What is the distance between, say, a cup of coffee and the table it rests on?
What is the distance between two touching hands?

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    $\begingroup$ Extension to your question: what is the distance between two separate objects? For instance how long would I walk from home to university vs how long would an ant walk for the same route? :D [I know it's a different question] $\endgroup$ – anderstood Oct 15 '14 at 14:20
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    $\begingroup$ If you like this question you may enjoy reading this Phys.SE post. $\endgroup$ – Qmechanic Oct 15 '14 at 14:31
  • $\begingroup$ I guess you mean the shortest possible distance, which is Plancks length. $1.616*10^{-35}$ $\endgroup$ – Rohinb97 Oct 15 '14 at 19:21
  • $\begingroup$ @Rohinb97: I don't think that's the shortest possible distance, wikipedia merely implies it's the shortest possible measurable distance, which is slightly different. $\endgroup$ – Mooing Duck Oct 16 '14 at 0:18
  • $\begingroup$ related discussion on exactly how you might define "touching": youtube.com/watch?v=P0TNJrTlbBQ $\endgroup$ – KutuluMike Oct 16 '14 at 3:25

This answer I once gave for What does it mean for two objects to "touch"? discusses what touching even means. It's not a direct answer to your question, but I think it may help you view the issue in a different way. Warning: It's one of my long, talky answers that some people love and others hate. The physics in it is accurate (and for many folks, unexpected) in any case.

The specific answer to your question is that the most fundamental distance between two touching objects is determined by Pauli exclusion surfaces between electrons in the touched and touching objects, with the surfaces being where there is zero probability of finding electrons from either of the objects. Thus how "close" the objects are depends on what level of normalized probability of finding either electron in the exclusion pair you are willing to tolerate. E.g., for some specific set of nearby atoms, "1%" gives one (very short, sub-Angstrom) distance, while "5%" gives another, somewhat larger distance.

Oddly, that also means that the simplest answer is that the objects really do "touch", specifically at the surface of zero probability due to Pauli exclusion.

There are other modifiers of course, such as thermal noise that bounces these surfaces apart at very high frequencies and so give various types of averaged distances. The deeper physics of actual repulsion always, for ordinary matter, goes back to those Pauli exclusion surfaces between individual pairs of electrons in the touched and touching objects.

  • $\begingroup$ Jim or whoever, a -1 for a relevant link minutes after my initial post, seriously? Given that my phone app dumped the rest of my answer (see above) before I could post it, that seems a bit unsporting. I had to use my laptop. $\endgroup$ – Terry Bollinger Oct 15 '14 at 15:21
  • $\begingroup$ I upvoted your answer, for the record. I also think it should be the accepted answer because I only intended mine to be a placeholder until a more thorough one came along $\endgroup$ – Jim Oct 16 '14 at 13:22
  • $\begingroup$ Jim... thank you, and my sincere apology for incorrectly and unfairly associating your comment with that vote. I was wrong to do that. The Android StackExchange app has a very unfortunate bug that completely erases an author's ongoing edits to an answer if someone else, as best I can tell, comments on it or downvotes it. Since I lost my entire first full draft to just that bug, I confess to having been unfairly grumpy... :) $\endgroup$ – Terry Bollinger Oct 16 '14 at 18:41

This very much depends on how you define the measurement. The simplest answer would be that the distance is zero since they are touching, but I assume you are making reference to the electromagnetic forces/Pauli exclusion principle that keep the electron clouds of the molecules and atoms from actually intersecting. However, since the electrons are in clouds and not at any specific positions, one cannot easily define how far apart one cloud is from another. Also, when objects touch, bonds are formed between molecules and electron clouds partially or fully merge. The other option is to specify the distance between the closest two nuclei, which is also tough to do because some nuclei will protrude into the other material. At an atomic level, this is not a very meaningful question.

So the summary is:

  • Macroscopically: The distance is zero

  • Molecularly: The measurement is hard to define, but Van der Waals forces, etc include the partial to full merging of molecular electron clouds. This would incline me to say zero again.

  • Nuclear: There is a distance between two "touching" atoms when measured from the nucleus, but for objects like hands and coffee cups and tables, it is difficult to speculate on the distance between surfaces. Atoms from one object will protrude past the mean surface of the other object. Cold welding, inter-molecular forces, and inclusion of different elements also makes it hard to determine at this level where one object ends and the next begins and what the mean distance between the nuclei of the outer surfaces of both objects is.

That all said, the only answer I can think of that even amounts to a hill of beans is that the distance between neutral touching objects is zero.

  • $\begingroup$ so, may atoms of touching materials get as close together as, say, two atoms in a molecule? $\endgroup$ – nir Oct 15 '14 at 14:24
  • $\begingroup$ @nir Yes, for instance, if two metals of the same type come into contact, they can spontaneously "cold weld", which means at that point they join as if they are the same object. Or when you put your hands together tightly, the oils on your palms mix together and bonds form between the proteins of your skin. The atoms can indeed get as close together as two atoms in a molecule. $\endgroup$ – Jim Oct 15 '14 at 14:33
  • $\begingroup$ @Jim yeah the metals will "cold weld" provided they don't have anything like an oxide layer or another layer of alien atoms on their contact surface. $\endgroup$ – Ruslan Oct 15 '14 at 15:24
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    $\begingroup$ @TerryBollinger, I found this in The Feynman Lectures volume 3, chapter 4, "The exclusion principle is also responsible for the stability of matter on a large scale. We explained earlier that the individual atoms in matter did not collapse because of the uncertainty principle; but this does not explain why it is that two hydrogen atoms can’t be squeezed together as close as you want—why it is that all the protons don’t get close together with one big smear of electrons around them."... $\endgroup$ – nir Oct 15 '14 at 20:18
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    $\begingroup$ (continued)..."The answer is, of course, that since no more than two electrons—with opposite spins—can be in roughly the same place, the hydrogen atoms must keep away from each other. So the stability of matter on a large scale is really a consequence of the Fermi particle nature of the electrons." $\endgroup$ – nir Oct 15 '14 at 20:18

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