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How could one go about deriving coulomb's law for an $n$-dimensional space. For example, a 9D space (I would like to know how for these higher dimensions)? I know in 3D space the volume and radius only change. Will this always be true for $n$-dimensional space.

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  • $\begingroup$ Coulomb's Law is a model of a physical law, so what experiment to do propose in n dimensional space that you can model? It also concerns the inverse square of the distance between charges, so I don't know what you mean by "in 3D space the volume and radius only change" $\endgroup$
    – Paul
    Feb 16, 2021 at 16:01

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Without visiting an $n$ dimensional space and measuring the fields there, there is no 'correct' answer.

Here are some possibilities:

  1. If we choose to transplant Coulomb's law, then

$$ E_n(r)=K\frac{q}{r^2} $$

Independently of dimension $n$. If we do this then Gauss' law $\oint \mathbf{E} \cdot da =q$ does not hold in any dimension $n \neq 3$, although it is possible to write down something similar and vastly less useful. This prescription has the advantage that it often makes other equations tractable (eg. Schrodinger equation for Hydrogen).

  1. If we choose to transplant Gauss' law, then we can find the field by applying Gauss' law, integrating over an $n-1$ dimensional hyper-sphere of area $A_{n-1}(r)$

$$ \oint \mathbf{E}_n \cdot da = E_n(r) A_{n-1}(r) =q $$

Thus we have

$$ E_n(r)=\frac{q}{A_{n-1}(r)}=\frac{q }{2r^{n-1}} \Gamma(n/2)\pi^{-n/2} $$

In this case, Coulomb's law (as we know it with $r^{-2}$) does not hold in any dimension other than $n=3$. This is often chosen as the 'correct' answer because it preserves the geometric nature of Maxwell's equations.

  1. Very general arguments such as rotational symmetry, etc can be used to argue for $E_n=E_n(r)$, the field is only dependent on radial distance to charge. You can then propose a series of constraints to narrow down the possible functions $E(r)$, for your given $n$-land.

  2. [Added for conceptual understanding] Write down a more general theory (eg. QED) that you think generalizes well to arbitrary dimensions, work that theory out in $n$ dimensions, then ask what falls out of it in the appropriate limits, and call that 'Electromagnetism in $n$ dimensions'.

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