# Electric force on spherical surface

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.