Derivation of Coulomb's Law in Higher Dimensions 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.
 A: Without visiting an $n$ dimensional space and measuring the fields there, there is no 'correct' answer.
Here are some possibilities:

*

*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).


*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.


*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.


*[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'.
