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I have a doubt about electrical forces on surfaces, for instance, on the surface of a sphere. I'll explain my point: let's say we have some spherical surface of unit radius and there's one point charge over it. My idea then is to represent the surface by the unit sphere $S^2\subset\mathbb{R}^3$ and then I say that on point $a\in S^2$ there's some charge $q$.

That's fine, then I add some other charge $q_1$ at another point $a_1$ and I want to know what's the electrical force on $q_1$ due to $q$. Since we're thinking of the sphere embedded in $3$-space it's fine to thake the vector $v=a_1-a$ and say that the distance is the euclidean norm of this vector. Then I use Coulomb's law and everything works fine.

But my thought is: both charges are on the sphere, so it seems to me that we could really forget about the ambient space and just use intrinsic measures, for instance, calculate the distance given by the geodesic that joins the points $a$ and $a_1$.

The point is, this gives rise to another distance, one that doesn't depend on the ambient space. So, this distance works also to calculate the force ? If not, is there a reason to depend on the ambient space ? Because for me it seems that everything is just happening on the sphere, without connection to the remaining of $3$-space.

Sorry if that's just a bunch of nonsense, it's just that I really don't know how to figure this out by myself.

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The fact that the charges are restricted to the $S^2$ does not change the fact that the remaining 3-space is still there and that Coulomb's law is defined with respect to its euclidean norm. The field lines will still go through 3D space, no matter how you confine the charges.

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Hi Frederic, thanks for your answer. So, the fact that the field lines will still go through 3D space no matter how the charges are confined is one experimental fact? And what if we're dealing with curved space time from GR, then the ambient space will be some curved manifold, in this case, the distance will then be the distance measured along the geodesic of the ambient space? Thanks again. –  user1620696 Feb 24 '13 at 16:47
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The point is that while you can constrain the particles to move on a sphere in 3-space while they interact using Coulomb's law, this is not the same as doing electromagnetism on a curved spacetime. Coulomb's law doesn't generalise to curved spacetime anyway –  alexarvanitakis Feb 24 '13 at 16:52
    
Hi @alexarvanitakis, so to study interactions between charges on curved spacetime another approach is needed right ? Can you point me where I can read more about the subject ? Thanks in advance. –  user1620696 Feb 24 '13 at 17:01
    
You should start with the covariant formulation of electromagnetism en.wikipedia.org/wiki/… in flat spacetime. Once you have a handle on that you will find that this generalises quite straightforwardly to curved spacetimes using the 'minimal coupling' prescription. –  alexarvanitakis Feb 24 '13 at 17:14
    
Thanks for the reference @alexarvanitakis. –  user1620696 Feb 24 '13 at 17:17
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