that by considering two different points "at infinity": one that is infinitely far from the wire and one that is inside the wire (there is such point since the wire "goes to infinity"). If all points at infinity are a reference for potential then the potential there is the same everywhere. Thus, a point on the wire and a point far from the wire have the same potential -- does not make sense.

The same thing happens with all infinite distributions of charge. This is more of a mathematical problem then physical. Math is rich with this kind of contradictions that are rooted in the formal concept of infinity.

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  • $\begingroup$ I wonder if you could have an infinite charge distribution that drops off in such a way where you could still define the potential at infinity? Or maybe I'm missing a simple proof of why this can't be. Just thinking about your claim that you can only do this for finite systems. $\endgroup$ Jul 13, 2019 at 23:56
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    $\begingroup$ That is a good point; I was assuming uniform distributions. When we consider distributions that drop fast enough the potential at infinity will not blow up. Exponentially decaying spherical charge distribution is one example. $\endgroup$
    – oleg
    Jul 14, 2019 at 0:50

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