electromagnetism - Gravity force strength in 1D, 2D, 3D and higher spatial dimensions

Let's say that we want to measure the gravity force in 1D, 2D, 3D and higher spatial dimensions.

Will we get the same force strength in the first 3 dimensions and then it will go up? How about if we do this with Electromagnetic force?

I've include the Electromagnetic force just to see if I can find an analogy to the gravity force behavior.

Answer

Let us begin with an example from electromagnetism, which you may be familiar with. Gauss' law is given in 3 dimensions by:

$\int\int E.ds= \frac{Q}{\epsilon}$, where $E$ is the electric field produced, $Q$ is the charge and the integral is performed over a surface that encloses the charge. In 3 dimensions, the simplest shape to enclose the charge is the sphere, so we will choose this to make the mathematics simpler.

Making the reasonable assumption that $E$ is constant along all points in the sphere, the above equation becomes:

$4\pi r^2E =\frac{Q}{\epsilon}$ and therefore, $E = \frac{Q}{4\pi r^2\epsilon}$.

It is not hard to see how to generalize this approach to 2 dimensions. Instead of considering a surface integral around a sphere, we simply need to consider a line integral around a circle. The modified 2-D Gauss' law will then become:

$\int E.dl= \frac{Q}{\epsilon}$, where the integral is performed around a circle enclosing the charge $Q$.

Again, evaluating this integral gives:

$2\pi rE = \frac{Q}{\epsilon}$, and therefore, $E = \frac{Q}{2\pi r\epsilon}$

Finally we can generalize to 1-D, where a circle in 1-D becomes a line, and line integral changes to simply adding together points. If we pick a line of length 2r, enclosing the charge, gauss' law will become:

$2E = \frac{Q}{\epsilon}$, that is, $E = \frac{Q}{2\epsilon}$

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Now we must see how to generalize this approach to gravity.

In 3-Dimensions, the gravitational field produced by a mass $M$ is given by:

$g = \frac{GM}{r^2}$

If we introduce a new variable, $k$, defined as $k = \frac{1}{4\pi G}$, then we can re-write the field as:

$g = \frac{M}{4\pi r^2k}$. Compare this to the electric field in 3-D, which is $E = \frac{Q}{4\pi r^2\epsilon}$

One could thus construct a "Gauss'" law for the gravitational field and construct the 2D and 1D fields, in the same way I did for the electric fields above.

The results will be the same as that for the electric field, but with $\epsilon$ replaced with $k = \frac{1}{4G\pi}$, and $Q$ replaced with $M$.