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The unit vector r shows that Coulomb's force is parallel to the line joining the charges. It could not be otherwise unless space itself has some built in directional property, for with two point charges alone in empty and isotropic space, no other direction could be singled out.

What is an isotropic space? And what does the sentences in italics mean?

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This is in fact an interesting question.

Isotropic, from Greek iso- =equal and. τρόπος trópos='direction' means that

all directions are equivalent.

And what does that mean? It basically means that all mechanical properties are the same in all directions. Mechanical properties refer to anything you can think of: density, electric charges, anything...

This leads to an easy consequence: if all directions have EXACTLY the same properties: they are indistinguishable. You just cannot tell whether you're looking to the North or to the East. Anywhere you look, you're seeing the same amount of things, separated by the same distance, with exactly the same force, whatever.

By the way, this implies you have spherical symmetry: you can turn around and you cannot know how many laps you've done. You don't have any reference in space, as ALL POINTS ARE EQUIVALENT. You also have translational symmetry: you can move anywhere and you keep seeing the same things.

There is not any privileged point, because you cannot know where you are.

Any examples of this situation are always ideal cases, but they must be considered.

  • Empty space: if the space is really empty, you cannot tell where you are, as there's nothing you can establish as your "reference", like "I'm at $10$ meters from that corner".

An empty space is isotropic, because all directions are equivalent: in all directions, there's just nothing.

  • An infinite tank of water is isotropic: anywhere you move, you keep seeing infinite water.

We could think of any uniform medium. Notice that this medium is required to be infinite, because if it were finite, you could tell where you are (I'm a little closer to the eastern edge).

So this situation cannot ever exist. Why should we think of it then? Because, in physics, we can approximate many situations using this.

Okay, there's not such an "infinite" solenoid, but we could make a very very long one. In such long solenoid, the solution isn't probably too different than the one of the infinite solenoid.

That's the key: in physics, as interactions usually decay with distance, we can make the assumption that, if something's too far, we can neglect its effects.

So if you're in the middle of something, and you move a little bit, so that you're still far enough to the borders, you can still say the tunnel is infinite, so the solution will be the one of an infinite tunnel. Why all this? Because the infinite tunnel is much easier to solve.

The same applies to isotropic problems. If something is isotropic, we can expect that many quantities do not depend on the angle, so the maths will be much easier. This simplifies problems, and gives very accurate solutions.

When you get closer to the edges, the solution starts to fail. This is called "edge effects".


So, back to your problem. We will never have two charges isolated in empty space, because real space is never empty. However, since interactions decrease with distance, we can say that "far enough from everything, everything works LIKE in empty space".

And once we're in empty space, we think...

"if the space is supposed to be completely empty... there's no possible reference". One could say that the force between two charges could make a $45$ degree angle. But the question would be: why $45º$ and not other value? And nobody could find a good argument for this.

It is senseless that a force makes $45º$ because... what determines $45º$, and not $47.61º$? or other value? Nothing. "The charge can't tell where it is", so it is impossible for it to choose a concrete value for no reason.

Okay, it could be random, but classical physics are deterministic.

So the only way to the force to exist is "along the line that joins them", because that's the only thing the charge can see.

If the force wanted to be perpendicular, the force would have "$360º$ to choose", and why one and not the other? No way. It must be along them two because that's the only possible reference. The charge can stick to it.

All this might sound "weird" or "little rigorous", but it can be proven mathematically. Nevertheless, thinking this way is an important learning for a physicists, as it simplifies the problem in a considerable way. In fact, it can reduce the number of hard integrals to only $1$ in certain cases, so this should be taken into account.

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  • $\begingroup$ Is it true that the universe must be infinite to be isotropic? $\endgroup$
    – Tom B.
    Commented Apr 1, 2018 at 0:12
  • $\begingroup$ @TomB. no. Think of being on the surface of a sphere. It's the same no matter where you look, but the sphere is finite. $\endgroup$ Commented Apr 1, 2018 at 0:36
  • $\begingroup$ @danielunderwood That's what I thought, but this answer seems to say the opposite. $\endgroup$
    – Tom B.
    Commented Apr 1, 2018 at 0:39
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    $\begingroup$ Ahh I see what you're saying now...guess I should have read the whole answer first. Maybe the answer meant infinite in the sense of never seeing an edge? $\endgroup$ Commented Apr 1, 2018 at 1:21
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This sentence is fairly abstract. The thought is not from our traditional thinking perspective, but from a philosophic or logic point of view.

Isotropic space means the space properties are not different in direction. In mechanics, when we say material properties are isotropic, it means, for example, its modulus is the same in all directions. For an empty space, it is trivial to show this. For an anisotropic space, let's take river for example, the flow velocity is direction dependent. In the river flow direction, the flow velocity is the maximum.

Back to the abstract sentence, for Coulomb force, we know it is a vector. In an empty space, there is nothing related to direction. So, in this empty space, the only direction we can think of is the line joining the two charges. We then conclude that the Coulomb force direction relates to the line joining the two charges. That may seems to be a useless statement, nothing miracle. Philosophers always think of something useless, aren't they?

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A space is isotropic when no direction is singled out.

For example, on the earth, the magnetic north and south pole are singled out since compasses point to either.

However, in the vacuum, no direction is singled out, thus we say the vacuum is isotropic.

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